============================== Prover9 =============================== Prover9 (64) version 2009-02A, February 2009. Process 4210 was started by veroff on proof, Sun Jan 31 18:00:06 2010 The command was "prover9 -f all.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file all.in op(500,infix,"V"). op(500,infix,"^"). op(500,infix,"@"). assign(order,kbo). set(lex_order_vars). clear(cac_redundancy). set(para_from_small). clear(back_demod). set(restrict_denials). assign(max_weight,5). formulas(assumptions). (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). (x V y) V z = x V (y V z) # label("Associativity-V"). x ^ x = x # label("Idempotence-^"). x V x = x # label("Idempotence-V"). x ^ y = y ^ x # label("Commutativity-^"). x V y = y V x # label("Commutativity-V"). (x ^ y) V x = x # label("Absorption a"). (x V y) ^ x = x # label("Absorption b"). (x * ((x \ z) ^ y)) V z = z # label("Residual a"). ((y ^ (z / x)) * x) V z = z # label("Residual b"). (x \ ((x * y) V z)) ^ y = y # label("Residual c"). (((y * x) V z) / x) ^ y = y # label("Residual d"). (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). 1 * x = x # label("Left monoid unit"). x * 1 = x # label("Right monoid unit"). x V y = x / ((x V y) \ x) # label("GMV a"). x V y = (x / (x V y)) \ x # label("GMV b"). x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). end_of_list. formulas(goals). x @ x = x # label("Goal 1"). (x @ y) @ z = x @ z # label("Goal 2"). x @ (y @ z) = x @ z # label("Goal 3"). (x ^ y) @ (z ^ u) = (x @ z) ^ (y @ u) # label("Goal 4"). (x V y) @ (z V u) = (x @ z) V (y @ u) # label("Goal 5"). (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). (x / y) @ (z / u) = (x @ z) / (y @ u) # label("Goal 7"). (x * (x \ 1)) @ 1 = x * (x \ 1) # label("Goal 8"). 1 @ (x * (x \ 1)) = 1 # label("Goal 9"). (x \ 1) @ 1 = 1 # label("Goal 10"). 1 @ (x \ 1) = x \ 1 # label("Goal 11"). (x / (y \ x)) @ (x V y) = x V y # label("Goal 12"). ((x / y) \ x) @ (x V y) = x V y # label("Goal 13"). (x V y) @ (x / (y \ x)) = x / (y \ x) # label("Goal 14"). (x V y) @ ((x / y) \ x) = (x / y) \ x # label("Goal 15"). end_of_list. formulas(hints). (x ^ y) ^ z = x ^ (y ^ z). (x V y) V z = x V (y V z). x ^ x = x. x V x = x. x ^ y = y ^ x. x V y = y V x. (x ^ y) V x = x. (x V y) ^ x = x. (x * ((x \ y) ^ z)) V y = y. ((x ^ (y / z)) * z) V y = y. (x \ ((x * y) V z)) ^ y = y. (((x * y) V z) / y) ^ x = x. (x * y) * z = x * (y * z). 1 * x = x. x * 1 = x. x V y = (x / (x V y)) \ x. (x / (x V y)) \ x = x V y. x @ y = (x * (x \ 1)) * ((y \ 1) \ 1). (x * (x \ 1)) * ((y \ 1) \ 1) = x @ y. 1 @ (c25 * (c25 \ 1)) != 1 # answer("Goal 9") # label("Goal 9"). x V (y V z) = y V (z V x). x V (y V z) = z V (x V y). (x ^ y) V y = y. x = x V (x ^ y). x V (x ^ y) = x. x = x ^ (x V y). x ^ (x V y) = x. (x V y) ^ y = y. (x * (y ^ (x \ z))) V z = z. x = x V (y * ((y \ x) ^ z)). x V (y * ((y \ x) ^ z)) = x. x = x V ((y ^ (x / z)) * z). x V ((y ^ (x / z)) * z) = x. x = x ^ (y \ ((y * x) V z)). x ^ (y \ ((y * x) V z)) = x. (x \ (y V (x * z))) ^ z = z. x = x ^ (((x * y) V z) / y). x ^ (((x * y) V z) / y) = x. ((x V (y * z)) / z) ^ y = y. (x / (x V x)) \ x = x. ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. x = x V (y ^ x). x V (y ^ x) = x. x V ((x ^ y) ^ z) = x. x ^ (y V x) = x. x ^ (y V (z V x)) = x. (x / x) \ x = x. x V (y ^ (x ^ z)) = x. x ^ (y V (x V z)) = x. x V y = (y ^ z) V (x V y). (x ^ y) V (z V x) = z V x. x = x V (y * (z ^ (y \ x))). x V (y * (z ^ (y \ x))) = x. x V (y * (y \ x)) = x. x V (1 \ x) = x. x ^ (y * (y \ x)) = y * (y \ x). x ^ (1 \ x) = 1 \ x. x V ((y ^ (z ^ (x / u))) * u) = x. x V ((x / y) * y) = x. x V (x / 1) = x. x ^ ((x / y) * y) = (x / y) * y. x ^ (x / 1) = x / 1. x ^ (y \ (y * x)) = x. x ^ (y \ (z V (y * x))) = x. 1 ^ (x \ (x V y)) = 1. x ^ ((y * z) \ (y * (z * x))) = x. x ^ (1 \ x) = x. 1 ^ (x \ x) = 1. x V (y \ (y * x)) = y \ (y * x). (x \ (x * y)) V y = x \ (x * y). x = 1 \ x. 1 \ x = x. (x ^ 1) ^ (y \ y) = x ^ 1. 1 V (x \ x) = x \ x. 1 ^ ((x ^ y) \ y) = 1. 1 ^ (x V (y \ y)) = 1. 1 ^ ((x \ x) V y) = 1. x ^ ((x * y) / y) = x. x ^ ((y V (x * z)) / z) = x. x ^ (y / (z ^ (x \ y))) = x. (x * y) ^ ((x * (y * z)) / z) = x * y. 1 ^ (x / x) = 1. x ^ (x / 1) = x. x V ((x * y) / y) = (x * y) / y. (x ^ 1) ^ (y / y) = x ^ 1. 1 V (x / x) = x / x. (x / x) V 1 = x / x. x = x / 1. x / 1 = x. (x / (y \ x)) ^ y = y. x = x ^ (y / (x \ y)). x ^ (y / (x \ y)) = x. x ^ ((y * z) \ (u V (y * (z * x)))) = x. x ^ ((y / x) \ y) = x. x V (x * (y ^ 1)) = x. x V (x * (1 ^ y)) = x. 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. x V ((y ^ 1) * x) = x. x V ((1 ^ y) * x) = x. 1 * ((x \ 1) \ 1) = 1 @ x. 1 @ x = (x \ 1) \ 1. (x \ 1) \ 1 = 1 @ x. x V (y * (z * (z \ (y \ x)))) = x. x V ((y * (y \ 1)) * x) = x. x V (y * ((y \ 1) * x)) = x. (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. x V ((y ^ z) * (z \ x)) = x. x V (y * ((y V z) \ x)) = x. x V ((1 V y) \ x) = x. (x \ y) ^ ((x V z) \ y) = (x V z) \ y. x V ((y / y) \ x) = x. x V (((x ^ y) / z) * z) = x. x V (y ^ ((x / z) * z)) = x. x V (x * ((1 / y) * y)) = x. x V (((1 / y) * y) * x) = x. x ^ (((y ^ z) / x) \ y) = x. x V ((1 / y) * (y * x)) = x. x ^ ((y / x) \ (y V z)) = x. (1 / x) ^ (y / (x * y)) = 1 / x. (x / y) ^ ((x V z) / y) = x / y. x ^ (y V (z \ (z * x))) = x. 1 ^ (x \ (x * (y \ y))) = 1. x ^ ((y / y) * x) = x. x = (x / x) * x. (x / x) * x = x. x ^ ((x * (y \ y)) / 1) = x. x ^ (x * (y \ y)) = x. x = x * (x \ x). x * (x \ x) = x. (x \ x) ^ ((y * x) \ (y * x)) = x \ x. x ^ (y \ (y * (z V x))) = x. x ^ ((x * (y V z)) / z) = x. x ^ (x * (y V 1)) = x. x V (x / (y V 1)) = x. x V (x / (y / y)) = x. x ^ (x / (y / y)) = x / (y / y). x ^ ((((x * y) * z) / z) / y) = x. x ^ (((x * (y * z)) / z) / y) = x. x ^ (((x * y) / y) / (y / y)) = x. (x / x) ^ ((x / x) / (x / x)) = x / x. ((x ^ y) / x) \ (x ^ y) = x. (1 ^ x) \ (1 ^ x) = 1. ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. x ^ ((y * (1 / x)) \ y) = x. (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). 1 ^ ((x * (1 / y)) \ (x / y)) = 1. (x / y) V ((x * (1 / y)) * 1) = x / y. ((x \ 1) * y) V (x \ y) = x \ y. x ^ ((y \ 1) \ (y \ x)) = x. (x \ 1) ^ ((x \ y) / y) = x \ 1. (x \ y) V ((x V z) \ y) = x \ y. x / x = (x / x) / (x / x). (x / x) / (x / x) = x / x. (x / x) \ (x / x) = x / x. ((1 / x) * x) \ ((1 / x) * x) = 1. (x \ x) ^ 1 = x \ x. 1 V (x \ x) = 1. 1 = x \ x. x \ x = 1. (x \ x) \ y = y. x ^ ((x \ 1) \ 1) = x. (x \ 1) ^ (1 / x) = x \ 1. 1 V ((x V y) \ x) = x \ x. 1 = x / x. x / x = 1. (x / (x V y)) ^ 1 = x / (x V y). 1 V ((x \ 1) * x) = 1. (1 / x) ^ (x \ 1) = x \ 1. 1 V ((x ^ (y \ 1)) * y) = 1. 1 V ((x V y) \ x) = 1. 1 V ((x V y) \ (y ^ z)) = 1. x ^ ((y ^ (x \ 1)) \ 1) = x. (1 / x) ^ (x \ 1) = 1 / x. 1 / x = x \ 1. x \ 1 = 1 / x. x V ((y / (y V z)) * x) = x. x V (((y ^ z) / y) * x) = x. 1 V ((x ^ y) / x) = 1. 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. 1 V (((x ^ y) / x) ^ z) = 1. 1 V (x ^ ((y ^ z) / y)) = 1. 1 V (x ^ (((y / z) * z) / y)) = 1. 1 V ((x / (y * x)) * y) = 1. (1 / x) ^ (y / (x * y)) = y / (x * y). (x / y) V (x * (1 / y)) = x / y. (x / y) V (x * (y \ 1)) = x / y. (x \ 1) ^ (y \ (y / x)) = x \ 1. x / (y * x) = 1 / y. x \ 1 = y / (x * y). x / (y * x) = y \ 1. ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. (x * y) \ 1 = (y \ (x \ 1)) ^ ((x * y) \ 1). (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. (x * y) \ 1 = y \ (x \ 1). ((x * (x \ 1)) \ 1) \ y = y. 1 = 1 @ (x * (x \ 1)). 1 @ (x * (x \ 1)) = 1. (c24 * (c24 \ 1)) @ 1 != c24 * (c24 \ 1) # answer("Goal 8") # label("Goal 8"). x * y = x * (y * 1). x * (y * 1) = x * y. x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). x * ((x \ 1) * ((y \ 1) \ 1)) = x @ y. x * (y * (z \ z)) = x * y. x * ((x \ 1) * 1) = x @ 1. x * (x \ 1) = x @ 1. (x * y) @ 1 = x * (y * ((x * y) \ 1)). x * (y * ((x * y) \ 1)) = (x * y) @ 1. x * (y * ((z * (z \ 1)) \ 1)) = x * y. x * (x \ 1) = (x * (x \ 1)) @ 1. (x * (x \ 1)) @ 1 = x * (x \ 1). 1 @ (c27 \ 1) != c27 \ 1 # answer("Goal 11") # label("Goal 11"). (x V y) V (y ^ z) = x V y. x V (y * (z * ((y * z) \ x))) = x. x ^ (y / ((x \ y) ^ z)) = x. (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). x V (x * (y * (y \ 1))) = x. x ^ (y / (((x \ y) / z) * z)) = x. (x \ 1) ^ ((y * x) \ y) = x \ 1. 1 V (x \ (x ^ y)) = 1. 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). 1 / (x \ 1) = 1 @ x. x ^ ((1 / y) \ (y \ x)) = x. (1 / x) ^ ((x \ y) / y) = 1 / x. x ^ (((x \ y) / y) \ 1) = x. (((x \ y) / y) * x) V 1 = 1. (1 / x) ^ ((x \ y) / y) = (x \ y) / y. (x \ y) / y = 1 / x. x \ 1 = (x \ y) / y. (x \ y) / y = x \ 1. 1 V ((x \ (x ^ y)) ^ z) = 1. 1 V (x * ((y * x) \ (y ^ z))) = 1. 1 V (x * ((y * x) \ y)) = 1. (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. (x * y) \ x = y \ 1. (x \ 1) / y = (y * x) \ 1. (x * y) \ 1 = (y \ 1) / x. (x \ 1) / y = x \ (y \ 1). x \ (y \ 1) = (x \ 1) / y. x \ ((x \ 1) \ 1) = 1. 1 / ((x \ 1) \ 1) = x \ 1. x \ 1 = 1 @ (x \ 1). 1 @ (x \ 1) = x \ 1. c5 @ (c6 @ c7) != c5 @ c7 # answer("Goal 3") # label("Goal 3"). x * ((x \ 1) * (1 @ y)) = x @ y. (x @ 1) * y = x * ((x \ 1) * y). x * ((x \ 1) * y) = (x @ 1) * y. (x @ 1) * (1 @ y) = x @ y. (x * (x \ 1)) \ 1 = 1. (x @ 1) \ 1 = 1. ((x @ 1) * y) \ 1 = y \ 1. x \ (1 @ x) = 1. 1 / (1 @ x) = x \ 1. (1 @ x) \ 1 = x \ 1. ((x @ 1) * (1 @ y)) \ 1 = y \ 1. (x @ y) \ 1 = y \ 1. (x \ 1) \ 1 = 1 @ (y @ x). x * ((x \ 1) * (1 @ (y @ z))) = x @ z. x @ y = x @ (z @ y). x @ (y @ z) = x @ z. x V y = x / ((x V y) \ x). x / ((x V y) \ x) = x V y. (c26 \ 1) @ 1 != 1 # answer("Goal 10") # label("Goal 10"). x / ((x V y) \ x) = y V x. (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. (x / (x V y)) \ x = y V x. x V (((x / y) ^ z) * y) = x. x ^ ((y ^ (z / x)) \ z) = x. 1 ^ ((x V y) \ (x V (y V z))) = 1. 1 ^ ((x ^ y) \ x) = 1. 1 V ((x ^ y) \ x) = (x ^ y) \ x. 1 ^ ((x V y) / x) = 1. 1 ^ (x / (x ^ y)) = 1. x / ((y V x) \ x) = y V x. (x / (y V x)) \ x = y V x. x ^ (((y / x) ^ z) \ y) = x. ((x / y) \ x) V y = (x / y) \ x. x ^ ((y ^ 1) \ x) = x. x V ((y ^ 1) \ x) = (y ^ 1) \ x. x ^ ((y * (y \ (z / x))) \ z) = x. (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). x ^ ((1 V y) * x) = x. x ^ (x / (y V 1)) = x / (y V 1). (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. x ^ ((y / (x V z)) \ y) = x. x V ((x / (y V z)) * y) = x. (x / y) ^ (x / (y V z)) = x / (y V z). (x ^ y) / (x \ (x ^ y)) = x. (x ^ y) / (y \ (y ^ x)) = y. (x ^ y) / (y \ (x ^ y)) = y. x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). ((x ^ y) / y) \ (y ^ x) = y. ((x ^ y) / y) \ (x ^ y) = y. (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. 1 ^ ((x \ y) / ((x V z) \ y)) = 1. 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. 1 ^ (x / (x V y)) = x / (x V y). (x \ (x / y)) ^ (1 / y) = 1 / y. (x V 1) ^ ((x ^ 1) \ x) = x V 1. (x ^ 1) ^ (x / (x V 1)) = x ^ 1. x ^ 1 = x ^ (1 ^ (x / (x V 1))). x ^ (1 ^ (x / (x V 1))) = x ^ 1. x ^ (x / (x V 1)) = x ^ 1. x ^ 1 = x / (x V 1). x / (x V 1) = x ^ 1. x / (1 V x) = x ^ 1. (x \ y) V (z * ((x * z) \ y)) = x \ y. (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. x ^ ((y \ (y / x)) \ 1) = x. (x \ (x / y)) ^ (1 / y) = x \ (x / y). x \ (x / y) = 1 / y. x \ 1 = y \ (y / x). x \ (x / y) = y \ 1. x \ (x ^ 1) = (x V 1) \ 1. (x V 1) \ 1 = x \ (x ^ 1). (x * y) \ z = y \ (x \ z). (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. ((1 V x) \ 1) \ ((1 V x) \ y) = y. (x / (y V 1)) / (x \ (x / (y V 1))) = x. (x / (1 V y)) / (x \ (x / (y V 1))) = x. (x / (1 V y)) / ((y V 1) \ 1) = x. x / ((x * (y V 1)) \ x) = x * (y V 1). x / ((y V 1) \ 1) = x * (y V 1). 1 ^ ((x \ 1) * (x V 1)) = 1. (x / (1 V y)) * (y V 1) = x. 1 ^ ((x \ 1) * (1 V x)) = 1. (x ^ 1) * (x V 1) = x. x * y = (x ^ 1) * ((x V 1) * y). (x ^ 1) * ((x V 1) * y) = x * y. 1 = ((x \ 1) * (1 V x)) ^ 1. ((x \ 1) * (1 V x)) ^ 1 = 1. (x / ((1 V y) * x)) \ x = (1 V y) * x. ((1 V x) \ 1) \ y = (1 V x) * y. (1 V x) * ((1 V x) \ y) = y. (1 V x) * ((x V 1) \ y) = y. (x V 1) * ((x V 1) \ y) = y. (x V 1) * (x \ (x ^ 1)) = 1. (x ^ 1) * 1 = x * (x \ (x ^ 1)). x * (x \ (x ^ 1)) = (x ^ 1) * 1. x * (x \ (x ^ 1)) = x ^ 1. ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. ((x \ 1) * (1 V x)) ^ 1 = (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))). (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. (x \ 1) * ((x \ 1) \ 1) = 1. 1 = (x \ 1) @ 1. (x \ 1) @ 1 = 1. c1 @ c1 != c1 # answer("Goal 1") # label("Goal 1"). x * 1 = x @ x. x @ x = x * 1. x @ x = x # label("Goal 1"). (c2 @ c3) @ c4 != c2 @ c4 # answer("Goal 2") # label("Goal 2"). (x @ y) * z = (x @ 1) * ((1 @ y) * z). (x @ 1) * ((1 @ y) * z) = (x @ y) * z. ((x \ 1) \ 1) \ 1 = x \ 1. (x @ y) * (y \ 1) = (x @ y) @ 1. ((x \ 1) \ 1) * (x \ 1) = 1. (1 @ x) * (x \ 1) = 1. x * ((1 @ y) * (y \ 1)) = x. x @ 1 = (x @ y) * (y \ 1). (x @ y) * (y \ 1) = x @ 1. x @ 1 = (x @ y) @ 1. (x @ y) @ 1 = x @ 1. (x @ 1) * (1 @ y) = (x @ z) @ y. (x @ y) @ z = (x @ 1) * (1 @ z). (x @ y) @ z = x @ z # label("Goal 2"). (c16 \ c17) @ (c18 \ c19) != (c16 @ c18) \ (c17 @ c19) # answer("Goal 6"). (c16 @ c18) \ (c17 @ c19) != (c16 \ c17) @ (c18 \ c19) # answer("Goal 6"). x / ((y V x) \ x) = x V y. x V y = (z ^ y) V (x V y). (x ^ y) V (z V y) = z V y. (x \ (x V y)) V 1 = x \ (x V y). 1 ^ (x \ (y \ (z V (y * x)))) = 1. x V (y / (x \ y)) = y / (x \ y). x V (y * (z * (u ^ ((y * z) \ x)))) = x. 1 ^ (x \ ((y / x) \ y)) = 1. ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). x V (y * ((z V y) \ x)) = x. x V ((y V 1) \ x) = x. x ^ ((y V 1) \ x) = (y V 1) \ x. x V (((y / z) * z) * (y \ x)) = x. x V ((((x / y) / z) * z) * y) = x. x ^ ((y V 1) * x) = x. x V ((y V 1) * x) = (y V 1) * x. x ^ ((x * y) / (z * (z \ y))) = x. 1 ^ (((x V y) \ z) \ (x \ z)) = 1. x V ((y / z) * (z * (y \ x))) = x. x V (((x / y) / z) * (z * y)) = x. 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. (1 @ x) V (y @ x) = 1 @ x. (x @ y) ^ (1 @ y) = x @ y. x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. x ^ (((y ^ (z / (u * x))) * u) \ z) = x. 1 ^ ((x V y) \ x) = (x V y) \ x. x ^ (1 @ x) = x. (1 @ x) V x = 1 @ x. x V (y V (1 @ x)) = y V (1 @ x). x ^ (1 @ (y V x)) = x. 1 ^ (1 @ (x \ (x V y))) = 1. 1 ^ (1 @ ((x ^ y) \ y)) = 1. (1 / x) \ 1 = 1 @ x. x V (x * (y * (1 / y))) = x. 1 @ (x V 1) = x V 1. 1 @ x = 1 / (1 / x). 1 / (1 / x) = 1 @ x. (1 / x) ^ ((y * x) \ y) = 1 / x. x V (((y V z) \ y) * x) = x. 1 V ((x ^ y) / y) = 1. 1 ^ ((x * y) \ (x / (1 / y))) = 1. x ^ ((x / (1 / y)) / (y * 1)) = x. x ^ ((x / (1 / y)) / y) = x. x ^ ((x / (1 @ y)) / (y \ 1)) = x. x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). (x \ y) ^ ((z * x) \ (z * y)) = x \ y. x ^ ((x / y) / (y \ 1)) = x. 1 V ((x \ 1) / (y \ (y / x))) = 1. x / ((y \ 1) * x) = 1 @ y. (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. ((x \ y) / y) \ 1 = 1 @ x. (x \ (y \ 1)) \ 1 = 1 @ (y * x). ((x \ 1) * y) \ 1 = y \ (1 @ x). x * (y * (y \ (x \ 1))) = (x * y) @ 1. (x \ (1 @ y)) \ 1 = 1 @ ((y \ 1) * x). ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. ((x \ 1) / y) \ 1 = 1 @ (y * x). (x \ 1) / (y \ 1) = x \ (1 @ y). (x \ 1) \ (y \ 1) = (1 @ x) / y. 1 V (x \ ((y \ (y / x)) \ 1)) = 1. 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. 1 @ (1 / x) = x \ 1. (x @ 1) * (y \ 1) = x @ (y \ 1). 1 @ ((x * y) \ x) = y \ 1. x \ 1 = 1 / (y @ x). 1 / (x @ y) = y \ 1. (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. (x @ y) \ (z \ 1) = (y \ 1) / z. 1 @ (x @ y) = 1 @ y. (x * y) \ 1 = (z @ y) \ (x \ 1). (x @ y) \ (z \ 1) = (z * y) \ 1. x \ (y \ 1) = (z @ x) \ (y \ 1). (x @ y) \ (z \ 1) = y \ (z \ 1). (x @ 1) \ (y \ 1) = y \ 1. (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. x \ (x / (y \ 1)) = 1 @ y. 1 @ (x \ (x / y)) = y \ 1. (x \ 1) / (y / x) = y \ 1. (x \ y) \ (x \ 1) = y \ 1. (x \ (y \ 1)) \ (x \ 1) = 1 @ y. (x \ 1) / (y \ 1) = (y \ x) \ 1. x \ ((y / x) \ 1) = y \ 1. (x \ 1) \ (y \ 1) = (y / x) \ 1. x \ (1 @ y) = (y \ x) \ 1. (x \ y) \ 1 = y \ (1 @ x). x \ (1 @ (x * y)) = (y \ 1) \ 1. (x \ (1 @ y)) \ 1 = 1 @ (y \ x). ((x V 1) \ y) \ 1 = y \ (x V 1). (x \ 1) \ (1 @ y) = 1 @ (x * y). (x \ 1) \ (y \ 1) = 1 @ (x / y). (x / y) \ 1 = (1 @ y) / x. (1 @ x) / y = 1 / (y / x). (1 @ x) / (y \ x) = 1 @ y. x \ ((1 @ x) / y) = y \ 1. (x \ 1) / (y \ (x \ 1)) = 1 @ y. x \ (1 @ (x * y)) = 1 @ y. 1 @ (x \ y) = 1 @ ((x \ 1) * y). 1 @ ((x \ 1) * y) = 1 @ (x \ y). 1 ^ ((1 @ x) \ (x V 1)) = 1. (x V 1) V ((1 @ x) * 1) = x V 1. 1 @ (x / y) = (1 @ x) / y. (1 @ x) / y = 1 @ (x / y). 1 @ (x / y) = (y / x) \ 1. (x / y) \ 1 = 1 @ (y / x). 1 @ (x / y) = 1 / (y / x). 1 / (x / y) = 1 @ (y / x). 1 @ (x / (y \ x)) = 1 @ y. x \ (1 @ (x / y)) = y \ 1. ((x / y) \ z) / z = 1 @ (y / x). 1 / (1 / x) = 1 @ ((x * y) / y). 1 @ ((x * y) / y) = 1 / (1 / x). (x \ 1) \ 1 = 1 @ ((y / x) \ y). 1 @ ((x / y) \ x) = (y \ 1) \ 1. (x \ 1) / (y \ 1) = 1 @ (x \ y). (x \ 1) \ (1 @ (x \ y)) = 1 @ y. (x V 1) V (1 @ x) = x V 1. x V 1 = x V (1 V (1 @ x)). x V (1 V (1 @ x)) = x V 1. x V 1 = 1 V (1 @ x). 1 V (1 @ x) = x V 1. 1 V (1 @ x) = (y @ x) V 1. (x @ y) V 1 = 1 V (1 @ y). (x @ y) V 1 = y V 1. x V 1 = 1 V (y @ x). 1 V (x @ y) = y V 1. 1 V (x @ y) = 1 V y. (x V y) / y = 1 @ ((x V y) / y). 1 @ ((x V y) / y) = (x V y) / y. 1 @ ((x * y) / y) = 1 @ x. 1 @ ((x / y) \ x) = 1 @ y. 1 @ (x \ y) = x \ (1 @ y). x \ (1 @ y) = 1 @ (x \ y). 1 @ (x \ y) = (y \ x) \ 1. (x \ y) \ 1 = 1 @ (y \ x). (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. x \ (1 @ y) = 1 @ (x \ (z @ y)). 1 @ (x \ (y @ z)) = x \ (1 @ z). 1 @ (x \ (x * y)) = 1 @ y. 1 @ (x \ y) = 1 / (y \ x). 1 / (x \ y) = 1 @ (y \ x). 1 V (x \ (1 @ ((y / x) \ y))) = 1. 1 @ x = 1 @ (y * (y \ x)). 1 @ (x * (x \ y)) = 1 @ y. 1 @ (x \ (y @ z)) = 1 @ (x \ z). 1 V (1 @ (x \ ((y / x) \ y))) = 1. 1 = 1 V (x \ ((y / x) \ y)). 1 V (x \ ((y / x) \ y)) = 1. 1 = x \ ((y / x) \ y). x \ ((y / x) \ y) = 1. 1 @ ((x @ y) * z) = 1 @ (y * z). (x @ y) \ (1 @ (y * z)) = 1 @ z. x @ (1 @ y) = x @ (z / (y \ z)). x @ (y / (z \ y)) = x @ (1 @ z). x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). x @ ((y \ 1) * z) = x @ (1 @ (y \ z)). x @ (1 @ y) = x @ ((y * z) / z). x @ ((y * z) / z) = x @ (1 @ y). x @ (1 @ y) = x @ ((z / y) \ z). x @ ((y / z) \ y) = x @ (1 @ z). x @ (1 @ y) = x @ (z \ (z * y)). x @ (y \ (y * z)) = x @ (1 @ z). x @ (1 @ y) = x @ (z * (z \ y)). x @ (y * (y \ z)) = x @ (1 @ z). x @ (1 @ (y \ z)) = x @ (y \ (u @ z)). x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). x @ (1 @ (y * z)) = x @ ((u @ y) * z). x @ (y / (z \ y)) = x @ z. x @ ((y * z) / z) = x @ y. x @ ((y / z) \ y) = x @ z. x @ (y \ (y * z)) = x @ z. x @ (y * (y \ z)) = x @ z. x @ (y * (1 / z)) = x @ (y / z). x @ (y * (z \ 1)) = x @ (y / z). x @ (y * (z * (1 / u))) = x @ ((y * z) / u). x @ (y * (z \ 1)) = x @ (y / (u @ z)). x @ (y / (z @ u)) = x @ (y * (u \ 1)). x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). (x @ y) \ (1 @ z) = y \ (1 @ z). x \ (1 @ y) = 1 @ ((z @ x) \ y). 1 @ ((x @ y) \ z) = y \ (1 @ z). 1 @ ((x @ y) \ z) = 1 @ (y \ z). (x \ (y @ z)) \ 1 = 1 @ (z \ x). x @ ((y \ 1) * z) = x @ (y \ z). x @ ((1 / y) * z) = x @ (y \ z). x @ ((y \ 1) * z) = x @ ((u @ y) \ z). x @ ((y @ z) \ u) = x @ ((z \ 1) * u). x @ (((y / z) \ 1) * y) = x @ z. x @ ((1 @ (y / z)) * z) = x @ y. x @ ((1 @ y) * z) = x @ (y * z). x @ ((y / z) * z) = x @ y. x @ (y \ (z @ u)) = x @ (y \ u). x @ ((y @ z) * u) = x @ (z * u). x @ (y / (z @ u)) = x @ (y / u). (1 @ x) / (y @ z) = 1 @ (x / z). (1 @ x) / (y @ (z \ x)) = 1 @ z. x @ ((y @ z) \ u) = x @ (z \ u). (x @ 1) \ y = (x \ 1) \ (x \ y). (x \ 1) \ (x \ y) = (x @ 1) \ y. (x @ y) \ z = (1 @ y) \ ((x @ 1) \ z). (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. x * (y @ (x \ 1)) = (x * (y @ 1)) @ 1. (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. x @ (y * (1 @ (y \ z))) = x @ z. x @ (y * (1 @ z)) = x @ (y * z). x @ (y * (1 @ z)) = x @ (y * (u @ z)). x @ (y * (z @ u)) = x @ (y * u). ((x \ 1) * x) * 1 = ((x \ 1) * x) @ 1. ((x \ 1) * x) @ 1 = (x \ 1) * x. (1 @ x) V y = 1 @ (y V (1 @ x)). 1 @ (x V (1 @ y)) = (1 @ y) V x. 1 @ (x \ (x * (1 @ y))) = (1 @ y) V (x \ (x * (1 @ y))). (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). (x * (y @ 1)) @ z = x * ((y @ (1 / x)) * (1 @ z)). x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. ((x * (x \ y)) / y) \ (x * (x \ y)) = y. x / (y * (y \ x)) = 1 @ (x / (y * (y \ x))). 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. (x ^ y) \ y = 1 @ ((x ^ y) \ y). 1 @ ((x ^ y) \ y) = (x ^ y) \ y. 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). x \ (x * (1 @ y)) = 1 @ (1 @ y). x \ (x * (1 @ y)) = 1 @ y. (x @ 1) \ (x @ y) = 1 @ y. (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). (x @ y) \ (x @ (y * z)) = 1 @ z. (x @ y) \ (x @ z) = 1 @ (y \ z). (x \ 1) * (1 @ x) = 1. x @ (y * (y \ x)) = x. (x \ 1) * y = ((x \ 1) * y) @ (x \ y). ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. (x @ y) * z = ((x @ y) * z) @ (y * z). ((x @ y) * z) @ (y * z) = (x @ y) * z. x / (y @ z) = (x / (y @ z)) @ (x / z). (x / (y @ z)) @ (x / z) = x / (y @ z). (x @ y) \ z = ((x @ y) \ z) @ (y \ z). ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. x \ (x @ (x * y)) = 1 @ y. x \ (x @ y) = 1 @ (x \ y). (x @ y) \ x = 1 @ (y \ x). x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. x @ (y * z) = x @ (y * (u * (u \ z))). x @ (y * (z * (z \ u))) = x @ (y * u). x \ (x @ y) = 1 @ (x \ (z @ y)). 1 ^ ((x @ (y ^ x)) \ x) = 1. (x \ (y @ z)) \ 1 = (x @ z) \ x. x \ (x @ y) = (y @ x) \ y. (x @ y) \ x = y \ (y @ x). 1 ^ ((x @ (x ^ y)) \ x) = 1. (x @ 1) \ y = (z \ 1) \ ((x @ z) \ y). (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. x V ((x @ (x ^ y)) * 1) = x. x V (x @ (x ^ y)) = x. x V (x @ ((y V 1) \ x)) = x. 1 @ (x / (x @ ((y V 1) \ x))) = (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)). (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. (x @ y) \ (z @ u) = ((x @ y) \ (z @ u)) @ (y \ u). ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). x V ((x / (1 @ y)) / (y \ 1)) = x. x = (x / (1 @ y)) / (y \ 1). (x / (1 @ y)) / (y \ 1) = x. x @ (y * (z * (1 / u))) = x @ (y * (z / u)). (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. (x \ (x @ y)) * ((x @ y) \ x) = 1. x V (((x @ y) / (x \ (x @ y))) * 1) = x. x V ((x @ y) / (x \ (x @ y))) = x. x = (x @ y) / (x \ (x @ y)). (x @ y) / (x \ (x @ y)) = x. (x @ (x * y)) / (1 @ y) = x. x @ (y * (z / u)) = x @ ((y * z) / u). x @ ((y * z) / u) = x @ (y * (z / u)). ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. (1 @ x) \ y = ((x \ 1) * y) ^ ((1 @ x) \ y). ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. (1 @ x) \ y = (x \ 1) * y. (x V 1) \ y = ((x V 1) \ 1) * y. ((x V 1) \ 1) * y = (x V 1) \ y. (x \ 1) \ y = ((x \ 1) \ 1) * y. ((x \ 1) \ 1) * y = (x \ 1) \ y. (1 @ (x / y)) \ z = (1 @ (y / x)) * z. (x \ 1) * (1 @ (x * y)) = 1 @ y. (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). ((x V 1) \ y) * z = ((x V 1) \ 1) * (y * z). ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. (1 @ x) * y = (x \ 1) \ y. (x \ 1) \ y = (1 @ x) * y. ((x \ 1) * y) \ z = y \ ((1 @ x) * z). (1 @ x) * (x \ y) = (x @ 1) \ y. (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. x * (1 @ y) = x @ (x * y). x @ (x * y) = x * (1 @ y). x * (1 @ (x \ y)) = x @ y. (x * (1 @ y)) @ z = x @ z. (x * (1 @ y)) / (1 @ y) = x. (x * y) * (y \ 1) = (x * y) @ x. (x * y) @ x = (x * y) * (y \ 1). x * (y \ 1) = x @ (x / y). x @ (x / y) = x * (y \ 1). (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. x * ((y @ x) \ y) = x @ y. x / (y \ 1) = x * (1 @ y). (x @ y) * z = x * (((y @ x) \ y) * z). x * (((y @ x) \ y) * z) = (x @ y) * z. (x @ y) * ((z @ (x @ y)) \ z) = x @ z. x ^ ((x / y) * (1 @ y)) = x. (x / (y \ 1)) @ z = x @ z. x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. (x / (1 @ y)) @ z = x @ z. x @ (x / y) = x / (1 @ y). x / (1 @ y) = x @ (x / y). x / (1 @ y) = x * (y \ 1). (x * y) @ x = x * (y * (y \ 1)). (x * y) @ x = x * (y @ 1). (x @ y) * (z @ 1) = ((x @ y) * z) @ y. ((x @ y) * z) @ y = (x @ y) * (z @ 1). (x * (y @ 1)) @ z = (x * y) @ z. (x / (y * (y \ x))) \ z = (1 @ ((y * (y \ x)) / x)) * z. (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. (x @ y) * ((z @ y) \ z) = x @ z. (x @ y) * (1 @ (y \ z)) = x @ z. (x @ y) * (y \ (y @ z)) = x @ z. (x @ y) * (1 @ z) = x @ (y * z). (x @ y) \ z = (u \ (u @ y)) \ ((x @ u) \ z). (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. x ^ ((x / y) @ x) = x. x ^ ((x V y) @ x) = x. x ^ (((y V 1) * x) @ x) = x. (x \ (x @ y)) \ z = ((x @ y) \ x) * z. ((x V 1) \ y) * z = (x V 1) \ (y * z). (x V 1) \ (y * z) = ((x V 1) \ y) * z. (x @ y) \ z = ((u @ y) \ u) * ((x @ u) \ z). ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. x * (y @ (x \ z)) = (x * (y @ 1)) @ z. (x * (y @ 1)) @ z = x * (y @ (x \ z)). x * (y @ (x \ z)) = (x * y) @ z. (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. (1 @ (x * (x \ 1))) * y = (z / (x * (x \ z))) \ y. (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. (x / (y * (y \ x))) \ z = 1 * z. (x / (y * (y \ x))) \ z = z. x / (y * (y \ x)) = 1. x / ((y ^ x) @ x) = 1. (x V y) / (y @ (x V y)) = 1. x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. x / ((((y V z) \ y) * x) @ x) = 1. x / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). x / (x @ ((y V 1) \ x)) = y V 1. x ^ ((y V 1) * (x @ ((y V 1) \ x))) = (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)). (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). (x @ y) * (z * ((z @ z) \ (y \ 1))) = ((x @ y) * (z @ u)) @ 1. ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. (x * (y @ z)) @ 1 = ((x @ x) * y) @ 1. ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. (x * y) @ 1 = (x * (y @ z)) @ 1. (x * (y @ z)) @ 1 = (x * y) @ 1. ((x * y) @ 1) @ z = (x * (y @ u)) @ z. (x * (y @ z)) @ u = (x * y) @ u. (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). (x V 1) * (y @ ((x V 1) \ y)) = y ^ (((x V 1) * y) @ y). x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. ((x V 1) * y) @ y = y. x @ y = ((z V 1) * x) @ y. ((x V 1) * y) @ z = y @ z. x @ y = ((z V 1) \ x) @ y. ((x V 1) \ y) @ z = y @ z. (((x V 1) \ y) * z) @ u = (y * z) @ u. x / ((y * x) @ x) = 1. x V (1 * ((y * x) @ x)) = x. ((x * y) @ y) \ (1 \ y) = 1. x V ((y * x) @ x) = x. x V ((y @ x) * (x @ 1)) = x. ((x * y) @ y) \ y = 1. (1 @ ((x * y) @ y)) * 1 = (((x * y) @ y) @ 1) \ y. (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. (((x * y) @ y) @ 1) \ y = 1 @ y. ((x * y) @ 1) \ y = 1 @ y. ((x \ 1) * x) \ x = 1 @ x. 1 @ x = x \ ((1 @ x) * x). x \ ((1 @ x) * x) = 1 @ x. x V (((1 @ x) * x) / (1 @ x)) = ((1 @ x) * x) / (x \ ((1 @ x) * x)). ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. ((1 @ x) * x) / (1 @ x) = x. x = ((1 @ x) * x) * (x \ 1). ((1 @ x) * x) * (x \ 1) = x. x = (1 @ x) * (x * (x \ 1)). (1 @ x) * (x * (x \ 1)) = x. (1 @ x) * (x @ 1) = x. (1 @ x) * ((y @ x) @ 1) = y @ x. (1 @ x) * (y @ 1) = y @ x. ((1 @ x) * (y @ 1)) @ z = y @ z. x @ y = ((1 @ z) * x) @ y. ((1 @ x) * y) @ z = y @ z. ((x \ 1) * y) @ z = y @ z. x @ (y * x) = (1 @ y) * x. ((x @ 1) \ y) @ z = (x \ y) @ z. ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. x @ (y \ x) = (y \ 1) * x. (1 @ (x / y)) * y = y @ x. ((x @ y) \ z) @ u = (x \ z) @ u. (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). (((x \ 1) * y) \ z) @ u = (y \ z) @ u. (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. (x \ (y @ z)) @ u = (x \ y) @ u. (x @ y) \ (z @ u) = (x \ z) @ (y \ u) # label("Goal 6"). (c8 ^ c9) @ (c10 ^ c11) != (c8 @ c10) ^ (c9 @ c11) # answer("Goal 4") # label("Goal 4"). x ^ (y ^ z) = y ^ (z ^ x). x ^ (y ^ z) = z ^ (x ^ y). (x ^ y) ^ z = x ^ (z ^ y). (x V y) V z = x V (z V y). x ^ y = y ^ (x ^ (y V z)). x ^ (y ^ (x V z)) = y ^ x. 1 V (x \ (x V y)) = x \ (x V y). x ^ (x * (1 V y)) = x. x ^ (x * (y \ (y V z))) = x. 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. (1 @ x) ^ (y @ x) = y @ x. x V (1 @ x) = 1 @ x. x ^ y = x ^ (y ^ (1 @ x)). x ^ (y ^ (1 @ x)) = x ^ y. (1 @ x) V y = x V (y V (1 @ x)). x V (y V (1 @ x)) = (1 @ x) V y. x ^ (1 @ (x V y)) = x. x V (1 @ (x V y)) = 1 @ (x V y). 1 V (x \ (y ^ x)) = 1. 1 = (x \ (y ^ x)) V 1. (x \ (y ^ x)) V 1 = 1. 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. x V ((y @ 1) * x) = x. x ^ ((y @ 1) * x) = (y @ 1) * x. 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. ((x @ y) \ z) \ 1 = z \ (1 @ y). ((x @ y) \ y) \ 1 = 1. ((x @ y) \ y) * 1 = ((x @ y) \ y) @ 1. 1 V (x @ (y \ (z ^ y))) = 1. ((x @ y) \ y) @ 1 = (x @ y) \ y. 1 V (x @ (y * z)) = ((u @ y) * z) V 1. ((x @ y) * z) V 1 = 1 V (u @ (y * z)). 1 V (x @ (y \ (z ^ (u @ y)))) = 1. 1 = (x \ (y ^ (z @ x))) V 1. (x \ (y ^ (z @ x))) V 1 = 1. 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). ((x @ y) * z) V 1 = 1 V (y * z). 1 V (x * y) = 1 V ((z @ x) * y). 1 V ((x @ y) * z) = 1 V (y * z). (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. 1 @ (x V (1 @ y)) = x V (1 @ y). 1 / ((x \ (x V y)) \ 1) = x \ (x V y). x \ (x V y) = 1 @ (x \ (x V y)). 1 @ (x \ (x V y)) = x \ (x V y). 1 * x = (y \ 1) * ((1 @ y) * x). (x \ 1) * ((1 @ x) * y) = 1 * y. x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. (x V y) V ((x V y) @ x) = x V y. x ^ (x @ (x ^ y)) = x @ (x ^ y). (x \ 1) * ((1 @ x) * y) = y. 1 @ x = (y \ 1) @ x. (x \ 1) @ y = 1 @ y. (1 @ x) @ y = 1 @ y. (x V (1 @ y)) @ z = 1 @ z. x V y = x V (y V ((x V y) @ x)). x V (y V ((x V y) @ x)) = x V y. (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. x V ((1 @ y) V (1 @ x)) = x V (1 @ y). 1 V ((x \ 1) * y) = 1 V (x \ y). 1 V ((1 / x) * y) = 1 V (x \ y). x V (1 @ y) = (1 @ y) V (1 @ x). (1 @ x) V (1 @ y) = y V (1 @ x). x V (1 @ y) = (1 @ x) V (1 @ y). (1 @ x) V (1 @ y) = x V (1 @ y). x V (1 @ y) = y V (1 @ x). (x V y) V (1 @ x) = 1 @ (x V y). x V (1 @ y) = (z @ y) V (1 @ x). (x @ y) V (1 @ z) = z V (1 @ y). 1 @ (x V y) = x V (y V (1 @ x)). x V (y V (1 @ x)) = 1 @ (x V y). 1 @ (x V y) = y V (1 @ x). 1 @ (x V y) = x V (1 @ y). x @ (y V (1 @ z)) = x @ (z V y). x @ (y V (1 @ z)) = x @ (y V z). x @ (y V (1 @ z)) = x @ (y V (u @ z)). x @ (y V (z @ u)) = x @ (u V y). x @ (y V (z @ u)) = x @ (y V u). x @ ((y @ z) V u) = x @ (z V u). (x V (y @ z)) @ (x V z) = x V (y @ z). x * (x \ (x V y)) = x @ (x V y). (x V y) V (x @ (x V y)) = x V y. x ^ (x @ (x V y)) = x. x V (x @ (x V y)) = x @ (x V y). x V (y V (x @ (x V z))) = y V (x @ (x V z)). x V y = x V (y V (x @ (x V y))). x V (y V (x @ (x V y))) = x V y. x V ((y ^ x) @ x) = x. x V ((x ^ y) @ x) = x. x V (y @ (y V x)) = y V x. x V (y @ ((y @ z) V x)) = (y @ z) V x. x V (y @ (x V (y @ z))) = (y @ z) V x. x V (y @ (x V z)) = (y @ z) V x. x V (y @ (x V z)) = x V (y @ z). x V (x @ y) = x @ (x V y). x @ (x V y) = x V (x @ y). (x V (x @ y)) @ z = x @ z. ((x @ y) V ((x @ y) @ z)) @ u = x @ u. ((x @ y) V (x @ z)) @ u = x @ u. x @ ((x @ y) V z) = (x @ y) V (x @ z). (x @ y) V (x @ z) = x @ (y V z). (x @ y) ^ (x @ (y V z)) = x @ y. (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). x @ (y ^ z) = (x @ z) ^ (x @ (y ^ z)). (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. x @ (y \ x) = (1 / y) * x. 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). (x \ y) @ 1 = (x @ y) \ y. (x @ y) \ y = (x \ y) @ 1. 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). x \ (y @ z) = (x \ y) @ (x \ z). (x \ y) @ (x \ z) = x \ (y @ z). (x \ y) @ 1 = x \ (y @ x). x \ (y @ x) = (x \ y) @ 1. 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). 1 V ((x \ y) * z) = 1 V (x \ (y * z)). 1 V (x \ (y * z)) = 1 V ((x \ y) * z). 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. 1 = 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x). 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. 1 = 1 V ((x ^ (y V ((z @ 1) * x))) \ x). 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. 1 = (x ^ (y V ((z @ 1) * x))) \ x. (x ^ (y V ((z @ 1) * x))) \ x = 1. ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. (x ^ y) \ (y ^ (1 @ x)) = 1. x * ((y ^ z) \ (z ^ (1 @ y))) = x. x @ (y ^ z) = x @ (z ^ (1 @ y)). x @ (y ^ (1 @ z)) = x @ (z ^ y). x @ (y ^ (1 @ z)) = x @ (y ^ z). x @ (y ^ (1 @ z)) = x @ ((u @ z) ^ y). x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). x @ ((y @ z) ^ u) = x @ (z ^ u). x @ ((y @ z) ^ u) = x @ (u ^ z). x @ (y ^ (z @ u)) = x @ (u ^ y). x @ (y ^ (z @ u)) = x @ (y ^ u). (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). x ^ (1 @ (x ^ y)) = (x ^ (1 @ y)) @ (x ^ y). (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). x ^ (1 @ (x ^ y)) = x ^ (1 @ y). (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. 1 = (x \ (x ^ (1 @ y))) @ 1. (x \ (x ^ (1 @ y))) @ 1 = 1. x \ ((x ^ (1 @ y)) @ x) = 1. ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. (x ^ (1 @ y)) @ x = x V ((x ^ (1 @ y)) @ x). x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. (x ^ (1 @ y)) @ x = x. x @ y = (x ^ (1 @ z)) @ y. (x ^ (1 @ y)) @ z = x @ z. x @ (y ^ x) = x ^ (1 @ y). x ^ (1 @ y) = x @ (y ^ x). (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). (x @ y) ^ (1 @ z) = x @ (y ^ z). (x @ y) ^ (1 @ z) = x @ (z ^ y). x ^ (y @ (z ^ x)) = x ^ (y @ z). x ^ ((y @ x) ^ (1 @ z)) = x ^ (y @ z). (x ^ (y @ x)) ^ (1 @ z) = x ^ (y @ z). (x ^ (y @ z)) @ u = (x ^ (y @ x)) @ u. (x ^ y) @ z = (x ^ (y @ x)) @ z. (x ^ (y @ x)) @ z = (x ^ y) @ z. (x ^ (y @ x)) @ (x ^ z) = x ^ (y @ z). x ^ (y @ z) = (x ^ y) @ (x ^ z). (x ^ y) @ (x ^ z) = x ^ (y @ z). (x ^ (y @ z)) @ u = (x ^ y) @ u. (x ^ (y @ z)) @ u = (y ^ x) @ u. ((x @ y) ^ z) @ u = (x ^ z) @ u. ((x @ y) ^ z) @ ((u @ w) ^ v5) = (x ^ z) @ (w ^ v5). (x ^ y) @ (z ^ u) = (x @ z) ^ (y @ u) # label("Goal 4"). (c12 V c13) @ (c14 V c15) != (c12 @ c14) V (c13 @ c15) # answer("Goal 5") # label("Goal 5"). (x V y) V z = y V (x V z). x V (y V z) = y V (x V z). x ^ ((1 ^ y) \ x) = x. 1 V (x @ 1) = 1. 1 ^ (x @ 1) = x @ 1. x ^ ((y @ 1) \ x) = x. x V (y @ x) = x V (y @ (x ^ z)). x V (y @ (x ^ z)) = x V (y @ x). x V (y V (y @ x)) = y V x. x V y = x V (y V (x @ y)). x V (y V (x @ y)) = x V y. x V ((y @ z) V (x @ z)) = x V (y @ z). (x \ y) @ (z \ y) = (x @ z) \ y. (x \ y) @ y = (x @ 1) \ y. (x @ 1) \ y = (x \ y) @ y. x ^ ((y \ x) @ x) = x. (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. (x @ y) ^ ((z \ x) @ y) = x @ y. (x @ y) ^ ((x V z) @ y) = x @ y. (x @ y) ^ ((z V x) @ y) = x @ y. (x @ y) V ((x V z) @ y) = (x V z) @ y. (x @ y) V ((z V x) @ y) = (z V x) @ y. (x V y) V ((x V y) @ z) = (x V y) V (x @ z). (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. (x V y) V (x @ z) = x V (y V ((x V y) @ z)). x V (y V ((x V y) @ z)) = (x V y) V (x @ z). ((x V y) V (x @ z)) @ u = (x V y) @ u. (x V (y V (x @ z))) @ u = (x V y) @ u. (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. x V (y V ((x V y) @ z)) = x V (y V (x @ z)). (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). (x V (y @ z)) V (y @ u) = x V ((y @ z) V ((x V (y @ z)) @ u)). x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). x V ((y @ z) V (x @ u)) = x V ((y @ z) V (y @ u)). x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). x V (y @ (z V u)) = x V ((y @ z) V (x @ u)). x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. (x V (y @ z)) @ u = (x V (y @ x)) @ u. (x V (y @ z)) @ u = (x V (y @ w)) @ u. (x V (y @ z)) @ (x V u) = x V (y @ u). (x V y) @ z = (y V (x @ u)) @ z. (x V (y @ z)) @ u = (y V x) @ u. ((x @ y) V z) @ u = (x V z) @ u. (x V y) @ (x V z) = x V (y @ z). (x @ y) V (z @ u) = ((x @ y) V z) @ (y V u). ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). (x @ y) V (z @ u) = (x V z) @ (y V u). (x V y) @ (z V u) = (x @ z) V (y @ u). (c34 V c35) @ ((c34 / c35) \ c34) != (c34 / c35) \ c34 # answer("Goal 15"). 1 ^ (x \ (y V x)) = 1. 1 ^ ((x * (x \ y)) \ y) = 1. x V ((y / x) \ y) = (y / x) \ y. 1 V ((x * (x \ y)) \ y) = (x * (x \ y)) \ y. (x * y) ^ ((z V (x * (y * u))) / u) = x * y. x V (y / (z ^ (x \ y))) = y / (z ^ (x \ y)). x ^ (y / (z * (z \ (x \ y)))) = x. x V (y * (((y \ x) / z) * z)) = x. x V (((y ^ x) / z) * z) = x. x ^ (((y ^ z) / x) \ z) = x. (x \ y) V ((x \ 1) * y) = x \ y. x ^ ((y / x) \ (1 @ y)) = x. x ^ ((y \ 1) \ (1 @ (y \ x))) = x. x / ((y / (z ^ (x \ y))) \ x) = x V (y / (z ^ (x \ y))). (x * y) \ x = 1 / y. 1 / (x @ y) = 1 / y. x \ (1 / y) = 1 / (y * x). x \ (1 / y) = 1 / ((z @ y) * x). 1 / ((x @ y) * z) = z \ (1 / y). 1 / ((x @ y) * z) = 1 / (y * z). x ^ (1 @ (y * (y \ x))) = x. 1 / ((x @ (y \ z)) * u) = u \ (1 @ (z \ y)). x @ (y / (1 / z)) = x @ (y * z). x @ (y * (y \ (z * (1 / u)))) = x @ (z / u). 1 = 1 V (x \ (y ^ (z @ x))). 1 V (x \ (y ^ (z @ x))) = 1. 1 V ((x * (x \ y)) \ y) = 1. 1 = (x * (x \ y)) \ y. (x * (x \ y)) \ y = 1. x ^ ((y \ (z / x)) \ (y \ z)) = x. 1 / ((x @ (y \ z)) * u) = 1 @ (u \ (z \ y)). x @ (y * (z \ u)) = x @ (y / (u \ z)). x @ (y / (z \ u)) = x @ (y * (u \ z)). 1 @ (x \ (y \ z)) = 1 / ((z \ y) * x). 1 / ((x \ y) * z) = 1 @ (z \ (y \ x)). x @ (y / (1 / ((z \ u) * w))) = x @ (y / (w \ (u \ z))). x / ((y / (z ^ (x \ y))) \ x) = y / (z ^ (x \ y)). (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / (z ^ ((x \ (y / z)) \ (x \ y))). x @ (y / (z \ (u \ w))) = x @ (y * ((w \ u) * z)). x @ (y / ((z * u) \ w)) = x @ (y * ((w \ z) * u)). x @ (y * ((z \ u) * w)) = x @ (y * (z \ (u * w))). x @ (y * (z \ (u * w))) = x @ (y * ((z \ u) * w)). x @ (y * ((y \ z) * (1 / u))) = x @ (z / u). x @ (y @ x) = x. (x / y) \ x = ((x / y) \ x) @ y. ((x / y) \ x) @ y = (x / y) \ x. x \ (x @ (x V y)) = x \ (x V y). x V (((y / x) \ y) @ x) = (y / x) \ y. ((1 @ x) * y) / y = 1 @ x. x @ (y * ((y \ z) / u)) = x @ (z / u). (x * ((x \ y) / z)) @ (y / z) = x * ((x \ y) / z). (1 @ x) \ ((1 @ x) * y) = y. x @ (x V (y @ z)) = x V (x @ z). x \ (x V (x @ y)) = x \ (x V y). x * (x \ (x V y)) = x V (x @ y). x \ (x V (x @ y)) = x \ (x V (z @ y)). x \ (x V (y @ z)) = x \ (x V (x @ z)). (x V (x @ y)) \ (x V y) = 1. x \ (x V (y @ z)) = x \ (x V z). (x V y) \ (x V (y V (z @ u))) = (x V y) \ ((x V y) V u). ((x * ((x \ y) / z)) V (x * ((x \ y) / z))) \ ((x * ((x \ y) / z)) V (y / z)) = 1. (x V y) \ (x V (y V (z @ u))) = (x V y) \ (x V (y V u)). (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V (y V y)). (x V y) \ (x V (y V y)) = (x V y) \ (x V ((z / y) \ z)). (x V y) \ (x V y) = (x V y) \ (x V ((z / y) \ z)). (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V y). (x V y) \ (x V ((z / y) \ z)) = 1. (x * ((x \ y) / z)) \ ((x * ((x \ y) / z)) V (y / z)) = 1. (x * ((x \ y) / z)) \ ((y / z) V (x * ((x \ y) / z))) = 1. 1 = ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))). ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))) = 1. (1 @ x) \ x = x @ 1. (x \ y) V (y @ (x \ y)) = x \ y. x / (x @ (y \ x)) = 1 @ y. (x @ y) / x = 1 @ (y / x). x / (x @ y) = 1 @ (x / y). ((x @ y) / x) \ (y / x) = (y / x) @ 1. (x @ y) V ((1 @ (x / y)) \ x) = (x / (x @ y)) \ x. (x @ y) / x = y / (y @ x). (x @ (y / z)) \ y = (x \ y) @ z. (x @ y) V ((1 \ x) @ y) = (x / (x @ y)) \ x. (x @ y) V (x @ y) = (x / (x @ y)) \ x. (x / (x @ y)) \ x = x @ y. ((x @ y) / x) \ y = y @ x. x @ (x ^ y) = x ^ (1 @ y). x ^ (1 @ y) = x @ (x ^ y). x ^ (y @ (x ^ z)) = x ^ (y @ z). x ^ (x @ y) = x @ (x ^ y). x @ (x ^ y) = x ^ (x @ y). x ^ (1 @ y) = x ^ (x @ y). x ^ (y ^ (y @ x)) = x ^ y. x ^ ((x @ y) ^ y) = y ^ x. x ^ (y ^ (x @ y)) = y ^ x. x ^ (y ^ (x @ y)) = x ^ y. (x ^ y) ^ (x @ y) = x ^ y. (x ^ y) V (x @ y) = x @ y. (x ^ y) V (y @ x) = y @ x. (x * y) V (((z V (x * (y * u))) / u) @ (x * y)) = ((z V (x * (y * u))) / u) @ (x * y). (x V y) @ z = (x V (y @ x)) @ z. (x V (y @ x)) @ z = (x V y) @ z. (x V y) @ (x V x) = x V (y @ x). (x V y) @ x = x V (y @ x). (x V y) @ y = y V (x @ y). (x V (y \ x)) @ (y \ x) = y \ x. (x \ y) @ z = (y V (x \ y)) @ z. (x V (y \ x)) @ z = (y \ x) @ z. (x V y) @ (z V x) = x V (y @ z). x V ((y \ x) @ z) = (y \ x) @ (x V z). (x \ y) @ (y V z) = y V ((x \ y) @ z). (x V y) \ (x V ((z \ x) @ y)) = ((x V y) \ (z \ x)) @ 1. (x @ y) V (z @ x) = x V (z @ y). x V (y @ z) = (y @ x) V (x @ z). (x @ y) V (y @ z) = y V (x @ z). (x V y) \ (x V ((x / y) \ x)) = ((x V y) \ ((x / y) \ x)) @ 1. ((x V y) \ ((x / y) \ x)) @ 1 = 1. ((x V y) @ ((x / y) \ x)) \ ((x / y) \ x) = 1. ((x V y) @ y) \ ((x / y) \ x) = 1. (x V (y @ x)) \ ((y / x) \ y) = 1. (x / y) ^ (x / ((y V (x @ y)) * 1)) = x / y. (x / y) ^ (x / (y V (x @ y))) = x / y. x / y = x / (y V (x @ y)). x / (y V (x @ y)) = x / y. x / ((y @ z) V (x @ z)) = x / (y @ z). x / (x V (y @ x)) = x / (y @ x). (x / (y @ x)) \ x = x V (y @ x). ((x @ y) / (y @ (x @ y))) \ (x @ y) = y V (x @ (x @ y)). ((x @ y) / y) \ (x @ y) = y V (x @ (x @ y)). (x * ((x \ y) / z)) V ((y / z) @ (x * ((x \ y) / z))) = ((y V (x * (((x \ y) / z) * z))) / z) @ (x * ((x \ y) / z)). ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (y * ((y \ x) / z))). (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / z. ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (x / z)). ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V (x / z). ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (x / z) V (y * ((y \ x) / z)). (x / y) @ (z * ((z \ x) / y)) = (x / y) V (z * ((z \ x) / y)). (x / y) V (z * ((z \ x) / y)) = (x / y) @ (x / y). (x / y) V (z * ((z \ x) / y)) = x / y. ((x \ y) / z) \ (x \ (y / z)) = 1. (x \ (y / z)) / 1 = (x \ y) / z. (x \ y) / z = x \ (y / z). x \ (y / z) = (x \ y) / z. (((x @ y) / x) \ y) / x = (y / x) @ 1. (x @ y) / y = (x / y) @ 1. x V (((y / x) @ 1) \ (y @ x)) = ((y @ x) / x) \ (y @ x). x V (((y / x) \ y) @ (1 \ x)) = ((y @ x) / x) \ (y @ x). x V (((y / x) \ y) @ x) = ((y @ x) / x) \ (y @ x). ((x @ y) / y) \ (x @ y) = y V (((x / y) \ x) @ y). ((x @ y) / y) \ (x @ y) = (x / y) \ x. (x / y) \ x = y V (x @ (x @ y)). x V (y @ (y @ x)) = (y / x) \ y. x V (y @ x) = (y / x) \ y. (x V y) @ y = (x / y) \ x. (x / y) \ x = (x V y) @ y. (x V y) @ ((z / y) \ z) = (x / y) \ x. (x / y) \ x = (x V y) @ ((z / y) \ z). ((c30 / c31) \ c30) @ (c30 V c31) != c30 V c31 # answer("Goal 13"). (x @ y) @ x = x. ((x / y) \ x) @ (x V y) = x V y # label("Goal 13"). (c32 V c33) @ (c32 / (c33 \ c32)) != c32 / (c33 \ c32) # answer("Goal 14"). x V y = y V (x V (y ^ z)). x V (y V (x ^ z)) = y V x. (x * (x \ y)) / ((y V (x * (x \ y))) \ (x * (x \ y))) = y. x = (1 @ x) ^ x. (1 @ x) ^ x = x. x ^ (1 @ ((y / x) \ y)) = x. (x @ y) \ (1 @ y) = 1. 1 @ ((x @ y) \ y) = 1. 1 ^ ((x @ y) \ y) = (x @ y) \ y. (x * (x \ y)) / (y \ (x * (x \ y))) = y. (x \ (x / y)) @ 1 = 1. (x \ (y V x)) @ 1 = 1. ((x V (y ^ z)) \ (x V y)) @ 1 = 1. x ^ ((y @ (y / x)) \ y) = x. (x @ (x / y)) ^ (x / y) = x @ (x / y). 1 ^ ((x @ (x / y)) \ (x / y)) = 1. ((x V (y ^ z)) \ (x V z)) @ 1 = 1. 1 = (x @ (x / y)) \ (x / y). (x @ (x / y)) \ (x / y) = 1. ((x V (y @ (y / z))) \ (x V (y / z))) @ 1 = 1. (1 @ (x / y)) * 1 = (x @ 1) \ (x / y). (x @ 1) \ (x / y) = (1 @ (x / y)) * 1. (x @ 1) \ (x / y) = 1 @ (x / y). ((x @ 1) * (1 @ (x / y))) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. (x @ (x / y)) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. (x @ (x / y)) / ((x / y) \ ((x @ 1) * (1 @ (x / y)))) = x / y. (x @ (x / y)) / ((x / y) \ (x @ (x / y))) = x / y. (x @ (x / y)) / (((x / y) \ x) @ 1) = x / y. (((x V y) @ z) \ u) @ w = ((y V (x @ v5)) \ u) @ w. ((x V (y @ z)) \ u) @ w = (((y V x) @ v5) \ u) @ w. (x V y) @ z = (x @ z) V (y @ z). (x @ y) V (z @ y) = (x V z) @ y. (x @ y) / (((x V z) @ y) \ (x @ y)) = (x @ y) V (z @ y). ((x V (y @ z)) \ u) @ w = ((y V x) \ u) @ w. ((x V y) \ (y V (x / z))) @ 1 = 1. ((x V y) \ (x / (y \ x))) @ 1 = 1. ((x V y) @ (x / (y \ x))) \ (x / (y \ x)) = 1. ((x V y) @ y) \ (x / (y \ x)) = 1. (x V (y @ x)) \ (y / (x \ y)) = 1. (x \ y) ^ (((x V (y @ x)) * 1) \ y) = x \ y. (x \ y) ^ ((x V (y @ x)) \ y) = x \ y. x \ y = (x V (y @ x)) \ y. (x V (y @ x)) \ y = x \ y. (x \ y) @ z = ((y V x) \ y) @ z. ((x V y) \ x) @ z = (y \ x) @ z. (x @ y) / (((x V z) \ x) @ (y \ y)) = (x @ y) V (z @ y). (x @ y) / (((x V z) \ x) @ (y \ y)) = (x V z) @ y. (x @ y) / (((x V z) \ x) @ 1) = (x V z) @ y. (x @ y) / ((z \ x) @ 1) = (x V z) @ y. x / ((y V x) @ z) = x / (y @ z). (x @ y) / ((x V z) @ u) = (x @ y) / (z @ u). ((x / y) \ x) @ z = (x V y) @ z. (x @ (x / y)) / ((x V y) @ 1) = x / y. x / y = (x @ (x / y)) / (y @ 1). (x @ (x / y)) / (y @ 1) = x / y. x / (y \ x) = (x V y) @ (x / (y \ x)). (x V y) @ (x / (y \ x)) = x / (y \ x). (c28 / (c29 \ c28)) @ (c28 V c29) != c28 V c29 # answer("Goal 12") # label("Goal 12"). (x / y) V (x @ (x / y)) = x / y. (x / y) @ z = ((x / y) V x) @ z. ((x / y) V x) @ z = (x / y) @ z. (x V (x / y)) @ z = (x / y) @ z. (x @ y) / (((x / z) \ x) @ 1) = (x / z) @ y. (x @ y) / ((x V z) @ 1) = (x / z) @ y. (x / y) @ z = (x @ z) / (y @ 1). (x @ y) / (z @ 1) = (x / z) @ y. (x / (y \ x)) @ z = (x V y) @ z. (x / (y \ x)) @ (x V y) = x V y # label("Goal 12"). (c20 / c21) @ (c22 / c23) != (c20 @ c22) / (c21 @ c23) # answer("Goal 7"). (c20 @ c22) / (c21 @ c23) != (c20 / c21) @ (c22 / c23) # answer("Goal 7"). (1 / x) / y = 1 / (y * x). (1 @ x) / y = 1 @ ((z @ x) / y). 1 @ ((x @ y) / z) = (1 @ y) / z. x @ ((1 @ y) / z) = x @ (y / z). x @ ((1 @ y) / z) = x @ ((u @ y) / z). x @ ((y @ z) / u) = x @ (z / u). x @ (y / z) = x @ ((u * (u \ y)) / z). x @ ((y * (y \ z)) / u) = x @ (z / u). x @ ((y * (1 / z)) / u) = x @ ((y / z) / u). (x @ y) / z = ((x @ y) / z) @ (y / z). ((x @ y) / z) @ (y / z) = (x @ y) / z. (x @ y) / (z @ u) = ((x @ y) / (z @ u)) @ (y / u). ((x @ y) / (z @ u)) @ (y / u) = (x @ y) / (z @ u). x @ (y * ((1 / z) / u)) = x @ ((y / z) / u). x @ (y * (1 / (z * u))) = x @ ((y / u) / z). x @ ((y / z) / u) = x @ (y / (u * z)). (((x / y) / ((x / y) @ z)) \ x) / y = (x / y) @ z. ((1 @ ((x / y) / z)) \ x) / y = (x / y) @ z. ((1 @ (x / (y * z))) \ x) / z = (x / z) @ y. ((1 \ x) @ (y * z)) / z = (x / z) @ y. (x @ (y * z)) / z = (x / z) @ y. (x @ y) / z = (x / z) @ (y / z). (x / y) @ (z / y) = (x @ z) / y. ((x @ y) / z) @ u = (x / z) @ u. (x / y) @ z = (x / (y @ 1)) @ z. (x / (y @ 1)) @ z = (x / y) @ z. (x / (y @ 1)) @ z = (x / (y @ u)) @ z. (x / (y @ z)) @ u = (x / y) @ u. ((x @ y) / z) @ (y / u) = (x @ y) / (z @ u). (x @ y) / (z @ u) = (x / z) @ (y / u) # label("Goal 7"). end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 x @ x = x # label("Goal 1") # label(non_clause) # label(goal). [goal]. 2 (x @ y) @ z = x @ z # label("Goal 2") # label(non_clause) # label(goal). [goal]. 3 x @ (y @ z) = x @ z # label("Goal 3") # label(non_clause) # label(goal). [goal]. 4 (x ^ y) @ (z ^ u) = (x @ z) ^ (y @ u) # label("Goal 4") # label(non_clause) # label(goal). [goal]. 5 (x V y) @ (z V u) = (x @ z) V (y @ u) # label("Goal 5") # label(non_clause) # label(goal). [goal]. 6 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6") # label(non_clause) # label(goal). [goal]. 7 (x / y) @ (z / u) = (x @ z) / (y @ u) # label("Goal 7") # label(non_clause) # label(goal). [goal]. 8 (x * (x \ 1)) @ 1 = x * (x \ 1) # label("Goal 8") # label(non_clause) # label(goal). [goal]. 9 1 @ (x * (x \ 1)) = 1 # label("Goal 9") # label(non_clause) # label(goal). [goal]. 10 (x \ 1) @ 1 = 1 # label("Goal 10") # label(non_clause) # label(goal). [goal]. 11 1 @ (x \ 1) = x \ 1 # label("Goal 11") # label(non_clause) # label(goal). [goal]. 12 (x / (y \ x)) @ (x V y) = x V y # label("Goal 12") # label(non_clause) # label(goal). [goal]. 13 ((x / y) \ x) @ (x V y) = x V y # label("Goal 13") # label(non_clause) # label(goal). [goal]. 14 (x V y) @ (x / (y \ x)) = x / (y \ x) # label("Goal 14") # label(non_clause) # label(goal). [goal]. 15 (x V y) @ ((x / y) \ x) = (x / y) \ x # label("Goal 15") # label(non_clause) # label(goal). [goal]. ============================== end of process non-clausal formulas === ============================== PROCESS INITIAL CLAUSES =============== % Clauses before input processing: formulas(usable). end_of_list. formulas(sos). (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. x ^ x = x # label("Idempotence-^"). [assumption]. x V x = x # label("Idempotence-V"). [assumption]. x ^ y = y ^ x # label("Commutativity-^"). [assumption]. x V y = y V x # label("Commutativity-V"). [assumption]. (x ^ y) V x = x # label("Absorption a"). [assumption]. (x V y) ^ x = x # label("Absorption b"). [assumption]. (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 1 * x = x # label("Left monoid unit"). [assumption]. x * 1 = x # label("Right monoid unit"). [assumption]. x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. c1 @ c1 != c1 # label("Goal 1"). [deny(1)]. (c2 @ c3) @ c4 != c2 @ c4 # label("Goal 2"). [deny(2)]. c5 @ (c6 @ c7) != c5 @ c7 # label("Goal 3"). [deny(3)]. (c8 @ c10) ^ (c9 @ c11) != (c8 ^ c9) @ (c10 ^ c11) # label("Goal 4"). [deny(4)]. (c12 @ c14) V (c13 @ c15) != (c12 V c13) @ (c14 V c15) # label("Goal 5"). [deny(5)]. (c16 @ c18) \ (c17 @ c19) != (c16 \ c17) @ (c18 \ c19) # label("Goal 6"). [deny(6)]. (c20 @ c22) / (c21 @ c23) != (c20 / c21) @ (c22 / c23) # label("Goal 7"). [deny(7)]. (c24 * (c24 \ 1)) @ 1 != c24 * (c24 \ 1) # label("Goal 8"). [deny(8)]. 1 @ (c25 * (c25 \ 1)) != 1 # label("Goal 9"). [deny(9)]. (c26 \ 1) @ 1 != 1 # label("Goal 10"). [deny(10)]. c27 \ 1 != 1 @ (c27 \ 1) # label("Goal 11"). [deny(11)]. c28 V c29 != (c28 / (c29 \ c28)) @ (c28 V c29) # label("Goal 12"). [deny(12)]. c30 V c31 != ((c30 / c31) \ c30) @ (c30 V c31) # label("Goal 13"). [deny(13)]. c32 / (c33 \ c32) != (c32 V c33) @ (c32 / (c33 \ c32)) # label("Goal 14"). [deny(14)]. (c34 / c35) \ c34 != (c34 V c35) @ ((c34 / c35) \ c34) # label("Goal 15"). [deny(15)]. end_of_list. formulas(demodulators). end_of_list. % 1291 hints input. ============================== PREDICATE ELIMINATION ================= No predicates eliminated. ============================== end predicate elimination =============  WARNING: denials share constants (see output). Auto_denials: % copying label "Goal 1" to answer in negative clause % copying label "Goal 2" to answer in negative clause % copying label "Goal 3" to answer in negative clause % copying label "Goal 4" to answer in negative clause % copying label "Goal 5" to answer in negative clause % copying label "Goal 6" to answer in negative clause % copying label "Goal 7" to answer in negative clause % copying label "Goal 8" to answer in negative clause % copying label "Goal 9" to answer in negative clause % copying label "Goal 10" to answer in negative clause % copying label "Goal 11" to answer in negative clause % copying label "Goal 12" to answer in negative clause % copying label "Goal 13" to answer in negative clause % copying label "Goal 14" to answer in negative clause % copying label "Goal 15" to answer in negative clause % assign(max_proofs, 15). % (Horn set with more than one neg. clause) WARNING, because some of the denials share constants, some of the denials or their descendents may be subsumed, preventing the target number of proofs from being found. The shared constants are: 1. Term ordering decisions: Function symbol KB weights: 1=1. c25=1. c24=1. c27=1. c5=1. c6=1. c7=1. c26=1. c1=1. c2=1. c3=1. c4=1. c16=1. c17=1. c18=1. c19=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. c34=1. c35=1. c30=1. c31=1. c32=1. c33=1. c28=1. c29=1. c20=1. c21=1. c22=1. c23=1. V=1. ^=1. *=1. \=1. /=1. @=1. Predicate symbol precedence: predicate_order([ = ]). Function symbol precedence: function_order([ 1, c25, c24, c27, c5, c6, c7, c26, c1, c2, c3, c4, c16, c17, c18, c19, c8, c9, c10, c11, c12, c13, c14, c15, c34, c35, c30, c31, c32, c33, c28, c29, c20, c21, c22, c23, V, ^, *, \, /, @ ]). Skipping inverse_order, because term ordering is KBO. Unfolding symbols: @/2. Auto_inference settings: % set(paramodulation). % (positive equality literals) Auto_process settings: (no changes). kept: 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. kept: 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. kept: 18 x ^ x = x # label("Idempotence-^"). [assumption]. kept: 19 x V x = x # label("Idempotence-V"). [assumption]. kept: 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. kept: 21 x V y = y V x # label("Commutativity-V"). [assumption]. kept: 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. kept: 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. kept: 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. kept: 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. kept: 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. kept: 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. kept: 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. kept: 29 1 * x = x # label("Left monoid unit"). [assumption]. kept: 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. kept: 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. kept: 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. kept: 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. kept: 36 c1 @ c1 != c1 # label("Goal 1") # answer("Goal 1"). [deny(1)]. kept: 37 (c2 @ c3) @ c4 != c2 @ c4 # label("Goal 2") # answer("Goal 2"). [deny(2)]. kept: 38 c5 @ (c6 @ c7) != c5 @ c7 # label("Goal 3") # answer("Goal 3"). [deny(3)]. 39 (c8 @ c10) ^ (c9 @ c11) != (c8 ^ c9) @ (c10 ^ c11) # label("Goal 4") # answer("Goal 4"). [deny(4)]. kept: 40 (c8 ^ c9) @ (c10 ^ c11) != (c8 @ c10) ^ (c9 @ c11) # answer("Goal 4") # label("Goal 4"). [copy(39),flip(a)]. 41 (c12 @ c14) V (c13 @ c15) != (c12 V c13) @ (c14 V c15) # label("Goal 5") # answer("Goal 5"). [deny(5)]. kept: 42 (c12 V c13) @ (c14 V c15) != (c12 @ c14) V (c13 @ c15) # answer("Goal 5") # label("Goal 5"). [copy(41),flip(a)]. 43 (c16 @ c18) \ (c17 @ c19) != (c16 \ c17) @ (c18 \ c19) # label("Goal 6") # answer("Goal 6"). [deny(6)]. kept: 44 (c16 \ c17) @ (c18 \ c19) != (c16 @ c18) \ (c17 @ c19) # answer("Goal 6"). [copy(43),flip(a)]. 45 (c20 @ c22) / (c21 @ c23) != (c20 / c21) @ (c22 / c23) # label("Goal 7") # answer("Goal 7"). [deny(7)]. kept: 46 (c20 / c21) @ (c22 / c23) != (c20 @ c22) / (c21 @ c23) # answer("Goal 7"). [copy(45),flip(a)]. kept: 47 (c24 * (c24 \ 1)) @ 1 != c24 * (c24 \ 1) # label("Goal 8") # answer("Goal 8"). [deny(8)]. kept: 48 1 @ (c25 * (c25 \ 1)) != 1 # label("Goal 9") # answer("Goal 9"). [deny(9)]. kept: 49 (c26 \ 1) @ 1 != 1 # label("Goal 10") # answer("Goal 10"). [deny(10)]. 50 c27 \ 1 != 1 @ (c27 \ 1) # label("Goal 11") # answer("Goal 11"). [deny(11)]. kept: 51 1 @ (c27 \ 1) != c27 \ 1 # answer("Goal 11") # label("Goal 11"). [copy(50),flip(a)]. 52 c28 V c29 != (c28 / (c29 \ c28)) @ (c28 V c29) # label("Goal 12") # answer("Goal 12"). [deny(12)]. kept: 53 (c28 / (c29 \ c28)) @ (c28 V c29) != c28 V c29 # answer("Goal 12") # label("Goal 12"). [copy(52),flip(a)]. 54 c30 V c31 != ((c30 / c31) \ c30) @ (c30 V c31) # label("Goal 13") # answer("Goal 13"). [deny(13)]. kept: 55 ((c30 / c31) \ c30) @ (c30 V c31) != c30 V c31 # answer("Goal 13"). [copy(54),flip(a)]. 56 c32 / (c33 \ c32) != (c32 V c33) @ (c32 / (c33 \ c32)) # label("Goal 14") # answer("Goal 14"). [deny(14)]. kept: 57 (c32 V c33) @ (c32 / (c33 \ c32)) != c32 / (c33 \ c32) # answer("Goal 14"). [copy(56),flip(a)]. 58 (c34 / c35) \ c34 != (c34 V c35) @ ((c34 / c35) \ c34) # label("Goal 15") # answer("Goal 15"). [deny(15)]. kept: 59 (c34 V c35) @ ((c34 / c35) \ c34) != (c34 / c35) \ c34 # answer("Goal 15"). [copy(58),flip(a)]. ============================== end of process initial clauses ======== ============================== CLAUSES FOR SEARCH ==================== % Clauses after input processing: formulas(usable). 36 c1 @ c1 != c1 # label("Goal 1") # answer("Goal 1"). [deny(1)]. 37 (c2 @ c3) @ c4 != c2 @ c4 # label("Goal 2") # answer("Goal 2"). [deny(2)]. 38 c5 @ (c6 @ c7) != c5 @ c7 # label("Goal 3") # answer("Goal 3"). [deny(3)]. 40 (c8 ^ c9) @ (c10 ^ c11) != (c8 @ c10) ^ (c9 @ c11) # answer("Goal 4") # label("Goal 4"). [copy(39),flip(a)]. 42 (c12 V c13) @ (c14 V c15) != (c12 @ c14) V (c13 @ c15) # answer("Goal 5") # label("Goal 5"). [copy(41),flip(a)]. 44 (c16 \ c17) @ (c18 \ c19) != (c16 @ c18) \ (c17 @ c19) # answer("Goal 6"). [copy(43),flip(a)]. 46 (c20 / c21) @ (c22 / c23) != (c20 @ c22) / (c21 @ c23) # answer("Goal 7"). [copy(45),flip(a)]. 47 (c24 * (c24 \ 1)) @ 1 != c24 * (c24 \ 1) # label("Goal 8") # answer("Goal 8"). [deny(8)]. 48 1 @ (c25 * (c25 \ 1)) != 1 # label("Goal 9") # answer("Goal 9"). [deny(9)]. 49 (c26 \ 1) @ 1 != 1 # label("Goal 10") # answer("Goal 10"). [deny(10)]. 51 1 @ (c27 \ 1) != c27 \ 1 # answer("Goal 11") # label("Goal 11"). [copy(50),flip(a)]. 53 (c28 / (c29 \ c28)) @ (c28 V c29) != c28 V c29 # answer("Goal 12") # label("Goal 12"). [copy(52),flip(a)]. 55 ((c30 / c31) \ c30) @ (c30 V c31) != c30 V c31 # answer("Goal 13"). [copy(54),flip(a)]. 57 (c32 V c33) @ (c32 / (c33 \ c32)) != c32 / (c33 \ c32) # answer("Goal 14"). [copy(56),flip(a)]. 59 (c34 V c35) @ ((c34 / c35) \ c34) != (c34 / c35) \ c34 # answer("Goal 15"). [copy(58),flip(a)]. end_of_list. formulas(sos). 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. end_of_list. formulas(demodulators). end_of_list. % 1067 hints (1291 processed, 224 redundant). ============================== end of clauses for search ============= ============================== SEARCH ================================ % Starting search at 0.04 seconds. given #1 (I,wt=11): 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. given #2 (I,wt=11): 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. given #3 (I,wt=5): 18 x ^ x = x # label("Idempotence-^"). [assumption]. given #4 (I,wt=5): 19 x V x = x # label("Idempotence-V"). [assumption]. given #5 (I,wt=7): 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. given #6 (I,wt=7): 21 x V y = y V x # label("Commutativity-V"). [assumption]. given #7 (I,wt=7): 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. given #8 (I,wt=7): 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. given #9 (I,wt=11): 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. given #10 (I,wt=11): 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. given #11 (I,wt=11): 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. given #12 (I,wt=11): 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. given #13 (I,wt=11): 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. given #14 (I,wt=5): 29 1 * x = x # label("Left monoid unit"). [assumption]. given #15 (I,wt=5): 30 x * 1 = x # label("Right monoid unit"). [assumption]. given #16 (I,wt=11): 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. given #17 (I,wt=11): 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. given #18 (I,wt=15): 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. given #19 (H,wt=7): 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. given #20 (H,wt=7): 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. given #21 (H,wt=7): 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. given #22 (H,wt=7): 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. given #23 (H,wt=7): 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. given #24 (H,wt=7): 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. given #25 (H,wt=9): 76 x * (y * 1) = x * y. [para(30(a,1),28(a,1)),flip(a)]. given #26 (H,wt=9): 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. given #27 (H,wt=7): 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. given #28 (H,wt=9): 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. given #29 (H,wt=9): 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. given #30 (H,wt=9): 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. given #31 (H,wt=9): 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. given #32 (H,wt=11): 60 x ^ (y ^ z) = z ^ (x ^ y). [para(20(a,1),16(a,1)),flip(a)]. given #33 (H,wt=11): 61 (x ^ y) ^ z = x ^ (z ^ y). [para(20(a,1),16(a,2,2))]. given #34 (H,wt=11): 62 (x V y) V z = y V (x V z). [para(21(a,1),17(a,1,1))]. given #35 (H,wt=11): 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. given #36 (H,wt=11): 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. given #37 (H,wt=11): 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. given #38 (H,wt=11): 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. given #39 (H,wt=9): 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. given #40 (H,wt=7): 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. given #41 (H,wt=9): 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. given #42 (H,wt=11): 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. given #43 (H,wt=9): 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. given #44 (H,wt=7): 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. given #45 (H,wt=9): 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. given #46 (H,wt=11): 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. given #47 (H,wt=9): 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. given #48 (H,wt=7): 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. given #49 (H,wt=5): 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. given #50 (H,wt=7): 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. given #51 (H,wt=9): 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. given #52 (H,wt=9): 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. given #53 (H,wt=9): 123 1 ^ (x \ (y V x)) = 1. [para(21(a,1),113(a,1,2,2))]. given #54 (H,wt=9): 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. given #55 (H,wt=9): 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. given #56 (H,wt=9): 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. given #57 (H,wt=9): 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. given #58 (H,wt=11): 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. given #59 (H,wt=11): 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. given #60 (H,wt=9): 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. given #61 (H,wt=7): 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. given #62 (H,wt=7): 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. given #63 (H,wt=5): 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. given #64 (H,wt=9): 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. given #65 (H,wt=9): 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. given #66 (H,wt=9): 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. given #67 (H,wt=9): 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. given #68 (H,wt=11): 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. given #69 (H,wt=9): 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. given #70 (H,wt=9): 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. given #71 (H,wt=11): 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. given #72 (H,wt=11): 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. given #73 (H,wt=11): 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. given #74 (H,wt=11): 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. given #75 (H,wt=11): 93 x ^ (y ^ (x V z)) = y ^ x. [para(23(a,1),60(a,1,2)),flip(a)]. given #76 (H,wt=11): 94 x V (y V z) = y V (x V z). [para(62(a,1),17(a,1))]. given #77 (H,wt=11): 95 x V (y V (x ^ z)) = y V x. [para(22(a,1),63(a,1,2)),flip(a)]. given #78 (H,wt=11): 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. given #79 (H,wt=11): 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. given #80 (H,wt=11): 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. given #81 (H,wt=11): 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. given #82 (H,wt=11): 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. given #83 (H,wt=9): 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. given #84 (H,wt=11): 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. given #85 (H,wt=11): 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. given #86 (H,wt=9): 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. given #87 (H,wt=9): 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. given #88 (H,wt=11): 129 1 ^ ((x * (x \ y)) \ y) = 1. [para(99(a,1),123(a,1,2,2))]. given #89 (H,wt=11): 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. given #90 (H,wt=11): 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. given #91 (H,wt=11): 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. given #92 (H,wt=11): 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. given #93 (H,wt=9): 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. given #94 (H,wt=9): 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. given #95 (H,wt=9): 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. given #96 (H,wt=9): 169 x ^ ((1 ^ y) \ x) = x. [para(20(a,1),167(a,1,2,1))]. given #97 (H,wt=11): 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. given #98 (H,wt=11): 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. given #99 (H,wt=11): 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. given #100 (H,wt=11): 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. given #101 (H,wt=13): 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. given #102 (H,wt=11): 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. given #103 (H,wt=9): 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. given #104 (H,wt=13): 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. given #105 (H,wt=13): 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. given #106 (H,wt=11): 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. given #107 (H,wt=11): 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. given #108 (H,wt=11): 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. given #109 (H,wt=13): 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. given #110 (H,wt=11): 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. given #111 (H,wt=11): 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. given #112 (H,wt=9): 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. given #113 (H,wt=9): 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. given #114 (H,wt=9): 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. given #115 (H,wt=11): 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. given #116 (H,wt=13): 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. given #117 (H,wt=11): 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. given #118 (H,wt=11): 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. given #119 (H,wt=11): 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. given #120 (H,wt=11): 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. given #121 (H,wt=11): 201 x V (((y ^ x) / z) * z) = x. [para(20(a,1),194(a,1,2,1,1))]. given #122 (H,wt=11): 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. given #123 (H,wt=11): 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. given #124 (H,wt=11): 204 x ^ (((y ^ z) / x) \ z) = x. [para(201(a,1),111(a,1,2,2))]. given #125 (H,wt=11): 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. given #126 (H,wt=13): 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. given #127 (H,wt=13): 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. given #128 (H,wt=11): 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. given #129 (H,wt=9): 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. given #130 (H,wt=9): 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. given #131 (H,wt=9): 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. given #132 (H,wt=7): 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. given #133 (H,wt=11): 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. given #134 (H,wt=11): 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. given #135 (H,wt=9): 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. given #136 (H,wt=7): 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. given #137 (H,wt=13): 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. given #138 (H,wt=11): 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. given #139 (H,wt=11): 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. given #140 (H,wt=9): 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. given #141 (H,wt=9): 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. given #142 (H,wt=9): 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. given #143 (H,wt=9): 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. given #144 (H,wt=13): 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. given #145 (H,wt=13): 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. given #146 (H,wt=11): 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. given #147 (H,wt=13): 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. given #148 (H,wt=13): 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. given #149 (H,wt=13): 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. given #150 (H,wt=13): 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. given #151 (H,wt=13): 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. given #152 (H,wt=13): 157 x V ((y / x) \ y) = (y / x) \ y. [para(155(a,1),65(a,1,1))]. given #153 (H,wt=13): 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. given #154 (H,wt=11): 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. given #155 (H,wt=11): 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. given #156 (H,wt=13): 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. given #157 (H,wt=13): 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. given #158 (H,wt=13): 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. given #159 (H,wt=13): 179 x ^ (y / (z * (z \ (x \ y)))) = x. [para(102(a,1),135(a,1,2,2))]. given #160 (H,wt=13): 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. given #161 (H,wt=13): 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. given #162 (H,wt=13): 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. given #163 (H,wt=13): 196 x V (y * (((y \ x) / z) * z)) = x. [para(108(a,1),70(a,1,2,2))]. given #164 (H,wt=13): 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. given #165 (H,wt=13): 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. given #166 (H,wt=13): 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. given #167 (H,wt=13): 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. given #168 (H,wt=13): 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. given #169 (H,wt=13): 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. given #170 (H,wt=13): 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. given #171 (H,wt=13): 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. given #172 (H,wt=13): 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. given #173 (H,wt=13): 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. given #174 (H,wt=13): 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. given #175 (H,wt=13): 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. given #176 (H,wt=13): 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. given #177 (H,wt=13): 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. given #178 (H,wt=15): 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. given #179 (H,wt=11): 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. given #180 (H,wt=11): 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. given #181 (H,wt=11): 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. given #182 (H,wt=15): 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. given #183 (H,wt=11): 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. given #184 (H,wt=9): 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. given #185 (H,wt=11): 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. given #186 (H,wt=11): 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. given #187 (H,wt=15): 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. given #188 (H,wt=13): 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. given #189 (H,wt=11): 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. given #190 (H,wt=11): 256 (1 @ x) ^ (y @ x) = y @ x. [para(255(a,1),68(a,1,1))]. given #191 (H,wt=11): 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. given #192 (H,wt=15): 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. given #193 (H,wt=15): 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. given #194 (H,wt=15): 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. given #195 (H,wt=11): 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. given #196 (H,wt=13): 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. given #197 (H,wt=13): 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. given #198 (H,wt=15): 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. given #199 (H,wt=13): 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. given #200 (H,wt=11): 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. given #201 (H,wt=13): 266 (x \ y) V ((x \ 1) * y) = x \ y. [para(183(a,1),66(a,1,2))]. given #202 (H,wt=13): 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. given #203 (H,wt=15): 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. given #204 (H,wt=13): 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. given #205 (H,wt=13): 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. given #206 (H,wt=15): 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. given #207 (H,wt=15): 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. given #208 (H,wt=13): 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. given #209 (H,wt=15): 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. given #210 (H,wt=15): 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. given #211 (H,wt=11): 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. given #212 (H,wt=11): 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. given #213 (H,wt=15): 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. given #214 (H,wt=13): 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. given #215 (H,wt=9): 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. given #216 (H,wt=7): 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. given #217 (H,wt=5): 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. given #218 (H,wt=5): 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. given #219 (H,wt=7): 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. given #220 (H,wt=9): 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. given #221 (H,wt=7): 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. given #222 (H,wt=7): 302 (1 @ x) ^ x = x. [para(301(a,1),20(a,1)),flip(a)]. given #223 (H,wt=9): 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. given #224 (H,wt=9): 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. given #225 (H,wt=9): 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. given #226 (H,wt=9): 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. given #227 (H,wt=9): 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. given #228 (H,wt=11): 280 x * (y * (z \ z)) = x * y. [para(278(a,2),76(a,1,2,2))]. given #229 (H,wt=11): 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. given #230 (H,wt=11): 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. given #231 (H,wt=9): 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. given #232 (H,wt=9): 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. given #233 (H,wt=9): 318 1 V (x \ (y ^ x)) = 1. [para(65(a,1),316(a,1,2,1))]. given #234 (H,wt=9): 322 (x \ (y ^ x)) V 1 = 1. [para(318(a,1),21(a,1)),flip(a)]. given #235 (H,wt=11): 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. given #236 (H,wt=7): 325 1 V (x @ 1) = 1. [para(299(a,1),182(a,1,2))]. given #237 (H,wt=9): 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. given #238 (H,wt=9): 326 1 ^ (x @ 1) = x @ 1. [para(325(a,1),68(a,1,1))]. given #239 (H,wt=9): 329 x V ((y @ 1) * x) = x. [para(324(a,1),180(a,1,2,1))]. given #240 (H,wt=9): 330 x ^ ((y @ 1) \ x) = x. [para(326(a,1),169(a,1,2,1))]. given #241 (H,wt=11): 305 x ^ (y ^ (1 @ x)) = x ^ y. [para(301(a,1),61(a,1,1)),flip(a)]. given #242 (H,wt=11): 307 x ^ ((y / x) \ (1 @ y)) = x. [para(301(a,1),204(a,1,2,1,1))]. given #243 (H,wt=11): 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. given #244 (H,wt=11): 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. given #245 (H,wt=11): 313 x ^ (1 @ ((y / x) \ y)) = x. [para(158(a,1),310(a,1,2,2))]. given #246 (H,wt=11): 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. given #247 (H,wt=11): 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. given #248 (H,wt=11): 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. given #249 (H,wt=11): 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. given #250 (H,wt=11): 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. given #251 (H,wt=7): 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. given #252 (H,wt=9): 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. given #253 (H,wt=9): 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. given #254 (H,wt=9): 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. given #255 (H,wt=9): 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. given #256 (H,wt=11): 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. given #257 (H,wt=11): 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. given #258 (H,wt=13): 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. given #259 (H,wt=11): 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. given #260 (H,wt=13): 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. given #261 (H,wt=11): 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. given #262 (H,wt=11): 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. given #263 (H,wt=9): 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. given #264 (H,wt=9): 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. given #265 (H,wt=13): 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. given #266 (H,wt=13): 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. given #267 (H,wt=13): 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. given #268 (H,wt=13): 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. given #269 (H,wt=13): 323 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. [para(93(a,1),318(a,1,2,2))]. given #270 (H,wt=13): 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. given #271 (H,wt=11): 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. given #272 (H,wt=13): 331 x ^ ((y @ 1) * x) = (y @ 1) * x. [para(329(a,1),68(a,1,1))]. given #273 (H,wt=13): 332 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. [para(305(a,1),126(a,1,2,1))]. given #274 (H,wt=13): 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. given #275 (H,wt=13): 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. given #276 (H,wt=13): 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. given #277 (H,wt=11): 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. given #278 (H,wt=11): 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. given #279 (H,wt=13): 354 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. [para(67(a,1),332(a,1,2,1))]. given #280 (H,wt=13): 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. given #281 (H,wt=13): 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. given #282 (H,wt=11): 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. given #283 (H,wt=15): 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. given #284 (H,wt=13): 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. given #285 (H,wt=11): 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. given #286 (H,wt=13): 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. given #287 (H,wt=15): 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. given #288 (H,wt=15): 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. given #289 (H,wt=13): 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. given #290 (H,wt=13): 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. given #291 (H,wt=13): 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. given #292 (H,wt=11): 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. given #293 (H,wt=13): 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. given #294 (H,wt=13): 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. given #295 (H,wt=15): 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. given #296 (H,wt=13): 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. given #297 (H,wt=13): 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. given #298 (H,wt=13): 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. given #299 (H,wt=11): 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. given #300 (H,wt=9): 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. given #301 (H,wt=9): 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. given #302 (H,wt=15): 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. given #303 (H,wt=15): 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. given #304 (H,wt=15): 359 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. [para(22(a,1),354(a,1,2,2,1))]. given #305 (H,wt=15): 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. given #306 (H,wt=9): 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. given #307 (H,wt=9): 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. given #308 (H,wt=11): 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. given #309 (H,wt=11): 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. given #310 (H,wt=11): 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. given #311 (H,wt=15): 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. given #312 (H,wt=15): 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. given #313 (H,wt=15): 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. given #314 (H,wt=9): 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. given #315 (H,wt=9): 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. given #316 (H,wt=11): 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. given #317 (H,wt=13): 393 x ^ ((y \ 1) \ (1 @ (y \ x))) = x. [para(390(a,1),307(a,1,2,1))]. given #318 (H,wt=17): 162 1 V ((x * (x \ y)) \ y) = (x * (x \ y)) \ y. [para(129(a,1),65(a,1,1))]. given #319 (H,wt=17): 163 (x * y) ^ ((z V (x * (y * u))) / u) = x * y. [para(28(a,1),134(a,1,2,1,2))]. given #320 (H,wt=17): 164 x V (y / (z ^ (x \ y))) = y / (z ^ (x \ y)). [para(137(a,1),65(a,1,1))]. given #321 (H,wt=17): 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. given #322 (H,wt=17): 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. given #323 (H,wt=17): 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. given #324 (H,wt=17): 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. given #325 (H,wt=11): 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. given #326 (H,wt=17): 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. given #327 (H,wt=17): 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. given #328 (H,wt=17): 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. given #329 (H,wt=11): 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. given #330 (H,wt=9): 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. given #331 (H,wt=7): 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. given #332 (H,wt=11): 402 ((x * (x \ 1)) \ 1) \ y = y. [para(398(a,2),281(a,1,1))]. -------- Proof 1 -------- "Goal 9". ============================== PROOF ================================= % Proof 1 at 1.20 (+ 0.04) seconds: "Goal 9". % Length of proof is 176. % Level of proof is 33. % Maximum clause weight is 17. % Given clauses 332. 9 1 @ (x * (x \ 1)) = 1 # label("Goal 9") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 48 1 @ (c25 * (c25 \ 1)) != 1 # label("Goal 9") # answer("Goal 9"). [deny(9)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 402 ((x * (x \ 1)) \ 1) \ y = y. [para(398(a,2),281(a,1,1))]. 407 1 @ (x * (x \ 1)) = 1. [para(402(a,1),175(a,1)),flip(a)]. 408 $F # answer("Goal 9"). [resolve(407,a,48,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 48 given #333 (H,wt=9): 407 1 @ (x * (x \ 1)) = 1. [para(402(a,1),175(a,1)),flip(a)]. given #334 (H,wt=11): 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. given #335 (H,wt=13): 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. given #336 (H,wt=13): 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. given #337 (H,wt=15): 403 x * (y * ((z * (z \ 1)) \ 1)) = x * y. [para(398(a,2),280(a,1,2,2))]. -------- Proof 2 -------- "Goal 8". ============================== PROOF ================================= % Proof 2 at 1.25 (+ 0.04) seconds: "Goal 8". % Length of proof is 179. % Level of proof is 33. % Maximum clause weight is 17. % Given clauses 337. 8 (x * (x \ 1)) @ 1 = x * (x \ 1) # label("Goal 8") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 47 (c24 * (c24 \ 1)) @ 1 != c24 * (c24 \ 1) # label("Goal 8") # answer("Goal 8"). [deny(8)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 76 x * (y * 1) = x * y. [para(30(a,1),28(a,1)),flip(a)]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 280 x * (y * (z \ z)) = x * y. [para(278(a,2),76(a,1,2,2))]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 403 x * (y * ((z * (z \ 1)) \ 1)) = x * y. [para(398(a,2),280(a,1,2,2))]. 411 (x * (x \ 1)) @ 1 = x * (x \ 1). [para(403(a,1),328(a,1)),flip(a)]. 412 $F # answer("Goal 8"). [resolve(411,a,47,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 47 given #338 (H,wt=13): 411 (x * (x \ 1)) @ 1 = x * (x \ 1). [para(403(a,1),328(a,1)),flip(a)]. given #339 (H,wt=15): 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. given #340 (H,wt=15): 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. given #341 (H,wt=19): 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. given #342 (H,wt=13): 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. given #343 (H,wt=11): 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. given #344 (H,wt=15): 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. given #345 (H,wt=15): 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. given #346 (H,wt=9): 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. given #347 (H,wt=9): 422 (x * y) \ x = 1 / y. [para(419(a,2),336(a,2)),flip(a)]. given #348 (H,wt=11): 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. given #349 (H,wt=11): 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. given #350 (H,wt=11): 425 (1 / x) / y = 1 / (y * x). [para(422(a,1),389(a,1,1))]. given #351 (H,wt=11): 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. given #352 (H,wt=9): 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. given #353 (H,wt=7): 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. given #354 (H,wt=9): 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. given #355 (H,wt=9): 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. given #356 (H,wt=11): 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. -------- Proof 3 -------- "Goal 11". ============================== PROOF ================================= % Proof 3 at 1.36 (+ 0.04) seconds: "Goal 11". % Length of proof is 201. % Level of proof is 38. % Maximum clause weight is 19. % Given clauses 356. 11 1 @ (x \ 1) = x \ 1 # label("Goal 11") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 50 c27 \ 1 != 1 @ (c27 \ 1) # label("Goal 11") # answer("Goal 11"). [deny(11)]. 51 1 @ (c27 \ 1) != c27 \ 1 # answer("Goal 11") # label("Goal 11"). [copy(50),flip(a)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 440 $F # answer("Goal 11"). [resolve(439,a,51,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 50x 51 given #357 (H,wt=9): 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. given #358 (H,wt=9): 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. given #359 (H,wt=11): 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. given #360 (H,wt=11): 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. given #361 (H,wt=13): 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. given #362 (H,wt=13): 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. given #363 (H,wt=13): 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. given #364 (H,wt=13): 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. given #365 (H,wt=13): 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. given #366 (H,wt=13): 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. given #367 (H,wt=9): 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. given #368 (H,wt=9): 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. given #369 (H,wt=9): 453 1 / (x @ y) = 1 / y. [para(447(a,2),336(a,2)),flip(a)]. given #370 (H,wt=11): 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. given #371 (H,wt=9): 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. given #372 (H,wt=9): 456 (x @ y) \ (1 @ y) = 1. [para(454(a,1),432(a,1,2))]. given #373 (H,wt=13): 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. given #374 (H,wt=13): 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. given #375 (H,wt=13): 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. given #376 (H,wt=13): 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. given #377 (H,wt=11): 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. given #378 (H,wt=13): 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. given #379 (H,wt=15): 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. given #380 (H,wt=9): 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. given #381 (H,wt=9): 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. given #382 (H,wt=11): 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. given #383 (H,wt=11): 466 1 / (x * y) = y \ (1 / x). [para(379(a,1),462(a,1,2)),flip(a)]. given #384 (H,wt=11): 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. given #385 (H,wt=11): 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. given #386 (H,wt=11): 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. given #387 (H,wt=11): 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. given #388 (H,wt=11): 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. given #389 (H,wt=11): 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. given #390 (H,wt=13): 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. given #391 (H,wt=13): 472 1 / ((x @ y) * z) = z \ (1 / y). [para(453(a,1),466(a,2,2))]. given #392 (H,wt=13): 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. given #393 (H,wt=13): 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. given #394 (H,wt=11): 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. given #395 (H,wt=11): 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. given #396 (H,wt=11): 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. given #397 (H,wt=11): 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. given #398 (H,wt=13): 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. given #399 (H,wt=11): 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. given #400 (H,wt=13): 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. given #401 (H,wt=13): 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. given #402 (H,wt=11): 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. given #403 (H,wt=13): 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. given #404 (H,wt=11): 493 x ^ (1 @ (y * (y \ x))) = x. [para(480(a,1),393(a,1,2))]. given #405 (H,wt=13): 481 ((x @ y) \ z) \ 1 = z \ (1 @ y). [para(454(a,1),476(a,2,2))]. given #406 (H,wt=9): 494 ((x @ y) \ y) \ 1 = 1. [para(481(a,2),432(a,1))]. given #407 (H,wt=13): 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. given #408 (H,wt=11): 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. given #409 (H,wt=11): 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. given #410 (H,wt=11): 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. given #411 (H,wt=11): 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. given #412 (H,wt=11): 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. given #413 (H,wt=13): 483 1 / ((x @ y) * z) = 1 / (y * z). [para(472(a,2),466(a,2)),flip(a)]. given #414 (H,wt=13): 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. given #415 (H,wt=13): 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. given #416 (H,wt=13): 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. given #417 (H,wt=11): 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. given #418 (H,wt=11): 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. given #419 (H,wt=9): 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. given #420 (H,wt=11): 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. given #421 (H,wt=9): 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. given #422 (H,wt=9): 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. given #423 (H,wt=9): 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. given #424 (H,wt=11): 515 1 V (x @ (y \ (z ^ y))) = 1. [para(512(a,2),322(a,1))]. given #425 (H,wt=13): 498 1 @ ((x @ y) / z) = (1 @ y) / z. [para(454(a,1),496(a,2,1))]. given #426 (H,wt=13): 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. given #427 (H,wt=13): 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. given #428 (H,wt=11): 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. given #429 (H,wt=13): 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. given #430 (H,wt=11): 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. given #431 (H,wt=13): 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. given #432 (H,wt=11): 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. given #433 (H,wt=9): 524 1 @ ((x @ y) \ y) = 1. [para(520(a,2),456(a,1))]. given #434 (H,wt=11): 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. given #435 (H,wt=11): 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. given #436 (H,wt=11): 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. given #437 (H,wt=13): 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. given #438 (H,wt=11): 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. given #439 (H,wt=13): 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. given #440 (H,wt=13): 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. given #441 (H,wt=13): 526 1 ^ ((x @ y) \ y) = (x @ y) \ y. [para(524(a,1),302(a,1,1))]. given #442 (H,wt=13): 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. given #443 (H,wt=13): 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. given #444 (H,wt=13): 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. given #445 (H,wt=11): 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. given #446 (H,wt=9): 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. given #447 (H,wt=15): 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. given #448 (H,wt=13): 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. given #449 (H,wt=13): 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. given #450 (H,wt=15): 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. -------- Proof 4 -------- "Goal 3". ============================== PROOF ================================= % Proof 4 at 1.96 (+ 0.05) seconds: "Goal 3". % Length of proof is 215. % Level of proof is 44. % Maximum clause weight is 19. % Given clauses 450. 3 x @ (y @ z) = x @ z # label("Goal 3") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 38 c5 @ (c6 @ c7) != c5 @ c7 # label("Goal 3") # answer("Goal 3"). [deny(3)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 540 $F # answer("Goal 3"). [resolve(539,a,38,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 38 given #451 (H,wt=9): 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. given #452 (H,wt=13): 541 x @ ((1 @ y) / z) = x @ (y / z). [para(496(a,1),539(a,1,2))]. given #453 (H,wt=13): 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. given #454 (H,wt=11): 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. given #455 (H,wt=13): 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. given #456 (H,wt=11): 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. given #457 (H,wt=13): 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. given #458 (H,wt=11): 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. given #459 (H,wt=13): 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. given #460 (H,wt=11): 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. given #461 (H,wt=13): 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. given #462 (H,wt=11): 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. given #463 (H,wt=13): 552 x @ (y / (1 / z)) = x @ (y * z). [para(422(a,1),551(a,1,2,2))]. given #464 (H,wt=13): 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. given #465 (H,wt=13): 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. given #466 (H,wt=15): 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. given #467 (H,wt=13): 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. given #468 (H,wt=13): 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. given #469 (H,wt=13): 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. given #470 (H,wt=13): 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. given #471 (H,wt=15): 495 ((x @ y) \ y) @ 1 = ((x @ y) \ y) * 1. [para(494(a,1),324(a,1,2)),flip(a)]. given #472 (H,wt=13): 574 ((x @ y) \ y) @ 1 = (x @ y) \ y. [para(495(a,2),30(a,1))]. given #473 (H,wt=15): 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. given #474 (H,wt=13): 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. given #475 (H,wt=13): 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. given #476 (H,wt=13): 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. given #477 (H,wt=13): 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. given #478 (H,wt=13): 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. given #479 (H,wt=11): 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. given #480 (H,wt=15): 544 x @ ((1 @ y) / z) = x @ ((u @ y) / z). [para(498(a,1),539(a,1,2))]. given #481 (H,wt=13): 583 x @ ((y @ z) / u) = x @ (z / u). [para(544(a,1),541(a,1))]. given #482 (H,wt=15): 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. given #483 (H,wt=13): 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. given #484 (H,wt=15): 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. given #485 (H,wt=13): 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. given #486 (H,wt=15): 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. given #487 (H,wt=13): 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. given #488 (H,wt=13): 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. given #489 (H,wt=13): 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. given #490 (H,wt=15): 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. given #491 (H,wt=13): 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. given #492 (H,wt=13): 595 1 V (x @ (y \ (z ^ (u @ y)))) = 1. [para(594(a,1),515(a,1,2))]. given #493 (H,wt=11): 596 (x \ (y ^ (z @ x))) V 1 = 1. [para(595(a,1),512(a,1)),flip(a)]. given #494 (H,wt=11): 597 1 V (x \ (y ^ (z @ x))) = 1. [para(595(a,1),514(a,1)),flip(a)]. given #495 (H,wt=11): 599 1 V ((x * (x \ y)) \ y) = 1. [para(493(a,1),597(a,1,2,2))]. given #496 (H,wt=9): 600 (x * (x \ y)) \ y = 1. [para(599(a,1),162(a,1)),flip(a)]. given #497 (H,wt=15): 588 ((x @ y) * z) V 1 = 1 V (u @ (y * z)). [para(587(a,1),512(a,1,2)),flip(a)]. given #498 (H,wt=13): 604 ((x @ y) * z) V 1 = 1 V (y * z). [para(588(a,2),514(a,1))]. given #499 (H,wt=13): 605 1 V ((x @ y) * z) = 1 V (y * z). [para(604(a,1),21(a,1)),flip(a)]. given #500 (H,wt=17): 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. given #501 (H,wt=11): 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. given #502 (H,wt=13): 607 x ^ ((y \ (z / x)) \ (y \ z)) = x. [para(606(a,1),181(a,1,2))]. given #503 (H,wt=13): 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. given #504 (H,wt=15): 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. given #505 (H,wt=17): 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. given #506 (H,wt=15): 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. given #507 (H,wt=15): 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. given #508 (H,wt=17): 529 1 / ((x @ (y \ z)) * u) = u \ (1 @ (z \ y)). [para(527(a,2),472(a,2,2))]. given #509 (H,wt=17): 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. given #510 (H,wt=17): 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. given #511 (H,wt=13): 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. given #512 (H,wt=13): 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. given #513 (H,wt=15): 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. given #514 (H,wt=13): 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. given #515 (H,wt=15): 617 x @ (y / (z \ u)) = x @ (y * (u \ z)). [para(615(a,1),566(a,1)),flip(a)]. given #516 (H,wt=17): 567 x @ (y * (y \ (z * (1 / u)))) = x @ (z / u). [para(561(a,1),560(a,2))]. given #517 (H,wt=17): 598 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). [para(596(a,1),23(a,1,1))]. given #518 (H,wt=13): 622 (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. [para(598(a,1),359(a,1))]. given #519 (H,wt=17): 613 1 / ((x @ (y \ z)) * u) = 1 @ (u \ (z \ y)). [para(529(a,2),520(a,2)),flip(a)]. given #520 (H,wt=15): 623 1 @ (x \ (y \ z)) = 1 / ((z \ y) * x). [para(613(a,1),483(a,1))]. given #521 (H,wt=19): 353 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. [para(331(a,1),323(a,1,2,2))]. given #522 (H,wt=19): 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. given #523 (H,wt=15): 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. given #524 (H,wt=13): 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. given #525 (H,wt=19): 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. given #526 (H,wt=19): 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. given #527 (H,wt=13): 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. given #528 (H,wt=13): 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. given #529 (H,wt=19): 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. given #530 (H,wt=19): 625 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. [para(21(a,1),353(a,1,2,1,2))]. given #531 (H,wt=19): 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. given #532 (H,wt=21): 171 (x * (x \ y)) / ((y V (x * (x \ y))) \ (x * (x \ y))) = y. [para(151(a,2),99(a,1))]. given #533 (H,wt=15): 633 (x * (x \ y)) / (y \ (x * (x \ y))) = y. [para(99(a,1),171(a,1,2,1))]. given #534 (H,wt=21): 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. given #535 (H,wt=15): 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. given #536 (H,wt=17): 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. given #537 (H,wt=21): 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. given #538 (H,wt=15): 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. given #539 (H,wt=13): 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. given #540 (H,wt=21): 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. given #541 (H,wt=15): 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. given #542 (H,wt=15): 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. given #543 (H,wt=13): 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. given #544 (H,wt=21): 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. given #545 (H,wt=15): 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. given #546 (H,wt=13): 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. given #547 (H,wt=21): 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. given #548 (H,wt=15): 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. given #549 (H,wt=13): 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. given #550 (H,wt=21): 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. given #551 (H,wt=15): 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. given #552 (H,wt=13): 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. given #553 (H,wt=11): 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. given #554 (H,wt=11): 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. given #555 (H,wt=9): 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. given #556 (H,wt=11): 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. given #557 (H,wt=11): 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. given #558 (H,wt=13): 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. given #559 (H,wt=21): 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. given #560 (H,wt=15): 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. given #561 (H,wt=13): 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. given #562 (H,wt=11): 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. given #563 (H,wt=11): 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. given #564 (H,wt=11): 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. given #565 (H,wt=11): 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. given #566 (H,wt=13): 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. given #567 (H,wt=11): 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. given #568 (H,wt=21): 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. given #569 (H,wt=21): 394 x / ((y / (z ^ (x \ y))) \ x) = x V (y / (z ^ (x \ y))). [para(164(a,1),32(a,1,2,1))]. given #570 (H,wt=19): 668 x / ((y / (z ^ (x \ y))) \ x) = y / (z ^ (x \ y)). [para(394(a,2),164(a,1))]. given #571 (H,wt=21): 624 x @ (y / (1 / ((z \ u) * w))) = x @ (y / (w \ (u \ z))). [para(623(a,1),591(a,1,2,2))]. given #572 (H,wt=19): 670 x @ (y / (z \ (u \ w))) = x @ (y * ((w \ u) * z)). [para(624(a,1),552(a,1))]. given #573 (H,wt=19): 671 x @ (y / ((z * u) \ w)) = x @ (y * ((w \ z) * u)). [para(606(a,2),670(a,1,2,2))]. given #574 (H,wt=19): 672 x @ (y * (z \ (u * w))) = x @ (y * ((z \ u) * w)). [para(671(a,1),617(a,1)),flip(a)]. given #575 (H,wt=17): 673 x @ (y * ((y \ z) * (1 / u))) = x @ (z / u). [para(672(a,1),567(a,1))]. given #576 (H,wt=15): 674 x @ ((y * (y \ z)) / u) = x @ (z / u). [para(673(a,1),564(a,1)),flip(a)]. given #577 (H,wt=17): 675 x @ ((y * (1 / z)) / u) = x @ ((y / z) / u). [para(462(a,1),674(a,1,2,1,2))]. given #578 (H,wt=21): 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. given #579 (H,wt=17): 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. given #580 (H,wt=13): 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. given #581 (H,wt=11): 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. given #582 (H,wt=11): 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. given #583 (H,wt=15): 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. given #584 (H,wt=13): 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. given #585 (H,wt=13): 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. given #586 (H,wt=27): 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. given #587 (H,wt=27): 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. given #588 (H,wt=27): 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. given #589 (H,wt=19): 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. given #590 (H,wt=11): 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. -------- Proof 5 -------- "Goal 10". ============================== PROOF ================================= % Proof 5 at 3.48 (+ 0.08) seconds: "Goal 10". % Length of proof is 270. % Level of proof is 46. % Maximum clause weight is 27. % Given clauses 590. 10 (x \ 1) @ 1 = 1 # label("Goal 10") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 49 (c26 \ 1) @ 1 != 1 # label("Goal 10") # answer("Goal 10"). [deny(10)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 693 (x \ 1) @ 1 = 1. [para(689(a,1),324(a,1)),flip(a)]. 694 $F # answer("Goal 10"). [resolve(693,a,49,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 49 given #591 (H,wt=7): 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. -------- Proof 6 -------- "Goal 1". ============================== PROOF ================================= % Proof 6 at 3.48 (+ 0.08) seconds: "Goal 1". % Length of proof is 268. % Level of proof is 47. % Maximum clause weight is 27. % Given clauses 591. 1 x @ x = x # label("Goal 1") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 36 c1 @ c1 != c1 # label("Goal 1") # answer("Goal 1"). [deny(1)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 698 $F # answer("Goal 1"). [resolve(697,a,36,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 36 given #592 (H,wt=5): 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. given #593 (H,wt=7): 693 (x \ 1) @ 1 = 1. [para(689(a,1),324(a,1)),flip(a)]. given #594 (H,wt=7): 699 x @ (y @ x) = x. [para(697(a,1),539(a,2))]. given #595 (H,wt=9): 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. given #596 (H,wt=9): 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. given #597 (H,wt=9): 711 (x \ (x / y)) @ 1 = 1. [para(463(a,2),693(a,1,1))]. given #598 (H,wt=9): 716 (x \ (y V x)) @ 1 = 1. [para(77(a,1),711(a,1,1,2))]. given #599 (H,wt=11): 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. given #600 (H,wt=9): 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. given #601 (H,wt=11): 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. given #602 (H,wt=11): 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. given #603 (H,wt=11): 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. given #604 (H,wt=11): 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. given #605 (H,wt=11): 723 x ^ ((y @ (y / x)) \ y) = x. [para(710(a,2),313(a,1,2))]. given #606 (H,wt=11): 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. given #607 (H,wt=11): 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. given #608 (H,wt=11): 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. given #609 (H,wt=9): 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. given #610 (H,wt=11): 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. given #611 (H,wt=11): 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. given #612 (H,wt=9): 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. given #613 (H,wt=11): 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. given #614 (H,wt=13): 700 ((x / y) \ x) @ y = (x / y) \ x. [para(697(a,1),556(a,1)),flip(a)]. given #615 (H,wt=13): 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. given #616 (H,wt=11): 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. given #617 (H,wt=9): 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. given #618 (H,wt=9): 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. given #619 (H,wt=11): 743 ((1 @ x) * y) / y = 1 @ x. [para(742(a,1),382(a,1,2))]. given #620 (H,wt=11): 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. given #621 (H,wt=13): 717 ((x V (y ^ z)) \ (x V y)) @ 1 = 1. [para(95(a,1),716(a,1,1,2))]. given #622 (H,wt=13): 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. given #623 (H,wt=13): 721 x \ (x @ (x V y)) = x \ (x V y). [para(709(a,2),645(a,1))]. given #624 (H,wt=13): 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. given #625 (H,wt=13): 729 1 ^ ((x @ (x / y)) \ (x / y)) = 1. [para(723(a,1),165(a,1,2,2,2))]. given #626 (H,wt=11): 749 (x @ (x / y)) \ (x / y) = 1. [para(729(a,1),526(a,1)),flip(a)]. given #627 (H,wt=13): 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. -------- Proof 7 -------- "Goal 2". ============================== PROOF ================================= % Proof 7 at 3.88 (+ 0.10) seconds: "Goal 2". % Length of proof is 301. % Level of proof is 52. % Maximum clause weight is 27. % Given clauses 627. 2 (x @ y) @ z = x @ z # label("Goal 2") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 37 (c2 @ c3) @ c4 != c2 @ c4 # label("Goal 2") # answer("Goal 2"). [deny(2)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 751 $F # answer("Goal 2"). [resolve(750,a,37,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 37 given #628 (H,wt=9): 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. given #629 (H,wt=7): 752 (x @ y) @ x = x. [para(750(a,2),697(a,1))]. given #630 (H,wt=13): 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. given #631 (H,wt=13): 738 x ^ (x @ (x ^ y)) = x @ (x ^ y). [para(736(a,1),68(a,1,1))]. given #632 (H,wt=13): 747 ((x V (y ^ z)) \ (x V z)) @ 1 = 1. [para(20(a,1),717(a,1,1,1,2))]. given #633 (H,wt=13): 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. given #634 (H,wt=15): 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. given #635 (H,wt=13): 756 1 V ((x \ 1) * y) = 1 V (x \ y). [para(703(a,1),514(a,1,2))]. given #636 (H,wt=13): 757 1 V ((1 / x) * y) = 1 V (x \ y). [para(336(a,2),756(a,1,2,1))]. given #637 (H,wt=15): 704 ((x @ y) / z) @ (y / z) = (x @ y) / z. [para(697(a,1),583(a,1)),flip(a)]. given #638 (H,wt=15): 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. given #639 (H,wt=15): 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. given #640 (H,wt=15): 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. given #641 (H,wt=15): 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. given #642 (H,wt=13): 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. given #643 (H,wt=11): 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. given #644 (H,wt=15): 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. given #645 (H,wt=15): 722 x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. [para(710(a,1),98(a,1,2,2,2))]. given #646 (H,wt=15): 728 (x @ (x / y)) ^ (x / y) = x @ (x / y). [para(723(a,1),137(a,1,2,2))]. given #647 (H,wt=13): 763 (x / y) V (x @ (x / y)) = x / y. [para(728(a,1),85(a,1,2))]. given #648 (H,wt=15): 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. given #649 (H,wt=15): 741 x V (((y / x) \ y) @ x) = (y / x) \ y. [para(700(a,2),157(a,1,2))]. given #650 (H,wt=15): 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. given #651 (H,wt=13): 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. given #652 (H,wt=13): 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. given #653 (H,wt=11): 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. given #654 (H,wt=13): 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. given #655 (H,wt=13): 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. given #656 (H,wt=13): 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. given #657 (H,wt=11): 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. given #658 (H,wt=11): 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. given #659 (H,wt=13): 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. given #660 (H,wt=13): 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. given #661 (H,wt=15): 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. given #662 (H,wt=13): 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. given #663 (H,wt=13): 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. given #664 (H,wt=13): 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. given #665 (H,wt=15): 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. given #666 (H,wt=17): 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. given #667 (H,wt=15): 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. given #668 (H,wt=13): 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. given #669 (H,wt=15): 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. given #670 (H,wt=13): 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. given #671 (H,wt=11): 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. given #672 (H,wt=11): 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. given #673 (H,wt=17): 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. given #674 (H,wt=15): 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. given #675 (H,wt=15): 793 x @ (y * ((y \ z) / u)) = x @ (z / u). [para(762(a,1),673(a,1))]. given #676 (H,wt=17): 764 ((x V (y @ (y / z))) \ (x V (y / z))) @ 1 = 1. [para(728(a,1),747(a,1,1,1,2))]. given #677 (H,wt=17): 794 x @ (y * ((1 / z) / u)) = x @ ((y / z) / u). [para(792(a,1),675(a,1))]. given #678 (H,wt=17): 796 x @ (y * (1 / (z * u))) = x @ ((y / u) / z). [para(425(a,1),794(a,1,2,2))]. given #679 (H,wt=15): 797 x @ ((y / z) / u) = x @ (y / (u * z)). [para(796(a,1),561(a,1))]. given #680 (H,wt=19): 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. given #681 (H,wt=17): 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. given #682 (H,wt=17): 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. given #683 (H,wt=11): 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. given #684 (H,wt=11): 806 (1 @ x) \ ((1 @ x) * y) = y. [para(800(a,2),742(a,1))]. given #685 (H,wt=13): 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. given #686 (H,wt=13): 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. given #687 (H,wt=11): 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. given #688 (H,wt=13): 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. given #689 (H,wt=11): 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. given #690 (H,wt=11): 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. given #691 (H,wt=11): 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. given #692 (H,wt=11): 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. given #693 (H,wt=11): 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. given #694 (H,wt=11): 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. given #695 (H,wt=11): 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. given #696 (H,wt=11): 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. given #697 (H,wt=11): 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. given #698 (H,wt=11): 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. given #699 (H,wt=11): 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. given #700 (H,wt=11): 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. given #701 (H,wt=13): 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. given #702 (H,wt=13): 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. given #703 (H,wt=13): 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. given #704 (H,wt=9): 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. given #705 (H,wt=13): 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. given #706 (H,wt=11): 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. given #707 (H,wt=13): 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. given #708 (H,wt=13): 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. given #709 (H,wt=13): 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. given #710 (H,wt=13): 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. given #711 (H,wt=15): 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. given #712 (H,wt=15): 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. given #713 (H,wt=15): 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. given #714 (H,wt=15): 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. given #715 (H,wt=9): 843 x V ((y ^ x) @ x) = x. [para(821(a,1),99(a,1,2))]. given #716 (H,wt=9): 844 x V ((x ^ y) @ x) = x. [para(20(a,1),843(a,1,2,1))]. given #717 (H,wt=15): 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. given #718 (H,wt=15): 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. given #719 (H,wt=13): 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. given #720 (H,wt=13): 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. given #721 (H,wt=13): 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. given #722 (H,wt=13): 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. given #723 (H,wt=15): 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. given #724 (H,wt=9): 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. given #725 (H,wt=9): 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. given #726 (H,wt=11): 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. given #727 (H,wt=15): 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. given #728 (H,wt=15): 842 (x @ 1) \ (x / y) = (1 @ (x / y)) * 1. [para(749(a,1),813(a,1,2)),flip(a)]. given #729 (H,wt=13): 856 (x @ 1) \ (x / y) = 1 @ (x / y). [para(842(a,2),30(a,1))]. given #730 (H,wt=17): 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. given #731 (H,wt=15): 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. given #732 (H,wt=17): 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. given #733 (H,wt=15): 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. given #734 (H,wt=17): 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. given #735 (H,wt=11): 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. given #736 (H,wt=15): 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. given #737 (H,wt=15): 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. given #738 (H,wt=13): 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. given #739 (H,wt=13): 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. given #740 (H,wt=11): 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. given #741 (H,wt=11): 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. given #742 (H,wt=11): 872 x V (y V (y @ x)) = y V x. [para(867(a,1),860(a,1,2))]. given #743 (H,wt=11): 875 x V (y V (x @ y)) = x V y. [para(872(a,1),94(a,1)),flip(a)]. given #744 (H,wt=13): 866 x V (y @ (x ^ z)) = x V (y @ x). [para(66(a,1),864(a,1,2,2)),flip(a)]. given #745 (H,wt=13): 868 x @ (x V (y @ z)) = x V (x @ z). [para(539(a,1),867(a,2,2))]. given #746 (H,wt=13): 869 x \ (x V (x @ y)) = x \ (x V y). [para(867(a,1),721(a,1,2))]. given #747 (H,wt=13): 871 x * (x \ (x V y)) = x V (x @ y). [para(867(a,1),820(a,2))]. given #748 (H,wt=11): 878 (x V (x @ y)) \ (x V y) = 1. [para(871(a,1),600(a,1,1))]. given #749 (H,wt=15): 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. given #750 (H,wt=13): 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. given #751 (H,wt=15): 876 x V ((y @ z) V (x @ z)) = x V (y @ z). [para(539(a,1),875(a,1,2,2))]. given #752 (H,wt=15): 877 x \ (x V (y @ z)) = x \ (x V (x @ z)). [para(868(a,1),721(a,1,2)),flip(a)]. given #753 (H,wt=13): 882 x \ (x V (y @ z)) = x \ (x V z). [para(877(a,2),869(a,1))]. given #754 (H,wt=15): 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. given #755 (H,wt=13): 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. given #756 (H,wt=13): 886 (x @ y) ^ (x @ (y V z)) = x @ y. [para(885(a,1),67(a,1,2))]. given #757 (H,wt=15): 888 (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). [para(65(a,1),886(a,1,2,2))]. given #758 (H,wt=15): 889 (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). [para(888(a,1),20(a,1)),flip(a)]. given #759 (H,wt=17): 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. given #760 (H,wt=17): 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. given #761 (H,wt=15): 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. given #762 (H,wt=13): 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. given #763 (H,wt=17): 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. given #764 (H,wt=19): 754 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. [para(738(a,1),622(a,1,1))]. given #765 (H,wt=17): 894 (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. [para(738(a,1),754(a,1,2,2,2))]. given #766 (H,wt=15): 895 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. [para(539(a,1),894(a,1,2,2))]. given #767 (H,wt=19): 758 ((x @ y) / (z @ u)) @ (y / u) = (x @ y) / (z @ u). [para(704(a,1),591(a,1)),flip(a)]. given #768 (H,wt=19): 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. given #769 (H,wt=19): 795 (x * ((x \ y) / z)) @ (y / z) = x * ((x \ y) / z). [para(793(a,1),697(a,1))]. given #770 (H,wt=21): 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. given #771 (H,wt=21): 884 (x V y) \ (x V (y V (z @ u))) = (x V y) \ ((x V y) V u). [para(17(a,1),882(a,1,2))]. given #772 (H,wt=21): 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. given #773 (H,wt=19): 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. given #774 (H,wt=13): 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. given #775 (H,wt=11): 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. given #776 (H,wt=9): 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. given #777 (H,wt=9): 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. given #778 (H,wt=11): 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. given #779 (H,wt=21): 899 (x V y) \ (x V (y V (z @ u))) = (x V y) \ (x V (y V u)). [para(17(a,1),884(a,2,2))]. given #780 (H,wt=21): 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. given #781 (H,wt=13): 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. given #782 (H,wt=21): 909 (x V y) \ (x V (y V y)) = (x V y) \ (x V ((z / y) \ z)). [para(741(a,1),899(a,1,2,2)),flip(a)]. given #783 (H,wt=19): 911 (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V y). [para(19(a,1),909(a,1,2,2)),flip(a)]. given #784 (H,wt=13): 912 (x V y) \ (x V ((z / y) \ z)) = 1. [para(911(a,2),278(a,1))]. given #785 (H,wt=29): 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. given #786 (H,wt=21): 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. given #787 (H,wt=21): 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. given #788 (H,wt=19): 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. given #789 (H,wt=15): 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. given #790 (H,wt=13): 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. given #791 (H,wt=29): 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. given #792 (H,wt=23): 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. given #793 (H,wt=21): 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. given #794 (H,wt=17): 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. given #795 (H,wt=15): 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. given #796 (H,wt=13): 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. given #797 (H,wt=15): 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. given #798 (H,wt=13): 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. given #799 (H,wt=29): 857 ((x @ 1) * (1 @ (x / y))) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. [para(856(a,1),633(a,1,1,2))]. given #800 (H,wt=25): 933 (x @ (x / y)) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. [para(351(a,1),857(a,1,1))]. given #801 (H,wt=23): 934 (x @ (x / y)) / ((x / y) \ ((x @ 1) * (1 @ (x / y)))) = x / y. [para(856(a,1),933(a,1,2,2,2))]. given #802 (H,wt=19): 935 (x @ (x / y)) / ((x / y) \ (x @ (x / y))) = x / y. [para(351(a,1),934(a,1,2,2))]. given #803 (H,wt=29): 897 ((x * ((x \ y) / z)) V (x * ((x \ y) / z))) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(795(a,1),878(a,1,1,2))]. given #804 (H,wt=21): 936 (x * ((x \ y) / z)) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(19(a,1),897(a,1,1))]. given #805 (H,wt=21): 937 (x * ((x \ y) / z)) \ ((y / z) V (x * ((x \ y) / z))) = 1. [para(21(a,1),936(a,1,2))]. given #806 (H,wt=21): 938 ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))) = 1. [para(937(a,1),606(a,1)),flip(a)]. given #807 (H,wt=31): 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. given #808 (H,wt=27): 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. given #809 (H,wt=21): 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. given #810 (H,wt=17): 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. given #811 (H,wt=9): 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. given #812 (H,wt=11): 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. given #813 (H,wt=11): 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. given #814 (H,wt=15): 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. given #815 (H,wt=9): 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. given #816 (H,wt=11): 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. given #817 (H,wt=9): 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. given #818 (H,wt=11): 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. given #819 (H,wt=9): 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. given #820 (H,wt=11): 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. given #821 (H,wt=19): 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. given #822 (H,wt=15): 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. given #823 (H,wt=13): 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. given #824 (H,wt=11): 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. given #825 (H,wt=11): 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. given #826 (H,wt=11): 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. given #827 (H,wt=25): 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. given #828 (H,wt=25): 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. given #829 (H,wt=25): 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. given #830 (H,wt=23): 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. given #831 (H,wt=15): 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. given #832 (H,wt=11): 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. given #833 (H,wt=11): 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. given #834 (H,wt=11): 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. given #835 (H,wt=9): 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. given #836 (H,wt=9): 972 (1 @ x) \ x = x @ 1. [para(970(a,1),806(a,1,2))]. given #837 (H,wt=13): 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. given #838 (H,wt=11): 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. given #839 (H,wt=13): 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. given #840 (H,wt=11): 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. given #841 (H,wt=11): 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. given #842 (H,wt=11): 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. given #843 (H,wt=11): 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. given #844 (H,wt=11): 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. given #845 (H,wt=11): 985 x @ (y \ x) = (1 / y) * x. [para(336(a,2),982(a,2,1))]. given #846 (H,wt=11): 986 x / (x @ (y \ x)) = 1 @ y. [para(982(a,2),382(a,1,2))]. given #847 (H,wt=11): 988 (x @ y) / x = 1 @ (y / x). [para(983(a,1),743(a,1,1))]. given #848 (H,wt=11): 990 x / (x @ y) = 1 @ (x / y). [para(556(a,1),986(a,1,2))]. given #849 (H,wt=11): 998 (x @ y) / x = y / (y @ x). [para(990(a,2),988(a,2))]. given #850 (H,wt=13): 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. given #851 (H,wt=13): 984 (x \ y) V (y @ (x \ y)) = x \ y. [para(982(a,2),266(a,1,2))]. given #852 (H,wt=15): 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. given #853 (H,wt=13): 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. given #854 (H,wt=11): 1001 (x @ y) \ y = (x \ y) @ 1. [para(999(a,1),574(a,1)),flip(a)]. given #855 (H,wt=13): 1002 (x \ y) @ (z \ y) = (x @ z) \ y. [para(999(a,1),707(a,1))]. given #856 (H,wt=11): 1006 (x @ 1) \ y = (x \ y) @ y. [para(119(a,1),1002(a,1,2)),flip(a)]. given #857 (H,wt=9): 1009 x ^ ((y \ x) @ x) = x. [para(1006(a,1),330(a,1,2))]. given #858 (H,wt=13): 1007 (x @ (y / z)) \ y = (x \ y) @ z. [para(1002(a,1),556(a,1))]. given #859 (H,wt=15): 993 ((x @ y) / x) \ (y / x) = (y / x) @ 1. [para(988(a,2),972(a,1,1))]. given #860 (H,wt=15): 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. given #861 (H,wt=15): 1010 (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. [para(539(a,1),1009(a,1,2))]. given #862 (H,wt=15): 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. given #863 (H,wt=13): 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. given #864 (H,wt=13): 1013 (x @ y) ^ ((z \ x) @ y) = x @ y. [para(1012(a,1),1010(a,1,2))]. given #865 (H,wt=13): 1014 (x @ y) ^ ((x V z) @ y) = x @ y. [para(34(a,1),1013(a,1,2,1))]. given #866 (H,wt=13): 1015 (x @ y) ^ ((z V x) @ y) = x @ y. [para(81(a,1),1013(a,1,2,1))]. given #867 (H,wt=15): 1017 (x @ y) V ((x V z) @ y) = (x V z) @ y. [para(1014(a,1),65(a,1,1))]. given #868 (H,wt=15): 1018 (x @ y) V ((z V x) @ y) = (z V x) @ y. [para(1015(a,1),65(a,1,1))]. given #869 (H,wt=17): 989 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). [para(985(a,1),605(a,1,2,1))]. given #870 (H,wt=17): 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. -------- Proof 8 -------- "Goal 6". ============================== PROOF ================================= % Proof 8 at 7.49 (+ 0.19) seconds: "Goal 6". % Length of proof is 652. % Level of proof is 108. % Maximum clause weight is 31. % Given clauses 870. 6 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 43 (c16 @ c18) \ (c17 @ c19) != (c16 \ c17) @ (c18 \ c19) # label("Goal 6") # answer("Goal 6"). [deny(6)]. 44 (c16 \ c17) @ (c18 \ c19) != (c16 @ c18) \ (c17 @ c19) # answer("Goal 6"). [copy(43),flip(a)]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. 1024 $F # answer("Goal 6"). [resolve(1023,a,44,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 43x 44 given #871 (H,wt=15): 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. given #872 (H,wt=13): 1025 (x \ y) @ (x \ z) = x \ (y @ z). [para(697(a,1),1023(a,2,1))]. given #873 (H,wt=11): 1027 x \ (y @ x) = (x \ y) @ 1. [para(278(a,1),1025(a,1,2)),flip(a)]. given #874 (H,wt=17): 1019 (x V y) V ((x V y) @ z) = (x V y) V (x @ z). [para(1017(a,1),876(a,1,2))]. given #875 (H,wt=15): 1031 ((x V y) V (x @ z)) @ u = (x V y) @ u. [para(1019(a,1),870(a,1,1))]. given #876 (H,wt=15): 1032 (x V (y V (x @ z))) @ u = (x V y) @ u. [para(17(a,1),1031(a,1,1))]. given #877 (H,wt=17): 1021 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). [para(28(a,1),989(a,1,2))]. given #878 (H,wt=15): 1034 1 V (x \ (y * z)) = 1 V ((x \ y) * z). [para(1021(a,1),757(a,1)),flip(a)]. given #879 (H,wt=17): 1028 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). [para(1025(a,1),605(a,1,2,1))]. given #880 (H,wt=17): 1029 (x @ (x / y)) / (((x / y) \ x) @ 1) = x / y. [para(1027(a,1),935(a,1,2))]. given #881 (H,wt=17): 1030 x V (y V ((x V y) @ z)) = (x V y) V (x @ z). [para(1019(a,1),17(a,1)),flip(a)]. given #882 (H,wt=17): 1036 x V (y V ((x V y) @ z)) = x V (y V (x @ z)). [para(1030(a,2),17(a,1))]. given #883 (H,wt=19): 994 (x @ y) V ((1 @ (x / y)) \ x) = (x / (x @ y)) \ x. [para(990(a,1),157(a,1,2,1))]. given #884 (H,wt=17): 1037 (x @ y) V ((1 \ x) @ y) = (x / (x @ y)) \ x. [para(1007(a,1),994(a,1,2))]. given #885 (H,wt=15): 1038 (x @ y) V (x @ y) = (x / (x @ y)) \ x. [para(119(a,1),1037(a,1,2,1))]. given #886 (H,wt=11): 1039 (x / (x @ y)) \ x = x @ y. [para(1038(a,1),19(a,1))]. given #887 (H,wt=11): 1040 ((x @ y) / x) \ y = y @ x. [para(998(a,2),1039(a,1,1))]. given #888 (H,wt=19): 1020 (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. [para(750(a,1),1018(a,1,1))]. given #889 (H,wt=19): 1035 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. [para(1034(a,1),625(a,1))]. given #890 (H,wt=17): 1042 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. [para(1035(a,1),1028(a,1)),flip(a)]. given #891 (H,wt=15): 1043 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. [para(1042(a,1),756(a,1)),flip(a)]. given #892 (H,wt=13): 1045 (x ^ (y V ((z @ 1) * x))) \ x = 1. [para(1043(a,1),130(a,1)),flip(a)]. given #893 (H,wt=19): 1046 ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(722(a,1),1045(a,1,1,2))]. given #894 (H,wt=17): 1047 ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(119(a,1),1046(a,1,1,1,2,2))]. given #895 (H,wt=17): 1048 (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. [para(20(a,1),1047(a,1,1))]. given #896 (H,wt=13): 1049 (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. [para(305(a,1),1048(a,1,1))]. given #897 (H,wt=11): 1050 (x ^ y) \ (y ^ (1 @ x)) = 1. [para(119(a,1),1049(a,1,2,2,2))]. given #898 (H,wt=13): 1051 x * ((y ^ z) \ (z ^ (1 @ y))) = x. [para(1050(a,2),30(a,1,2))]. given #899 (H,wt=13): 1052 x @ (y ^ (1 @ z)) = x @ (z ^ y). [para(1051(a,1),560(a,1,2)),flip(a)]. given #900 (H,wt=13): 1053 x @ (y ^ (1 @ z)) = x @ (y ^ z). [para(20(a,1),1052(a,2,2))]. given #901 (H,wt=15): 1054 x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). [para(539(a,1),1052(a,1,2,2)),flip(a)]. given #902 (H,wt=13): 1055 x @ ((y @ z) ^ u) = x @ (z ^ u). [para(1054(a,2),1052(a,1))]. given #903 (H,wt=13): 1056 x @ ((y @ z) ^ u) = x @ (u ^ z). [para(1054(a,2),1053(a,1))]. given #904 (H,wt=13): 1057 x @ (y ^ (z @ u)) = x @ (u ^ y). [para(20(a,1),1055(a,1,2))]. given #905 (H,wt=13): 1058 x @ (y ^ (z @ u)) = x @ (y ^ u). [para(20(a,1),1056(a,1,2))]. given #906 (H,wt=15): 1060 (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). [para(1057(a,1),697(a,1))]. given #907 (H,wt=15): 1061 (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). [para(1058(a,1),697(a,1))]. given #908 (H,wt=21): 1033 (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. [para(866(a,1),1032(a,2,1))]. given #909 (H,wt=21): 1062 (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. [para(1058(a,1),1001(a,1,1))]. given #910 (H,wt=21): 1063 (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). [para(1061(a,1),256(a,1,2))]. given #911 (H,wt=21): 1064 x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). [para(1063(a,1),60(a,2))]. given #912 (H,wt=17): 1065 (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). [para(889(a,1),1064(a,1,2)),flip(a)]. given #913 (H,wt=13): 1066 x ^ (1 @ (x ^ y)) = x ^ (1 @ y). [para(1065(a,1),1061(a,1))]. given #914 (H,wt=13): 1067 (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. [para(1066(a,1),895(a,1,2))]. given #915 (H,wt=11): 1068 (x \ (x ^ (1 @ y))) @ 1 = 1. [para(1067(a,1),1062(a,1)),flip(a)]. given #916 (H,wt=11): 1069 x \ ((x ^ (1 @ y)) @ x) = 1. [para(1068(a,1),1027(a,2))]. given #917 (H,wt=19): 1070 ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. [para(1069(a,1),99(a,1,2,2))]. given #918 (H,wt=17): 1071 ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. [para(30(a,1),1070(a,1,2))]. given #919 (H,wt=17): 1072 x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. [para(1071(a,1),21(a,1)),flip(a)]. given #920 (H,wt=9): 1073 (x ^ (1 @ y)) @ x = x. [para(1072(a,1),844(a,1))]. given #921 (H,wt=11): 1074 (x ^ (1 @ y)) @ z = x @ z. [para(1073(a,1),750(a,1,1)),flip(a)]. given #922 (H,wt=11): 1075 x ^ (1 @ y) = x @ (y ^ x). [para(1074(a,1),1060(a,1)),flip(a)]. given #923 (H,wt=11): 1076 x ^ (1 @ y) = x @ (x ^ y). [para(1074(a,1),1061(a,1)),flip(a)]. given #924 (H,wt=15): 1077 (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). [para(1075(a,2),750(a,1))]. given #925 (H,wt=13): 1078 (x @ y) ^ (1 @ z) = x @ (y ^ z). [para(1077(a,2),1057(a,1))]. given #926 (H,wt=13): 1079 (x @ y) ^ (1 @ z) = x @ (z ^ y). [para(1077(a,2),1058(a,1))]. given #927 (H,wt=13): 1080 x ^ (y @ (z ^ x)) = x ^ (y @ z). [para(1078(a,1),305(a,1,2))]. given #928 (H,wt=13): 1082 x ^ (y @ (x ^ z)) = x ^ (y @ z). [para(1079(a,1),305(a,1,2))]. given #929 (H,wt=11): 1084 x @ (x ^ y) = x ^ (x @ y). [para(1082(a,1),738(a,1)),flip(a)]. given #930 (H,wt=11): 1086 x ^ (1 @ y) = x ^ (x @ y). [para(1084(a,1),1076(a,2))]. given #931 (H,wt=11): 1087 x ^ (y ^ (y @ x)) = x ^ y. [para(1086(a,1),305(a,1,2))]. given #932 (H,wt=11): 1088 x ^ ((x @ y) ^ y) = y ^ x. [para(1087(a,1),60(a,2))]. given #933 (H,wt=11): 1089 x ^ (y ^ (x @ y)) = y ^ x. [para(20(a,1),1088(a,1,2))]. given #934 (H,wt=11): 1090 x ^ (y ^ (x @ y)) = x ^ y. [para(1089(a,2),20(a,1))]. given #935 (H,wt=11): 1091 (x ^ y) ^ (x @ y) = x ^ y. [para(1090(a,1),16(a,2))]. given #936 (H,wt=11): 1092 (x ^ y) V (x @ y) = x @ y. [para(1091(a,1),65(a,1,1))]. given #937 (H,wt=11): 1093 (x ^ y) V (y @ x) = y @ x. [para(20(a,1),1092(a,1,1))]. given #938 (H,wt=15): 1083 x ^ ((y @ x) ^ (1 @ z)) = x ^ (y @ z). [para(1079(a,2),1080(a,1,2))]. given #939 (H,wt=15): 1095 (x ^ (y @ x)) ^ (1 @ z) = x ^ (y @ z). [para(1083(a,1),16(a,2))]. given #940 (H,wt=15): 1097 (x ^ (y @ z)) @ u = (x ^ (y @ x)) @ u. [para(1095(a,1),1074(a,1,1))]. given #941 (H,wt=13): 1098 (x ^ (y @ x)) @ z = (x ^ y) @ z. [para(697(a,1),1097(a,1,1,2)),flip(a)]. given #942 (H,wt=15): 1099 (x ^ (y @ x)) @ (x ^ z) = x ^ (y @ z). [para(1097(a,1),1061(a,1))]. given #943 (H,wt=13): 1101 (x ^ y) @ (x ^ z) = x ^ (y @ z). [para(1099(a,1),1098(a,1)),flip(a)]. given #944 (H,wt=13): 1102 (x ^ (y @ z)) @ u = (x ^ y) @ u. [para(1101(a,1),750(a,1,1))]. given #945 (H,wt=13): 1104 (x ^ (y @ z)) @ u = (y ^ x) @ u. [para(20(a,1),1102(a,2,1))]. given #946 (H,wt=13): 1105 ((x @ y) ^ z) @ u = (x ^ z) @ u. [para(20(a,1),1104(a,1,1))]. given #947 (H,wt=19): 1106 ((x @ y) ^ z) @ ((u @ w) ^ v5) = (x ^ z) @ (w ^ v5). [para(1105(a,1),1055(a,2))]. -------- Proof 9 -------- "Goal 4". ============================== PROOF ================================= % Proof 9 at 8.70 (+ 0.20) seconds: "Goal 4". % Length of proof is 800. % Level of proof is 150. % Maximum clause weight is 31. % Given clauses 947. 4 (x ^ y) @ (z ^ u) = (x @ z) ^ (y @ u) # label("Goal 4") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 39 (c8 @ c10) ^ (c9 @ c11) != (c8 ^ c9) @ (c10 ^ c11) # label("Goal 4") # answer("Goal 4"). [deny(4)]. 40 (c8 ^ c9) @ (c10 ^ c11) != (c8 @ c10) ^ (c9 @ c11) # answer("Goal 4") # label("Goal 4"). [copy(39),flip(a)]. 60 x ^ (y ^ z) = z ^ (x ^ y). [para(20(a,1),16(a,1)),flip(a)]. 61 (x ^ y) ^ z = x ^ (z ^ y). [para(20(a,1),16(a,2,2))]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 93 x ^ (y ^ (x V z)) = y ^ x. [para(23(a,1),60(a,1,2)),flip(a)]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 256 (1 @ x) ^ (y @ x) = y @ x. [para(255(a,1),68(a,1,1))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 305 x ^ (y ^ (1 @ x)) = x ^ y. [para(301(a,1),61(a,1,1)),flip(a)]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 318 1 V (x \ (y ^ x)) = 1. [para(65(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 322 (x \ (y ^ x)) V 1 = 1. [para(318(a,1),21(a,1)),flip(a)]. 323 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. [para(93(a,1),318(a,1,2,2))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 329 x V ((y @ 1) * x) = x. [para(324(a,1),180(a,1,2,1))]. 331 x ^ ((y @ 1) * x) = (y @ 1) * x. [para(329(a,1),68(a,1,1))]. 332 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. [para(305(a,1),126(a,1,2,1))]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 353 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. [para(331(a,1),323(a,1,2,2))]. 354 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. [para(67(a,1),332(a,1,2,1))]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 359 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. [para(22(a,1),354(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 481 ((x @ y) \ z) \ 1 = z \ (1 @ y). [para(454(a,1),476(a,2,2))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 494 ((x @ y) \ y) \ 1 = 1. [para(481(a,2),432(a,1))]. 495 ((x @ y) \ y) @ 1 = ((x @ y) \ y) * 1. [para(494(a,1),324(a,1,2)),flip(a)]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 515 1 V (x @ (y \ (z ^ y))) = 1. [para(512(a,2),322(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 574 ((x @ y) \ y) @ 1 = (x @ y) \ y. [para(495(a,2),30(a,1))]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 588 ((x @ y) * z) V 1 = 1 V (u @ (y * z)). [para(587(a,1),512(a,1,2)),flip(a)]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 595 1 V (x @ (y \ (z ^ (u @ y)))) = 1. [para(594(a,1),515(a,1,2))]. 596 (x \ (y ^ (z @ x))) V 1 = 1. [para(595(a,1),512(a,1)),flip(a)]. 598 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). [para(596(a,1),23(a,1,1))]. 604 ((x @ y) * z) V 1 = 1 V (y * z). [para(588(a,2),514(a,1))]. 605 1 V ((x @ y) * z) = 1 V (y * z). [para(604(a,1),21(a,1)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 622 (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. [para(598(a,1),359(a,1))]. 625 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. [para(21(a,1),353(a,1,2,1,2))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 722 x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. [para(710(a,1),98(a,1,2,2,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. 738 x ^ (x @ (x ^ y)) = x @ (x ^ y). [para(736(a,1),68(a,1,1))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. 754 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. [para(738(a,1),622(a,1,1))]. 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. 756 1 V ((x \ 1) * y) = 1 V (x \ y). [para(703(a,1),514(a,1,2))]. 757 1 V ((1 / x) * y) = 1 V (x \ y). [para(336(a,2),756(a,1,2,1))]. 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 843 x V ((y ^ x) @ x) = x. [para(821(a,1),99(a,1,2))]. 844 x V ((x ^ y) @ x) = x. [para(20(a,1),843(a,1,2,1))]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. 886 (x @ y) ^ (x @ (y V z)) = x @ y. [para(885(a,1),67(a,1,2))]. 888 (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). [para(65(a,1),886(a,1,2,2))]. 889 (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). [para(888(a,1),20(a,1)),flip(a)]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 894 (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. [para(738(a,1),754(a,1,2,2,2))]. 895 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. [para(539(a,1),894(a,1,2,2))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 985 x @ (y \ x) = (1 / y) * x. [para(336(a,2),982(a,2,1))]. 989 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). [para(985(a,1),605(a,1,2,1))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1001 (x @ y) \ y = (x \ y) @ 1. [para(999(a,1),574(a,1)),flip(a)]. 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1021 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). [para(28(a,1),989(a,1,2))]. 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. 1025 (x \ y) @ (x \ z) = x \ (y @ z). [para(697(a,1),1023(a,2,1))]. 1027 x \ (y @ x) = (x \ y) @ 1. [para(278(a,1),1025(a,1,2)),flip(a)]. 1028 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). [para(1025(a,1),605(a,1,2,1))]. 1034 1 V (x \ (y * z)) = 1 V ((x \ y) * z). [para(1021(a,1),757(a,1)),flip(a)]. 1035 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. [para(1034(a,1),625(a,1))]. 1042 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. [para(1035(a,1),1028(a,1)),flip(a)]. 1043 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. [para(1042(a,1),756(a,1)),flip(a)]. 1045 (x ^ (y V ((z @ 1) * x))) \ x = 1. [para(1043(a,1),130(a,1)),flip(a)]. 1046 ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(722(a,1),1045(a,1,1,2))]. 1047 ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(119(a,1),1046(a,1,1,1,2,2))]. 1048 (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. [para(20(a,1),1047(a,1,1))]. 1049 (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. [para(305(a,1),1048(a,1,1))]. 1050 (x ^ y) \ (y ^ (1 @ x)) = 1. [para(119(a,1),1049(a,1,2,2,2))]. 1051 x * ((y ^ z) \ (z ^ (1 @ y))) = x. [para(1050(a,2),30(a,1,2))]. 1052 x @ (y ^ (1 @ z)) = x @ (z ^ y). [para(1051(a,1),560(a,1,2)),flip(a)]. 1053 x @ (y ^ (1 @ z)) = x @ (y ^ z). [para(20(a,1),1052(a,2,2))]. 1054 x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). [para(539(a,1),1052(a,1,2,2)),flip(a)]. 1055 x @ ((y @ z) ^ u) = x @ (z ^ u). [para(1054(a,2),1052(a,1))]. 1056 x @ ((y @ z) ^ u) = x @ (u ^ z). [para(1054(a,2),1053(a,1))]. 1057 x @ (y ^ (z @ u)) = x @ (u ^ y). [para(20(a,1),1055(a,1,2))]. 1058 x @ (y ^ (z @ u)) = x @ (y ^ u). [para(20(a,1),1056(a,1,2))]. 1060 (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). [para(1057(a,1),697(a,1))]. 1061 (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). [para(1058(a,1),697(a,1))]. 1062 (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. [para(1058(a,1),1001(a,1,1))]. 1063 (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). [para(1061(a,1),256(a,1,2))]. 1064 x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). [para(1063(a,1),60(a,2))]. 1065 (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). [para(889(a,1),1064(a,1,2)),flip(a)]. 1066 x ^ (1 @ (x ^ y)) = x ^ (1 @ y). [para(1065(a,1),1061(a,1))]. 1067 (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. [para(1066(a,1),895(a,1,2))]. 1068 (x \ (x ^ (1 @ y))) @ 1 = 1. [para(1067(a,1),1062(a,1)),flip(a)]. 1069 x \ ((x ^ (1 @ y)) @ x) = 1. [para(1068(a,1),1027(a,2))]. 1070 ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. [para(1069(a,1),99(a,1,2,2))]. 1071 ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. [para(30(a,1),1070(a,1,2))]. 1072 x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. [para(1071(a,1),21(a,1)),flip(a)]. 1073 (x ^ (1 @ y)) @ x = x. [para(1072(a,1),844(a,1))]. 1074 (x ^ (1 @ y)) @ z = x @ z. [para(1073(a,1),750(a,1,1)),flip(a)]. 1075 x ^ (1 @ y) = x @ (y ^ x). [para(1074(a,1),1060(a,1)),flip(a)]. 1077 (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). [para(1075(a,2),750(a,1))]. 1078 (x @ y) ^ (1 @ z) = x @ (y ^ z). [para(1077(a,2),1057(a,1))]. 1079 (x @ y) ^ (1 @ z) = x @ (z ^ y). [para(1077(a,2),1058(a,1))]. 1080 x ^ (y @ (z ^ x)) = x ^ (y @ z). [para(1078(a,1),305(a,1,2))]. 1083 x ^ ((y @ x) ^ (1 @ z)) = x ^ (y @ z). [para(1079(a,2),1080(a,1,2))]. 1095 (x ^ (y @ x)) ^ (1 @ z) = x ^ (y @ z). [para(1083(a,1),16(a,2))]. 1097 (x ^ (y @ z)) @ u = (x ^ (y @ x)) @ u. [para(1095(a,1),1074(a,1,1))]. 1098 (x ^ (y @ x)) @ z = (x ^ y) @ z. [para(697(a,1),1097(a,1,1,2)),flip(a)]. 1099 (x ^ (y @ x)) @ (x ^ z) = x ^ (y @ z). [para(1097(a,1),1061(a,1))]. 1101 (x ^ y) @ (x ^ z) = x ^ (y @ z). [para(1099(a,1),1098(a,1)),flip(a)]. 1102 (x ^ (y @ z)) @ u = (x ^ y) @ u. [para(1101(a,1),750(a,1,1))]. 1104 (x ^ (y @ z)) @ u = (y ^ x) @ u. [para(20(a,1),1102(a,2,1))]. 1105 ((x @ y) ^ z) @ u = (x ^ z) @ u. [para(20(a,1),1104(a,1,1))]. 1106 ((x @ y) ^ z) @ ((u @ w) ^ v5) = (x ^ z) @ (w ^ v5). [para(1105(a,1),1055(a,2))]. 1107 (x ^ y) @ (z ^ u) = (x @ z) ^ (y @ u) # label("Goal 4"). [para(1106(a,1),1101(a,1))]. 1108 $F # answer("Goal 4"). [resolve(1107,a,40,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 39x 40 given #948 (H,wt=15): 1107 (x ^ y) @ (z ^ u) = (x @ z) ^ (y @ u) # label("Goal 4"). [para(1106(a,1),1101(a,1))]. given #949 (H,wt=23): 1041 (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). [para(1020(a,1),876(a,1,2))]. given #950 (H,wt=23): 1109 x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). [para(1041(a,1),17(a,1)),flip(a)]. given #951 (H,wt=19): 1110 (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). [para(1109(a,1),1036(a,1))]. given #952 (H,wt=19): 1111 x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). [para(1110(a,1),17(a,1)),flip(a)]. given #953 (H,wt=17): 1112 x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). [para(885(a,1),1111(a,1,2)),flip(a)]. given #954 (H,wt=19): 1113 (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. [para(1112(a,1),1033(a,1,1))]. given #955 (H,wt=15): 1114 (x V (y @ z)) @ u = (x V (y @ x)) @ u. [para(65(a,1),1113(a,1,1,2,2))]. given #956 (H,wt=13): 1115 (x V (y @ x)) @ z = (x V y) @ z. [para(697(a,1),1114(a,1,1,2)),flip(a)]. given #957 (H,wt=13): 1119 ((x / y) V x) @ z = (x / y) @ z. [para(763(a,1),1115(a,1,1)),flip(a)]. given #958 (H,wt=13): 1120 (x V y) @ (x V x) = x V (y @ x). [para(1115(a,1),785(a,1))]. given #959 (H,wt=11): 1122 (x V y) @ x = x V (y @ x). [para(19(a,1),1120(a,1,2))]. given #960 (H,wt=11): 1123 (x V y) @ y = y V (x @ y). [para(21(a,1),1122(a,1,1))]. given #961 (H,wt=13): 1121 (x V (x / y)) @ z = (x / y) @ z. [para(21(a,1),1119(a,1,1))]. given #962 (H,wt=13): 1124 (x V (y \ x)) @ (y \ x) = y \ x. [para(1123(a,2),984(a,1))]. given #963 (H,wt=13): 1125 (x V (y \ x)) @ z = (y \ x) @ z. [para(1124(a,1),750(a,1,1)),flip(a)]. given #964 (H,wt=15): 1118 (x V (y @ z)) @ u = (x V (y @ w)) @ u. [para(1114(a,2),1114(a,2))]. given #965 (H,wt=13): 1128 (x V (y @ z)) @ u = (y V x) @ u. [para(860(a,1),1118(a,1,1)),flip(a)]. given #966 (H,wt=13): 1133 ((x @ y) V z) @ u = (x V z) @ u. [para(21(a,1),1128(a,1,1))]. given #967 (H,wt=15): 1127 (x V (y @ z)) @ (x V u) = x V (y @ u). [para(1118(a,1),785(a,1))]. given #968 (H,wt=13): 1135 (x V y) @ (x V z) = x V (y @ z). [para(697(a,1),1127(a,1,1,2))]. given #969 (H,wt=13): 1136 (x V y) @ (z V x) = x V (y @ z). [para(21(a,1),1135(a,1,2))]. given #970 (H,wt=15): 1138 (x \ y) @ (y V z) = y V ((x \ y) @ z). [para(1135(a,1),1125(a,1)),flip(a)]. given #971 (H,wt=17): 1137 ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). [para(1135(a,1),784(a,1)),flip(a)]. -------- Proof 10 -------- "Goal 5". ============================== PROOF ================================= % Proof 10 at 9.12 (+ 0.20) seconds: "Goal 5". % Length of proof is 738. % Level of proof is 124. % Maximum clause weight is 31. % Given clauses 971. 5 (x V y) @ (z V u) = (x @ z) V (y @ u) # label("Goal 5") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 41 (c12 @ c14) V (c13 @ c15) != (c12 V c13) @ (c14 V c15) # label("Goal 5") # answer("Goal 5"). [deny(5)]. 42 (c12 V c13) @ (c14 V c15) != (c12 @ c14) V (c13 @ c15) # answer("Goal 5") # label("Goal 5"). [copy(41),flip(a)]. 62 (x V y) V z = y V (x V z). [para(21(a,1),17(a,1,1))]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 94 x V (y V z) = y V (x V z). [para(62(a,1),17(a,1))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 169 x ^ ((1 ^ y) \ x) = x. [para(20(a,1),167(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 325 1 V (x @ 1) = 1. [para(299(a,1),182(a,1,2))]. 326 1 ^ (x @ 1) = x @ 1. [para(325(a,1),68(a,1,1))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 330 x ^ ((y @ 1) \ x) = x. [para(326(a,1),169(a,1,2,1))]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. 866 x V (y @ (x ^ z)) = x V (y @ x). [para(66(a,1),864(a,1,2,2)),flip(a)]. 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. 872 x V (y V (y @ x)) = y V x. [para(867(a,1),860(a,1,2))]. 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. 875 x V (y V (x @ y)) = x V y. [para(872(a,1),94(a,1)),flip(a)]. 876 x V ((y @ z) V (x @ z)) = x V (y @ z). [para(539(a,1),875(a,1,2,2))]. 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1002 (x \ y) @ (z \ y) = (x @ z) \ y. [para(999(a,1),707(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1006 (x @ 1) \ y = (x \ y) @ y. [para(119(a,1),1002(a,1,2)),flip(a)]. 1009 x ^ ((y \ x) @ x) = x. [para(1006(a,1),330(a,1,2))]. 1010 (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. [para(539(a,1),1009(a,1,2))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1013 (x @ y) ^ ((z \ x) @ y) = x @ y. [para(1012(a,1),1010(a,1,2))]. 1014 (x @ y) ^ ((x V z) @ y) = x @ y. [para(34(a,1),1013(a,1,2,1))]. 1015 (x @ y) ^ ((z V x) @ y) = x @ y. [para(81(a,1),1013(a,1,2,1))]. 1017 (x @ y) V ((x V z) @ y) = (x V z) @ y. [para(1014(a,1),65(a,1,1))]. 1018 (x @ y) V ((z V x) @ y) = (z V x) @ y. [para(1015(a,1),65(a,1,1))]. 1019 (x V y) V ((x V y) @ z) = (x V y) V (x @ z). [para(1017(a,1),876(a,1,2))]. 1020 (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. [para(750(a,1),1018(a,1,1))]. 1030 x V (y V ((x V y) @ z)) = (x V y) V (x @ z). [para(1019(a,1),17(a,1)),flip(a)]. 1031 ((x V y) V (x @ z)) @ u = (x V y) @ u. [para(1019(a,1),870(a,1,1))]. 1032 (x V (y V (x @ z))) @ u = (x V y) @ u. [para(17(a,1),1031(a,1,1))]. 1033 (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. [para(866(a,1),1032(a,2,1))]. 1036 x V (y V ((x V y) @ z)) = x V (y V (x @ z)). [para(1030(a,2),17(a,1))]. 1041 (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). [para(1020(a,1),876(a,1,2))]. 1109 x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). [para(1041(a,1),17(a,1)),flip(a)]. 1110 (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). [para(1109(a,1),1036(a,1))]. 1111 x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). [para(1110(a,1),17(a,1)),flip(a)]. 1112 x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). [para(885(a,1),1111(a,1,2)),flip(a)]. 1113 (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. [para(1112(a,1),1033(a,1,1))]. 1114 (x V (y @ z)) @ u = (x V (y @ x)) @ u. [para(65(a,1),1113(a,1,1,2,2))]. 1118 (x V (y @ z)) @ u = (x V (y @ w)) @ u. [para(1114(a,2),1114(a,2))]. 1127 (x V (y @ z)) @ (x V u) = x V (y @ u). [para(1118(a,1),785(a,1))]. 1128 (x V (y @ z)) @ u = (y V x) @ u. [para(860(a,1),1118(a,1,1)),flip(a)]. 1133 ((x @ y) V z) @ u = (x V z) @ u. [para(21(a,1),1128(a,1,1))]. 1135 (x V y) @ (x V z) = x V (y @ z). [para(697(a,1),1127(a,1,1,2))]. 1137 ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). [para(1135(a,1),784(a,1)),flip(a)]. 1140 (x V y) @ (z V u) = (x @ z) V (y @ u). [para(1137(a,1),1133(a,1)),flip(a)]. 1141 $F # answer("Goal 5"). [resolve(1140,a,42,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 41x 42 given #972 (H,wt=15): 1140 (x V y) @ (z V u) = (x @ z) V (y @ u). [para(1137(a,1),1133(a,1)),flip(a)]. given #973 (H,wt=13): 1142 (x @ y) V (z @ y) = (x V z) @ y. [para(19(a,1),1140(a,1,2)),flip(a)]. given #974 (H,wt=13): 1144 (x @ y) V (z @ x) = x V (z @ y). [para(1140(a,1),1136(a,1))]. given #975 (H,wt=13): 1146 (x @ y) V (y @ z) = y V (x @ z). [para(1144(a,1),21(a,1)),flip(a)]. given #976 (H,wt=19): 1134 ((x V (y @ z)) \ u) @ w = (((y V x) @ v5) \ u) @ w. [para(1128(a,1),999(a,1,1,1)),flip(a)]. given #977 (H,wt=17): 1147 ((x V (y @ z)) \ u) @ w = ((y V x) \ u) @ w. [para(697(a,1),1134(a,2,1,1))]. given #978 (H,wt=13): 1148 ((x V y) \ (y V (x / z))) @ 1 = 1. [para(1147(a,1),764(a,1))]. given #979 (H,wt=13): 1149 ((x V y) \ (x / (y \ x))) @ 1 = 1. [para(150(a,1),1148(a,1,1,2))]. given #980 (H,wt=17): 1151 ((x V y) @ (x / (y \ x))) \ (x / (y \ x)) = 1. [para(1149(a,1),1001(a,2))]. given #981 (H,wt=13): 1152 ((x V y) @ y) \ (x / (y \ x)) = 1. [para(551(a,1),1151(a,1,1))]. given #982 (H,wt=13): 1154 (x V (y @ x)) \ (y / (x \ y)) = 1. [para(1123(a,1),1152(a,1,1))]. given #983 (H,wt=17): 1155 (x \ y) ^ (((x V (y @ x)) * 1) \ y) = x \ y. [para(1154(a,1),181(a,1,2,1,2))]. given #984 (H,wt=15): 1156 (x \ y) ^ ((x V (y @ x)) \ y) = x \ y. [para(30(a,1),1155(a,1,2,1))]. given #985 (H,wt=11): 1157 (x V (y @ x)) \ y = x \ y. [para(1156(a,1),188(a,1)),flip(a)]. given #986 (H,wt=13): 1158 ((x V y) \ x) @ z = (y \ x) @ z. [para(1157(a,1),1147(a,1,1)),flip(a)]. given #987 (H,wt=21): 1139 (x V y) \ (x V ((z \ x) @ y)) = ((x V y) \ (z \ x)) @ 1. [para(1138(a,1),1027(a,1,2))]. given #988 (H,wt=21): 1145 (x @ y) / (((x V z) @ y) \ (x @ y)) = (x @ y) V (z @ y). [para(1142(a,1),32(a,1,2,1))]. given #989 (H,wt=21): 1160 (x @ y) / (((x V z) \ x) @ (y \ y)) = (x @ y) V (z @ y). [para(1023(a,2),1145(a,1,2))]. given #990 (H,wt=19): 1161 (x @ y) / (((x V z) \ x) @ (y \ y)) = (x V z) @ y. [para(1160(a,2),1142(a,1))]. given #991 (H,wt=17): 1162 (x @ y) / (((x V z) \ x) @ 1) = (x V z) @ y. [para(278(a,1),1161(a,1,2,2))]. given #992 (H,wt=15): 1164 (x @ y) / ((z \ x) @ 1) = (x V z) @ y. [para(1158(a,1),1162(a,1,2))]. given #993 (H,wt=17): 1165 (x @ y) / (((x / z) \ x) @ 1) = (x / z) @ y. [para(1164(a,2),1121(a,1))]. given #994 (H,wt=23): 1159 (x V y) \ (x V ((x / y) \ x)) = ((x V y) \ ((x / y) \ x)) @ 1. [para(700(a,1),1139(a,1,2,2))]. given #995 (H,wt=13): 1166 ((x V y) \ ((x / y) \ x)) @ 1 = 1. [para(1159(a,1),912(a,1))]. given #996 (H,wt=17): 1168 ((x V y) @ ((x / y) \ x)) \ ((x / y) \ x) = 1. [para(1166(a,1),1001(a,2))]. given #997 (H,wt=13): 1169 ((x V y) @ y) \ ((x / y) \ x) = 1. [para(556(a,1),1168(a,1,1))]. given #998 (H,wt=13): 1171 (x V (y @ x)) \ ((y / x) \ y) = 1. [para(1123(a,1),1169(a,1,1))]. given #999 (H,wt=17): 1172 (x / y) ^ (x / ((y V (x @ y)) * 1)) = x / y. [para(1171(a,1),179(a,1,2,2,2))]. given #1000 (H,wt=15): 1173 (x / y) ^ (x / (y V (x @ y))) = x / y. [para(30(a,1),1172(a,1,2,2))]. given #1001 (H,wt=11): 1174 x / (y V (x @ y)) = x / y. [para(1173(a,1),235(a,1)),flip(a)]. given #1002 (H,wt=15): 1175 x / ((y @ z) V (x @ z)) = x / (y @ z). [para(539(a,1),1174(a,1,2,2))]. given #1003 (H,wt=13): 1176 x / ((y V x) @ z) = x / (y @ z). [para(1142(a,1),1175(a,1,2))]. given #1004 (H,wt=13): 1177 x / (x V (y @ x)) = x / (y @ x). [para(1146(a,1),1175(a,1,2))]. given #1005 (H,wt=13): 1179 (x / (y @ x)) \ x = x V (y @ x). [para(1177(a,1),34(a,1,1))]. given #1006 (H,wt=17): 1178 (x @ y) / ((x V z) @ u) = (x @ y) / (z @ u). [para(1128(a,1),1176(a,1,2))]. given #1007 (H,wt=21): 1180 ((x @ y) / (y @ (x @ y))) \ (x @ y) = y V (x @ (x @ y)). [para(1179(a,2),1146(a,1))]. given #1008 (H,wt=17): 1181 ((x @ y) / y) \ (x @ y) = y V (x @ (x @ y)). [para(699(a,1),1180(a,1,1,2))]. given #1009 (H,wt=31): 1094 (x * y) V (((z V (x * (y * u))) / u) @ (x * y)) = ((z V (x * (y * u))) / u) @ (x * y). [para(163(a,1),1093(a,1,1))]. given #1010 (H,wt=33): 669 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / (z ^ ((x \ (y / z)) \ (x \ y))). [para(607(a,1),668(a,1,2,1,2))]. given #1011 (H,wt=23): 1183 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / z. [para(607(a,1),669(a,2,2))]. given #1012 (H,wt=41): 1182 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (y * ((y \ x) / z))). [para(196(a,1),1094(a,1,2,1,1)),flip(a)]. given #1013 (H,wt=37): 1184 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (x / z)). [para(793(a,1),1182(a,2,2))]. given #1014 (H,wt=33): 1185 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V (x / z). [para(697(a,1),1184(a,2,2))]. given #1015 (H,wt=33): 1187 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (x / z) V (y * ((y \ x) / z)). [para(1185(a,2),21(a,1))]. given #1016 (H,wt=23): 1188 (x / y) @ (z * ((z \ x) / y)) = (x / y) V (z * ((z \ x) / y)). [para(196(a,1),1187(a,1,1,1))]. given #1017 (H,wt=19): 1189 (x / y) V (z * ((z \ x) / y)) = (x / y) @ (x / y). [para(1188(a,1),793(a,1))]. given #1018 (H,wt=15): 1190 (x / y) V (z * ((z \ x) / y)) = x / y. [para(1189(a,2),697(a,1))]. given #1019 (H,wt=13): 1192 ((x \ y) / z) \ (x \ (y / z)) = 1. [para(1190(a,1),938(a,1,2,2))]. given #1020 (H,wt=13): 1193 (x \ (y / z)) / 1 = (x \ y) / z. [para(1192(a,1),1183(a,1,2))]. given #1021 (H,wt=11): 1195 (x \ y) / z = x \ (y / z). [para(1193(a,1),146(a,1))]. given #1022 (H,wt=15): 1197 (((x @ y) / x) \ y) / x = (y / x) @ 1. [para(1195(a,2),993(a,1))]. given #1023 (H,wt=11): 1199 (x @ y) / y = (x / y) @ 1. [para(1040(a,1),1197(a,1,1))]. given #1024 (H,wt=19): 1198 (((x / y) / ((x / y) @ z)) \ x) / y = (x / y) @ z. [para(1195(a,2),1039(a,1))]. given #1025 (H,wt=17): 1201 ((1 @ ((x / y) / z)) \ x) / y = (x / y) @ z. [para(990(a,1),1198(a,1,1,1))]. given #1026 (H,wt=17): 1202 ((1 @ (x / (y * z))) \ x) / z = (x / z) @ y. [para(797(a,1),1201(a,1,1,1))]. given #1027 (H,wt=15): 1203 ((1 \ x) @ (y * z)) / z = (x / z) @ y. [para(1007(a,1),1202(a,1,1))]. given #1028 (H,wt=13): 1204 (x @ (y * z)) / z = (x / z) @ y. [para(119(a,1),1203(a,1,1,1))]. given #1029 (H,wt=13): 1205 (x / y) @ (z / y) = (x @ z) / y. [para(582(a,1),1204(a,1,1)),flip(a)]. given #1030 (H,wt=13): 1206 ((x @ y) / z) @ u = (x / z) @ u. [para(1205(a,1),750(a,1,1))]. given #1031 (H,wt=21): 1200 x V (((y / x) @ 1) \ (y @ x)) = ((y @ x) / x) \ (y @ x). [para(1199(a,1),157(a,1,2,1))]. given #1032 (H,wt=21): 1208 x V (((y / x) \ y) @ (1 \ x)) = ((y @ x) / x) \ (y @ x). [para(1023(a,2),1200(a,1,2))]. given #1033 (H,wt=19): 1209 ((x @ y) / y) \ (x @ y) = y V (((x / y) \ x) @ y). [para(119(a,1),1208(a,1,2,2)),flip(a)]. given #1034 (H,wt=15): 1210 ((x @ y) / y) \ (x @ y) = (x / y) \ x. [para(1209(a,2),741(a,1))]. given #1035 (H,wt=13): 1211 x V (y @ (y @ x)) = (y / x) \ y. [para(1210(a,1),1181(a,1)),flip(a)]. given #1036 (H,wt=11): 1212 x V (y @ x) = (y / x) \ y. [para(539(a,1),1211(a,1,2))]. given #1037 (H,wt=11): 1216 (x / y) \ x = (x V y) @ y. [para(1212(a,1),1123(a,2)),flip(a)]. -------- Proof 11 -------- "Goal 15". ============================== PROOF ================================= % Proof 11 at 10.36 (+ 0.23) seconds: "Goal 15". % Length of proof is 957. % Level of proof is 170. % Maximum clause weight is 41. % Given clauses 1037. 15 (x V y) @ ((x / y) \ x) = (x / y) \ x # label("Goal 15") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 58 (c34 / c35) \ c34 != (c34 V c35) @ ((c34 / c35) \ c34) # label("Goal 15") # answer("Goal 15"). [deny(15)]. 59 (c34 V c35) @ ((c34 / c35) \ c34) != (c34 / c35) \ c34 # answer("Goal 15"). [copy(58),flip(a)]. 60 x ^ (y ^ z) = z ^ (x ^ y). [para(20(a,1),16(a,1)),flip(a)]. 61 (x ^ y) ^ z = x ^ (z ^ y). [para(20(a,1),16(a,2,2))]. 62 (x V y) V z = y V (x V z). [para(21(a,1),17(a,1,1))]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 93 x ^ (y ^ (x V z)) = y ^ x. [para(23(a,1),60(a,1,2)),flip(a)]. 94 x V (y V z) = y V (x V z). [para(62(a,1),17(a,1))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 123 1 ^ (x \ (y V x)) = 1. [para(21(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 129 1 ^ ((x * (x \ y)) \ y) = 1. [para(99(a,1),123(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 157 x V ((y / x) \ y) = (y / x) \ y. [para(155(a,1),65(a,1,1))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 162 1 V ((x * (x \ y)) \ y) = (x * (x \ y)) \ y. [para(129(a,1),65(a,1,1))]. 163 (x * y) ^ ((z V (x * (y * u))) / u) = x * y. [para(28(a,1),134(a,1,2,1,2))]. 164 x V (y / (z ^ (x \ y))) = y / (z ^ (x \ y)). [para(137(a,1),65(a,1,1))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 169 x ^ ((1 ^ y) \ x) = x. [para(20(a,1),167(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 179 x ^ (y / (z * (z \ (x \ y)))) = x. [para(102(a,1),135(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 196 x V (y * (((y \ x) / z) * z)) = x. [para(108(a,1),70(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 201 x V (((y ^ x) / z) * z) = x. [para(20(a,1),194(a,1,2,1,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 204 x ^ (((y ^ z) / x) \ z) = x. [para(201(a,1),111(a,1,2,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 256 (1 @ x) ^ (y @ x) = y @ x. [para(255(a,1),68(a,1,1))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 266 (x \ y) V ((x \ 1) * y) = x \ y. [para(183(a,1),66(a,1,2))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 305 x ^ (y ^ (1 @ x)) = x ^ y. [para(301(a,1),61(a,1,1)),flip(a)]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 307 x ^ ((y / x) \ (1 @ y)) = x. [para(301(a,1),204(a,1,2,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 318 1 V (x \ (y ^ x)) = 1. [para(65(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 322 (x \ (y ^ x)) V 1 = 1. [para(318(a,1),21(a,1)),flip(a)]. 323 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. [para(93(a,1),318(a,1,2,2))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 325 1 V (x @ 1) = 1. [para(299(a,1),182(a,1,2))]. 326 1 ^ (x @ 1) = x @ 1. [para(325(a,1),68(a,1,1))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 329 x V ((y @ 1) * x) = x. [para(324(a,1),180(a,1,2,1))]. 330 x ^ ((y @ 1) \ x) = x. [para(326(a,1),169(a,1,2,1))]. 331 x ^ ((y @ 1) * x) = (y @ 1) * x. [para(329(a,1),68(a,1,1))]. 332 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. [para(305(a,1),126(a,1,2,1))]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 353 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. [para(331(a,1),323(a,1,2,2))]. 354 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. [para(67(a,1),332(a,1,2,1))]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 359 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. [para(22(a,1),354(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 393 x ^ ((y \ 1) \ (1 @ (y \ x))) = x. [para(390(a,1),307(a,1,2,1))]. 394 x / ((y / (z ^ (x \ y))) \ x) = x V (y / (z ^ (x \ y))). [para(164(a,1),32(a,1,2,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 422 (x * y) \ x = 1 / y. [para(419(a,2),336(a,2)),flip(a)]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 453 1 / (x @ y) = 1 / y. [para(447(a,2),336(a,2)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 466 1 / (x * y) = y \ (1 / x). [para(379(a,1),462(a,1,2)),flip(a)]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 472 1 / ((x @ y) * z) = z \ (1 / y). [para(453(a,1),466(a,2,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 481 ((x @ y) \ z) \ 1 = z \ (1 @ y). [para(454(a,1),476(a,2,2))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 483 1 / ((x @ y) * z) = 1 / (y * z). [para(472(a,2),466(a,2)),flip(a)]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 493 x ^ (1 @ (y * (y \ x))) = x. [para(480(a,1),393(a,1,2))]. 494 ((x @ y) \ y) \ 1 = 1. [para(481(a,2),432(a,1))]. 495 ((x @ y) \ y) @ 1 = ((x @ y) \ y) * 1. [para(494(a,1),324(a,1,2)),flip(a)]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 515 1 V (x @ (y \ (z ^ y))) = 1. [para(512(a,2),322(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 529 1 / ((x @ (y \ z)) * u) = u \ (1 @ (z \ y)). [para(527(a,2),472(a,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 552 x @ (y / (1 / z)) = x @ (y * z). [para(422(a,1),551(a,1,2,2))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 567 x @ (y * (y \ (z * (1 / u)))) = x @ (z / u). [para(561(a,1),560(a,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 574 ((x @ y) \ y) @ 1 = (x @ y) \ y. [para(495(a,2),30(a,1))]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 588 ((x @ y) * z) V 1 = 1 V (u @ (y * z)). [para(587(a,1),512(a,1,2)),flip(a)]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 595 1 V (x @ (y \ (z ^ (u @ y)))) = 1. [para(594(a,1),515(a,1,2))]. 596 (x \ (y ^ (z @ x))) V 1 = 1. [para(595(a,1),512(a,1)),flip(a)]. 597 1 V (x \ (y ^ (z @ x))) = 1. [para(595(a,1),514(a,1)),flip(a)]. 598 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). [para(596(a,1),23(a,1,1))]. 599 1 V ((x * (x \ y)) \ y) = 1. [para(493(a,1),597(a,1,2,2))]. 600 (x * (x \ y)) \ y = 1. [para(599(a,1),162(a,1)),flip(a)]. 604 ((x @ y) * z) V 1 = 1 V (y * z). [para(588(a,2),514(a,1))]. 605 1 V ((x @ y) * z) = 1 V (y * z). [para(604(a,1),21(a,1)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 607 x ^ ((y \ (z / x)) \ (y \ z)) = x. [para(606(a,1),181(a,1,2))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 613 1 / ((x @ (y \ z)) * u) = 1 @ (u \ (z \ y)). [para(529(a,2),520(a,2)),flip(a)]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 617 x @ (y / (z \ u)) = x @ (y * (u \ z)). [para(615(a,1),566(a,1)),flip(a)]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 622 (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. [para(598(a,1),359(a,1))]. 623 1 @ (x \ (y \ z)) = 1 / ((z \ y) * x). [para(613(a,1),483(a,1))]. 624 x @ (y / (1 / ((z \ u) * w))) = x @ (y / (w \ (u \ z))). [para(623(a,1),591(a,1,2,2))]. 625 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. [para(21(a,1),353(a,1,2,1,2))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 668 x / ((y / (z ^ (x \ y))) \ x) = y / (z ^ (x \ y)). [para(394(a,2),164(a,1))]. 669 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / (z ^ ((x \ (y / z)) \ (x \ y))). [para(607(a,1),668(a,1,2,1,2))]. 670 x @ (y / (z \ (u \ w))) = x @ (y * ((w \ u) * z)). [para(624(a,1),552(a,1))]. 671 x @ (y / ((z * u) \ w)) = x @ (y * ((w \ z) * u)). [para(606(a,2),670(a,1,2,2))]. 672 x @ (y * (z \ (u * w))) = x @ (y * ((z \ u) * w)). [para(671(a,1),617(a,1)),flip(a)]. 673 x @ (y * ((y \ z) * (1 / u))) = x @ (z / u). [para(672(a,1),567(a,1))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 699 x @ (y @ x) = x. [para(697(a,1),539(a,2))]. 700 ((x / y) \ x) @ y = (x / y) \ x. [para(697(a,1),556(a,1)),flip(a)]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 721 x \ (x @ (x V y)) = x \ (x V y). [para(709(a,2),645(a,1))]. 722 x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. [para(710(a,1),98(a,1,2,2,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. 738 x ^ (x @ (x ^ y)) = x @ (x ^ y). [para(736(a,1),68(a,1,1))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 741 x V (((y / x) \ y) @ x) = (y / x) \ y. [para(700(a,2),157(a,1,2))]. 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. 743 ((1 @ x) * y) / y = 1 @ x. [para(742(a,1),382(a,1,2))]. 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. 754 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. [para(738(a,1),622(a,1,1))]. 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. 756 1 V ((x \ 1) * y) = 1 V (x \ y). [para(703(a,1),514(a,1,2))]. 757 1 V ((1 / x) * y) = 1 V (x \ y). [para(336(a,2),756(a,1,2,1))]. 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 793 x @ (y * ((y \ z) / u)) = x @ (z / u). [para(762(a,1),673(a,1))]. 795 (x * ((x \ y) / z)) @ (y / z) = x * ((x \ y) / z). [para(793(a,1),697(a,1))]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 806 (1 @ x) \ ((1 @ x) * y) = y. [para(800(a,2),742(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 843 x V ((y ^ x) @ x) = x. [para(821(a,1),99(a,1,2))]. 844 x V ((x ^ y) @ x) = x. [para(20(a,1),843(a,1,2,1))]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. 866 x V (y @ (x ^ z)) = x V (y @ x). [para(66(a,1),864(a,1,2,2)),flip(a)]. 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. 868 x @ (x V (y @ z)) = x V (x @ z). [para(539(a,1),867(a,2,2))]. 869 x \ (x V (x @ y)) = x \ (x V y). [para(867(a,1),721(a,1,2))]. 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. 871 x * (x \ (x V y)) = x V (x @ y). [para(867(a,1),820(a,2))]. 872 x V (y V (y @ x)) = y V x. [para(867(a,1),860(a,1,2))]. 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. 875 x V (y V (x @ y)) = x V y. [para(872(a,1),94(a,1)),flip(a)]. 876 x V ((y @ z) V (x @ z)) = x V (y @ z). [para(539(a,1),875(a,1,2,2))]. 877 x \ (x V (y @ z)) = x \ (x V (x @ z)). [para(868(a,1),721(a,1,2)),flip(a)]. 878 (x V (x @ y)) \ (x V y) = 1. [para(871(a,1),600(a,1,1))]. 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. 882 x \ (x V (y @ z)) = x \ (x V z). [para(877(a,2),869(a,1))]. 884 (x V y) \ (x V (y V (z @ u))) = (x V y) \ ((x V y) V u). [para(17(a,1),882(a,1,2))]. 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. 886 (x @ y) ^ (x @ (y V z)) = x @ y. [para(885(a,1),67(a,1,2))]. 888 (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). [para(65(a,1),886(a,1,2,2))]. 889 (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). [para(888(a,1),20(a,1)),flip(a)]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 894 (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. [para(738(a,1),754(a,1,2,2,2))]. 895 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. [para(539(a,1),894(a,1,2,2))]. 897 ((x * ((x \ y) / z)) V (x * ((x \ y) / z))) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(795(a,1),878(a,1,1,2))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 899 (x V y) \ (x V (y V (z @ u))) = (x V y) \ (x V (y V u)). [para(17(a,1),884(a,2,2))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 909 (x V y) \ (x V (y V y)) = (x V y) \ (x V ((z / y) \ z)). [para(741(a,1),899(a,1,2,2)),flip(a)]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 911 (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V y). [para(19(a,1),909(a,1,2,2)),flip(a)]. 912 (x V y) \ (x V ((z / y) \ z)) = 1. [para(911(a,2),278(a,1))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 936 (x * ((x \ y) / z)) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(19(a,1),897(a,1,1))]. 937 (x * ((x \ y) / z)) \ ((y / z) V (x * ((x \ y) / z))) = 1. [para(21(a,1),936(a,1,2))]. 938 ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))) = 1. [para(937(a,1),606(a,1)),flip(a)]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 972 (1 @ x) \ x = x @ 1. [para(970(a,1),806(a,1,2))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 984 (x \ y) V (y @ (x \ y)) = x \ y. [para(982(a,2),266(a,1,2))]. 985 x @ (y \ x) = (1 / y) * x. [para(336(a,2),982(a,2,1))]. 986 x / (x @ (y \ x)) = 1 @ y. [para(982(a,2),382(a,1,2))]. 988 (x @ y) / x = 1 @ (y / x). [para(983(a,1),743(a,1,1))]. 989 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). [para(985(a,1),605(a,1,2,1))]. 990 x / (x @ y) = 1 @ (x / y). [para(556(a,1),986(a,1,2))]. 993 ((x @ y) / x) \ (y / x) = (y / x) @ 1. [para(988(a,2),972(a,1,1))]. 994 (x @ y) V ((1 @ (x / y)) \ x) = (x / (x @ y)) \ x. [para(990(a,1),157(a,1,2,1))]. 998 (x @ y) / x = y / (y @ x). [para(990(a,2),988(a,2))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1001 (x @ y) \ y = (x \ y) @ 1. [para(999(a,1),574(a,1)),flip(a)]. 1002 (x \ y) @ (z \ y) = (x @ z) \ y. [para(999(a,1),707(a,1))]. 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1006 (x @ 1) \ y = (x \ y) @ y. [para(119(a,1),1002(a,1,2)),flip(a)]. 1007 (x @ (y / z)) \ y = (x \ y) @ z. [para(1002(a,1),556(a,1))]. 1009 x ^ ((y \ x) @ x) = x. [para(1006(a,1),330(a,1,2))]. 1010 (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. [para(539(a,1),1009(a,1,2))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1013 (x @ y) ^ ((z \ x) @ y) = x @ y. [para(1012(a,1),1010(a,1,2))]. 1014 (x @ y) ^ ((x V z) @ y) = x @ y. [para(34(a,1),1013(a,1,2,1))]. 1015 (x @ y) ^ ((z V x) @ y) = x @ y. [para(81(a,1),1013(a,1,2,1))]. 1017 (x @ y) V ((x V z) @ y) = (x V z) @ y. [para(1014(a,1),65(a,1,1))]. 1018 (x @ y) V ((z V x) @ y) = (z V x) @ y. [para(1015(a,1),65(a,1,1))]. 1019 (x V y) V ((x V y) @ z) = (x V y) V (x @ z). [para(1017(a,1),876(a,1,2))]. 1020 (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. [para(750(a,1),1018(a,1,1))]. 1021 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). [para(28(a,1),989(a,1,2))]. 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. 1025 (x \ y) @ (x \ z) = x \ (y @ z). [para(697(a,1),1023(a,2,1))]. 1027 x \ (y @ x) = (x \ y) @ 1. [para(278(a,1),1025(a,1,2)),flip(a)]. 1028 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). [para(1025(a,1),605(a,1,2,1))]. 1030 x V (y V ((x V y) @ z)) = (x V y) V (x @ z). [para(1019(a,1),17(a,1)),flip(a)]. 1031 ((x V y) V (x @ z)) @ u = (x V y) @ u. [para(1019(a,1),870(a,1,1))]. 1032 (x V (y V (x @ z))) @ u = (x V y) @ u. [para(17(a,1),1031(a,1,1))]. 1033 (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. [para(866(a,1),1032(a,2,1))]. 1034 1 V (x \ (y * z)) = 1 V ((x \ y) * z). [para(1021(a,1),757(a,1)),flip(a)]. 1035 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. [para(1034(a,1),625(a,1))]. 1036 x V (y V ((x V y) @ z)) = x V (y V (x @ z)). [para(1030(a,2),17(a,1))]. 1037 (x @ y) V ((1 \ x) @ y) = (x / (x @ y)) \ x. [para(1007(a,1),994(a,1,2))]. 1038 (x @ y) V (x @ y) = (x / (x @ y)) \ x. [para(119(a,1),1037(a,1,2,1))]. 1039 (x / (x @ y)) \ x = x @ y. [para(1038(a,1),19(a,1))]. 1040 ((x @ y) / x) \ y = y @ x. [para(998(a,2),1039(a,1,1))]. 1041 (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). [para(1020(a,1),876(a,1,2))]. 1042 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. [para(1035(a,1),1028(a,1)),flip(a)]. 1043 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. [para(1042(a,1),756(a,1)),flip(a)]. 1045 (x ^ (y V ((z @ 1) * x))) \ x = 1. [para(1043(a,1),130(a,1)),flip(a)]. 1046 ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(722(a,1),1045(a,1,1,2))]. 1047 ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(119(a,1),1046(a,1,1,1,2,2))]. 1048 (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. [para(20(a,1),1047(a,1,1))]. 1049 (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. [para(305(a,1),1048(a,1,1))]. 1050 (x ^ y) \ (y ^ (1 @ x)) = 1. [para(119(a,1),1049(a,1,2,2,2))]. 1051 x * ((y ^ z) \ (z ^ (1 @ y))) = x. [para(1050(a,2),30(a,1,2))]. 1052 x @ (y ^ (1 @ z)) = x @ (z ^ y). [para(1051(a,1),560(a,1,2)),flip(a)]. 1053 x @ (y ^ (1 @ z)) = x @ (y ^ z). [para(20(a,1),1052(a,2,2))]. 1054 x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). [para(539(a,1),1052(a,1,2,2)),flip(a)]. 1055 x @ ((y @ z) ^ u) = x @ (z ^ u). [para(1054(a,2),1052(a,1))]. 1056 x @ ((y @ z) ^ u) = x @ (u ^ z). [para(1054(a,2),1053(a,1))]. 1057 x @ (y ^ (z @ u)) = x @ (u ^ y). [para(20(a,1),1055(a,1,2))]. 1058 x @ (y ^ (z @ u)) = x @ (y ^ u). [para(20(a,1),1056(a,1,2))]. 1060 (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). [para(1057(a,1),697(a,1))]. 1061 (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). [para(1058(a,1),697(a,1))]. 1062 (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. [para(1058(a,1),1001(a,1,1))]. 1063 (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). [para(1061(a,1),256(a,1,2))]. 1064 x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). [para(1063(a,1),60(a,2))]. 1065 (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). [para(889(a,1),1064(a,1,2)),flip(a)]. 1066 x ^ (1 @ (x ^ y)) = x ^ (1 @ y). [para(1065(a,1),1061(a,1))]. 1067 (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. [para(1066(a,1),895(a,1,2))]. 1068 (x \ (x ^ (1 @ y))) @ 1 = 1. [para(1067(a,1),1062(a,1)),flip(a)]. 1069 x \ ((x ^ (1 @ y)) @ x) = 1. [para(1068(a,1),1027(a,2))]. 1070 ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. [para(1069(a,1),99(a,1,2,2))]. 1071 ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. [para(30(a,1),1070(a,1,2))]. 1072 x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. [para(1071(a,1),21(a,1)),flip(a)]. 1073 (x ^ (1 @ y)) @ x = x. [para(1072(a,1),844(a,1))]. 1074 (x ^ (1 @ y)) @ z = x @ z. [para(1073(a,1),750(a,1,1)),flip(a)]. 1075 x ^ (1 @ y) = x @ (y ^ x). [para(1074(a,1),1060(a,1)),flip(a)]. 1076 x ^ (1 @ y) = x @ (x ^ y). [para(1074(a,1),1061(a,1)),flip(a)]. 1077 (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). [para(1075(a,2),750(a,1))]. 1079 (x @ y) ^ (1 @ z) = x @ (z ^ y). [para(1077(a,2),1058(a,1))]. 1082 x ^ (y @ (x ^ z)) = x ^ (y @ z). [para(1079(a,1),305(a,1,2))]. 1084 x @ (x ^ y) = x ^ (x @ y). [para(1082(a,1),738(a,1)),flip(a)]. 1086 x ^ (1 @ y) = x ^ (x @ y). [para(1084(a,1),1076(a,2))]. 1087 x ^ (y ^ (y @ x)) = x ^ y. [para(1086(a,1),305(a,1,2))]. 1088 x ^ ((x @ y) ^ y) = y ^ x. [para(1087(a,1),60(a,2))]. 1089 x ^ (y ^ (x @ y)) = y ^ x. [para(20(a,1),1088(a,1,2))]. 1090 x ^ (y ^ (x @ y)) = x ^ y. [para(1089(a,2),20(a,1))]. 1091 (x ^ y) ^ (x @ y) = x ^ y. [para(1090(a,1),16(a,2))]. 1092 (x ^ y) V (x @ y) = x @ y. [para(1091(a,1),65(a,1,1))]. 1093 (x ^ y) V (y @ x) = y @ x. [para(20(a,1),1092(a,1,1))]. 1094 (x * y) V (((z V (x * (y * u))) / u) @ (x * y)) = ((z V (x * (y * u))) / u) @ (x * y). [para(163(a,1),1093(a,1,1))]. 1109 x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). [para(1041(a,1),17(a,1)),flip(a)]. 1110 (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). [para(1109(a,1),1036(a,1))]. 1111 x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). [para(1110(a,1),17(a,1)),flip(a)]. 1112 x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). [para(885(a,1),1111(a,1,2)),flip(a)]. 1113 (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. [para(1112(a,1),1033(a,1,1))]. 1114 (x V (y @ z)) @ u = (x V (y @ x)) @ u. [para(65(a,1),1113(a,1,1,2,2))]. 1115 (x V (y @ x)) @ z = (x V y) @ z. [para(697(a,1),1114(a,1,1,2)),flip(a)]. 1118 (x V (y @ z)) @ u = (x V (y @ w)) @ u. [para(1114(a,2),1114(a,2))]. 1120 (x V y) @ (x V x) = x V (y @ x). [para(1115(a,1),785(a,1))]. 1122 (x V y) @ x = x V (y @ x). [para(19(a,1),1120(a,1,2))]. 1123 (x V y) @ y = y V (x @ y). [para(21(a,1),1122(a,1,1))]. 1124 (x V (y \ x)) @ (y \ x) = y \ x. [para(1123(a,2),984(a,1))]. 1125 (x V (y \ x)) @ z = (y \ x) @ z. [para(1124(a,1),750(a,1,1)),flip(a)]. 1127 (x V (y @ z)) @ (x V u) = x V (y @ u). [para(1118(a,1),785(a,1))]. 1128 (x V (y @ z)) @ u = (y V x) @ u. [para(860(a,1),1118(a,1,1)),flip(a)]. 1133 ((x @ y) V z) @ u = (x V z) @ u. [para(21(a,1),1128(a,1,1))]. 1135 (x V y) @ (x V z) = x V (y @ z). [para(697(a,1),1127(a,1,1,2))]. 1136 (x V y) @ (z V x) = x V (y @ z). [para(21(a,1),1135(a,1,2))]. 1137 ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). [para(1135(a,1),784(a,1)),flip(a)]. 1138 (x \ y) @ (y V z) = y V ((x \ y) @ z). [para(1135(a,1),1125(a,1)),flip(a)]. 1139 (x V y) \ (x V ((z \ x) @ y)) = ((x V y) \ (z \ x)) @ 1. [para(1138(a,1),1027(a,1,2))]. 1140 (x V y) @ (z V u) = (x @ z) V (y @ u). [para(1137(a,1),1133(a,1)),flip(a)]. 1144 (x @ y) V (z @ x) = x V (z @ y). [para(1140(a,1),1136(a,1))]. 1146 (x @ y) V (y @ z) = y V (x @ z). [para(1144(a,1),21(a,1)),flip(a)]. 1159 (x V y) \ (x V ((x / y) \ x)) = ((x V y) \ ((x / y) \ x)) @ 1. [para(700(a,1),1139(a,1,2,2))]. 1166 ((x V y) \ ((x / y) \ x)) @ 1 = 1. [para(1159(a,1),912(a,1))]. 1168 ((x V y) @ ((x / y) \ x)) \ ((x / y) \ x) = 1. [para(1166(a,1),1001(a,2))]. 1169 ((x V y) @ y) \ ((x / y) \ x) = 1. [para(556(a,1),1168(a,1,1))]. 1171 (x V (y @ x)) \ ((y / x) \ y) = 1. [para(1123(a,1),1169(a,1,1))]. 1172 (x / y) ^ (x / ((y V (x @ y)) * 1)) = x / y. [para(1171(a,1),179(a,1,2,2,2))]. 1173 (x / y) ^ (x / (y V (x @ y))) = x / y. [para(30(a,1),1172(a,1,2,2))]. 1174 x / (y V (x @ y)) = x / y. [para(1173(a,1),235(a,1)),flip(a)]. 1175 x / ((y @ z) V (x @ z)) = x / (y @ z). [para(539(a,1),1174(a,1,2,2))]. 1177 x / (x V (y @ x)) = x / (y @ x). [para(1146(a,1),1175(a,1,2))]. 1179 (x / (y @ x)) \ x = x V (y @ x). [para(1177(a,1),34(a,1,1))]. 1180 ((x @ y) / (y @ (x @ y))) \ (x @ y) = y V (x @ (x @ y)). [para(1179(a,2),1146(a,1))]. 1181 ((x @ y) / y) \ (x @ y) = y V (x @ (x @ y)). [para(699(a,1),1180(a,1,1,2))]. 1182 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (y * ((y \ x) / z))). [para(196(a,1),1094(a,1,2,1,1)),flip(a)]. 1183 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / z. [para(607(a,1),669(a,2,2))]. 1184 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (x / z)). [para(793(a,1),1182(a,2,2))]. 1185 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V (x / z). [para(697(a,1),1184(a,2,2))]. 1187 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (x / z) V (y * ((y \ x) / z)). [para(1185(a,2),21(a,1))]. 1188 (x / y) @ (z * ((z \ x) / y)) = (x / y) V (z * ((z \ x) / y)). [para(196(a,1),1187(a,1,1,1))]. 1189 (x / y) V (z * ((z \ x) / y)) = (x / y) @ (x / y). [para(1188(a,1),793(a,1))]. 1190 (x / y) V (z * ((z \ x) / y)) = x / y. [para(1189(a,2),697(a,1))]. 1192 ((x \ y) / z) \ (x \ (y / z)) = 1. [para(1190(a,1),938(a,1,2,2))]. 1193 (x \ (y / z)) / 1 = (x \ y) / z. [para(1192(a,1),1183(a,1,2))]. 1195 (x \ y) / z = x \ (y / z). [para(1193(a,1),146(a,1))]. 1197 (((x @ y) / x) \ y) / x = (y / x) @ 1. [para(1195(a,2),993(a,1))]. 1199 (x @ y) / y = (x / y) @ 1. [para(1040(a,1),1197(a,1,1))]. 1200 x V (((y / x) @ 1) \ (y @ x)) = ((y @ x) / x) \ (y @ x). [para(1199(a,1),157(a,1,2,1))]. 1208 x V (((y / x) \ y) @ (1 \ x)) = ((y @ x) / x) \ (y @ x). [para(1023(a,2),1200(a,1,2))]. 1209 ((x @ y) / y) \ (x @ y) = y V (((x / y) \ x) @ y). [para(119(a,1),1208(a,1,2,2)),flip(a)]. 1210 ((x @ y) / y) \ (x @ y) = (x / y) \ x. [para(1209(a,2),741(a,1))]. 1211 x V (y @ (y @ x)) = (y / x) \ y. [para(1210(a,1),1181(a,1)),flip(a)]. 1212 x V (y @ x) = (y / x) \ y. [para(539(a,1),1211(a,1,2))]. 1216 (x / y) \ x = (x V y) @ y. [para(1212(a,1),1123(a,2)),flip(a)]. 1217 (x / y) \ x = (x V y) @ ((z / y) \ z). [para(1216(a,2),556(a,2)),flip(a)]. 1218 $F # answer("Goal 15"). [resolve(1217,a,59,a(flip))]. ============================== end of proof ========================== -------- Proof 12 -------- "Goal 13". ============================== PROOF ================================= % Proof 12 at 10.38 (+ 0.23) seconds: "Goal 13". % Length of proof is 958. % Level of proof is 170. % Maximum clause weight is 41. % Given clauses 1037. 13 ((x / y) \ x) @ (x V y) = x V y # label("Goal 13") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 54 c30 V c31 != ((c30 / c31) \ c30) @ (c30 V c31) # label("Goal 13") # answer("Goal 13"). [deny(13)]. 55 ((c30 / c31) \ c30) @ (c30 V c31) != c30 V c31 # answer("Goal 13"). [copy(54),flip(a)]. 60 x ^ (y ^ z) = z ^ (x ^ y). [para(20(a,1),16(a,1)),flip(a)]. 61 (x ^ y) ^ z = x ^ (z ^ y). [para(20(a,1),16(a,2,2))]. 62 (x V y) V z = y V (x V z). [para(21(a,1),17(a,1,1))]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 93 x ^ (y ^ (x V z)) = y ^ x. [para(23(a,1),60(a,1,2)),flip(a)]. 94 x V (y V z) = y V (x V z). [para(62(a,1),17(a,1))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 123 1 ^ (x \ (y V x)) = 1. [para(21(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 129 1 ^ ((x * (x \ y)) \ y) = 1. [para(99(a,1),123(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 157 x V ((y / x) \ y) = (y / x) \ y. [para(155(a,1),65(a,1,1))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 162 1 V ((x * (x \ y)) \ y) = (x * (x \ y)) \ y. [para(129(a,1),65(a,1,1))]. 163 (x * y) ^ ((z V (x * (y * u))) / u) = x * y. [para(28(a,1),134(a,1,2,1,2))]. 164 x V (y / (z ^ (x \ y))) = y / (z ^ (x \ y)). [para(137(a,1),65(a,1,1))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 169 x ^ ((1 ^ y) \ x) = x. [para(20(a,1),167(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 179 x ^ (y / (z * (z \ (x \ y)))) = x. [para(102(a,1),135(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 196 x V (y * (((y \ x) / z) * z)) = x. [para(108(a,1),70(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 201 x V (((y ^ x) / z) * z) = x. [para(20(a,1),194(a,1,2,1,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 204 x ^ (((y ^ z) / x) \ z) = x. [para(201(a,1),111(a,1,2,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 256 (1 @ x) ^ (y @ x) = y @ x. [para(255(a,1),68(a,1,1))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 266 (x \ y) V ((x \ 1) * y) = x \ y. [para(183(a,1),66(a,1,2))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 305 x ^ (y ^ (1 @ x)) = x ^ y. [para(301(a,1),61(a,1,1)),flip(a)]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 307 x ^ ((y / x) \ (1 @ y)) = x. [para(301(a,1),204(a,1,2,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 318 1 V (x \ (y ^ x)) = 1. [para(65(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 322 (x \ (y ^ x)) V 1 = 1. [para(318(a,1),21(a,1)),flip(a)]. 323 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. [para(93(a,1),318(a,1,2,2))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 325 1 V (x @ 1) = 1. [para(299(a,1),182(a,1,2))]. 326 1 ^ (x @ 1) = x @ 1. [para(325(a,1),68(a,1,1))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 329 x V ((y @ 1) * x) = x. [para(324(a,1),180(a,1,2,1))]. 330 x ^ ((y @ 1) \ x) = x. [para(326(a,1),169(a,1,2,1))]. 331 x ^ ((y @ 1) * x) = (y @ 1) * x. [para(329(a,1),68(a,1,1))]. 332 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. [para(305(a,1),126(a,1,2,1))]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 353 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. [para(331(a,1),323(a,1,2,2))]. 354 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. [para(67(a,1),332(a,1,2,1))]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 359 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. [para(22(a,1),354(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 393 x ^ ((y \ 1) \ (1 @ (y \ x))) = x. [para(390(a,1),307(a,1,2,1))]. 394 x / ((y / (z ^ (x \ y))) \ x) = x V (y / (z ^ (x \ y))). [para(164(a,1),32(a,1,2,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 422 (x * y) \ x = 1 / y. [para(419(a,2),336(a,2)),flip(a)]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 453 1 / (x @ y) = 1 / y. [para(447(a,2),336(a,2)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 466 1 / (x * y) = y \ (1 / x). [para(379(a,1),462(a,1,2)),flip(a)]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 472 1 / ((x @ y) * z) = z \ (1 / y). [para(453(a,1),466(a,2,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 481 ((x @ y) \ z) \ 1 = z \ (1 @ y). [para(454(a,1),476(a,2,2))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 483 1 / ((x @ y) * z) = 1 / (y * z). [para(472(a,2),466(a,2)),flip(a)]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 493 x ^ (1 @ (y * (y \ x))) = x. [para(480(a,1),393(a,1,2))]. 494 ((x @ y) \ y) \ 1 = 1. [para(481(a,2),432(a,1))]. 495 ((x @ y) \ y) @ 1 = ((x @ y) \ y) * 1. [para(494(a,1),324(a,1,2)),flip(a)]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 515 1 V (x @ (y \ (z ^ y))) = 1. [para(512(a,2),322(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 529 1 / ((x @ (y \ z)) * u) = u \ (1 @ (z \ y)). [para(527(a,2),472(a,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 552 x @ (y / (1 / z)) = x @ (y * z). [para(422(a,1),551(a,1,2,2))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 567 x @ (y * (y \ (z * (1 / u)))) = x @ (z / u). [para(561(a,1),560(a,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 574 ((x @ y) \ y) @ 1 = (x @ y) \ y. [para(495(a,2),30(a,1))]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 588 ((x @ y) * z) V 1 = 1 V (u @ (y * z)). [para(587(a,1),512(a,1,2)),flip(a)]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 595 1 V (x @ (y \ (z ^ (u @ y)))) = 1. [para(594(a,1),515(a,1,2))]. 596 (x \ (y ^ (z @ x))) V 1 = 1. [para(595(a,1),512(a,1)),flip(a)]. 597 1 V (x \ (y ^ (z @ x))) = 1. [para(595(a,1),514(a,1)),flip(a)]. 598 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). [para(596(a,1),23(a,1,1))]. 599 1 V ((x * (x \ y)) \ y) = 1. [para(493(a,1),597(a,1,2,2))]. 600 (x * (x \ y)) \ y = 1. [para(599(a,1),162(a,1)),flip(a)]. 604 ((x @ y) * z) V 1 = 1 V (y * z). [para(588(a,2),514(a,1))]. 605 1 V ((x @ y) * z) = 1 V (y * z). [para(604(a,1),21(a,1)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 607 x ^ ((y \ (z / x)) \ (y \ z)) = x. [para(606(a,1),181(a,1,2))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 613 1 / ((x @ (y \ z)) * u) = 1 @ (u \ (z \ y)). [para(529(a,2),520(a,2)),flip(a)]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 617 x @ (y / (z \ u)) = x @ (y * (u \ z)). [para(615(a,1),566(a,1)),flip(a)]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 622 (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. [para(598(a,1),359(a,1))]. 623 1 @ (x \ (y \ z)) = 1 / ((z \ y) * x). [para(613(a,1),483(a,1))]. 624 x @ (y / (1 / ((z \ u) * w))) = x @ (y / (w \ (u \ z))). [para(623(a,1),591(a,1,2,2))]. 625 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. [para(21(a,1),353(a,1,2,1,2))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 668 x / ((y / (z ^ (x \ y))) \ x) = y / (z ^ (x \ y)). [para(394(a,2),164(a,1))]. 669 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / (z ^ ((x \ (y / z)) \ (x \ y))). [para(607(a,1),668(a,1,2,1,2))]. 670 x @ (y / (z \ (u \ w))) = x @ (y * ((w \ u) * z)). [para(624(a,1),552(a,1))]. 671 x @ (y / ((z * u) \ w)) = x @ (y * ((w \ z) * u)). [para(606(a,2),670(a,1,2,2))]. 672 x @ (y * (z \ (u * w))) = x @ (y * ((z \ u) * w)). [para(671(a,1),617(a,1)),flip(a)]. 673 x @ (y * ((y \ z) * (1 / u))) = x @ (z / u). [para(672(a,1),567(a,1))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 699 x @ (y @ x) = x. [para(697(a,1),539(a,2))]. 700 ((x / y) \ x) @ y = (x / y) \ x. [para(697(a,1),556(a,1)),flip(a)]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 721 x \ (x @ (x V y)) = x \ (x V y). [para(709(a,2),645(a,1))]. 722 x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. [para(710(a,1),98(a,1,2,2,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. 738 x ^ (x @ (x ^ y)) = x @ (x ^ y). [para(736(a,1),68(a,1,1))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 741 x V (((y / x) \ y) @ x) = (y / x) \ y. [para(700(a,2),157(a,1,2))]. 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. 743 ((1 @ x) * y) / y = 1 @ x. [para(742(a,1),382(a,1,2))]. 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 752 (x @ y) @ x = x. [para(750(a,2),697(a,1))]. 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. 754 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. [para(738(a,1),622(a,1,1))]. 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. 756 1 V ((x \ 1) * y) = 1 V (x \ y). [para(703(a,1),514(a,1,2))]. 757 1 V ((1 / x) * y) = 1 V (x \ y). [para(336(a,2),756(a,1,2,1))]. 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 793 x @ (y * ((y \ z) / u)) = x @ (z / u). [para(762(a,1),673(a,1))]. 795 (x * ((x \ y) / z)) @ (y / z) = x * ((x \ y) / z). [para(793(a,1),697(a,1))]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 806 (1 @ x) \ ((1 @ x) * y) = y. [para(800(a,2),742(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 843 x V ((y ^ x) @ x) = x. [para(821(a,1),99(a,1,2))]. 844 x V ((x ^ y) @ x) = x. [para(20(a,1),843(a,1,2,1))]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. 866 x V (y @ (x ^ z)) = x V (y @ x). [para(66(a,1),864(a,1,2,2)),flip(a)]. 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. 868 x @ (x V (y @ z)) = x V (x @ z). [para(539(a,1),867(a,2,2))]. 869 x \ (x V (x @ y)) = x \ (x V y). [para(867(a,1),721(a,1,2))]. 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. 871 x * (x \ (x V y)) = x V (x @ y). [para(867(a,1),820(a,2))]. 872 x V (y V (y @ x)) = y V x. [para(867(a,1),860(a,1,2))]. 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. 875 x V (y V (x @ y)) = x V y. [para(872(a,1),94(a,1)),flip(a)]. 876 x V ((y @ z) V (x @ z)) = x V (y @ z). [para(539(a,1),875(a,1,2,2))]. 877 x \ (x V (y @ z)) = x \ (x V (x @ z)). [para(868(a,1),721(a,1,2)),flip(a)]. 878 (x V (x @ y)) \ (x V y) = 1. [para(871(a,1),600(a,1,1))]. 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. 882 x \ (x V (y @ z)) = x \ (x V z). [para(877(a,2),869(a,1))]. 884 (x V y) \ (x V (y V (z @ u))) = (x V y) \ ((x V y) V u). [para(17(a,1),882(a,1,2))]. 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. 886 (x @ y) ^ (x @ (y V z)) = x @ y. [para(885(a,1),67(a,1,2))]. 888 (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). [para(65(a,1),886(a,1,2,2))]. 889 (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). [para(888(a,1),20(a,1)),flip(a)]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 894 (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. [para(738(a,1),754(a,1,2,2,2))]. 895 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. [para(539(a,1),894(a,1,2,2))]. 897 ((x * ((x \ y) / z)) V (x * ((x \ y) / z))) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(795(a,1),878(a,1,1,2))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 899 (x V y) \ (x V (y V (z @ u))) = (x V y) \ (x V (y V u)). [para(17(a,1),884(a,2,2))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 909 (x V y) \ (x V (y V y)) = (x V y) \ (x V ((z / y) \ z)). [para(741(a,1),899(a,1,2,2)),flip(a)]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 911 (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V y). [para(19(a,1),909(a,1,2,2)),flip(a)]. 912 (x V y) \ (x V ((z / y) \ z)) = 1. [para(911(a,2),278(a,1))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 936 (x * ((x \ y) / z)) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(19(a,1),897(a,1,1))]. 937 (x * ((x \ y) / z)) \ ((y / z) V (x * ((x \ y) / z))) = 1. [para(21(a,1),936(a,1,2))]. 938 ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))) = 1. [para(937(a,1),606(a,1)),flip(a)]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 972 (1 @ x) \ x = x @ 1. [para(970(a,1),806(a,1,2))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 984 (x \ y) V (y @ (x \ y)) = x \ y. [para(982(a,2),266(a,1,2))]. 985 x @ (y \ x) = (1 / y) * x. [para(336(a,2),982(a,2,1))]. 986 x / (x @ (y \ x)) = 1 @ y. [para(982(a,2),382(a,1,2))]. 988 (x @ y) / x = 1 @ (y / x). [para(983(a,1),743(a,1,1))]. 989 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). [para(985(a,1),605(a,1,2,1))]. 990 x / (x @ y) = 1 @ (x / y). [para(556(a,1),986(a,1,2))]. 993 ((x @ y) / x) \ (y / x) = (y / x) @ 1. [para(988(a,2),972(a,1,1))]. 994 (x @ y) V ((1 @ (x / y)) \ x) = (x / (x @ y)) \ x. [para(990(a,1),157(a,1,2,1))]. 998 (x @ y) / x = y / (y @ x). [para(990(a,2),988(a,2))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1001 (x @ y) \ y = (x \ y) @ 1. [para(999(a,1),574(a,1)),flip(a)]. 1002 (x \ y) @ (z \ y) = (x @ z) \ y. [para(999(a,1),707(a,1))]. 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1006 (x @ 1) \ y = (x \ y) @ y. [para(119(a,1),1002(a,1,2)),flip(a)]. 1007 (x @ (y / z)) \ y = (x \ y) @ z. [para(1002(a,1),556(a,1))]. 1009 x ^ ((y \ x) @ x) = x. [para(1006(a,1),330(a,1,2))]. 1010 (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. [para(539(a,1),1009(a,1,2))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1013 (x @ y) ^ ((z \ x) @ y) = x @ y. [para(1012(a,1),1010(a,1,2))]. 1014 (x @ y) ^ ((x V z) @ y) = x @ y. [para(34(a,1),1013(a,1,2,1))]. 1015 (x @ y) ^ ((z V x) @ y) = x @ y. [para(81(a,1),1013(a,1,2,1))]. 1017 (x @ y) V ((x V z) @ y) = (x V z) @ y. [para(1014(a,1),65(a,1,1))]. 1018 (x @ y) V ((z V x) @ y) = (z V x) @ y. [para(1015(a,1),65(a,1,1))]. 1019 (x V y) V ((x V y) @ z) = (x V y) V (x @ z). [para(1017(a,1),876(a,1,2))]. 1020 (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. [para(750(a,1),1018(a,1,1))]. 1021 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). [para(28(a,1),989(a,1,2))]. 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. 1025 (x \ y) @ (x \ z) = x \ (y @ z). [para(697(a,1),1023(a,2,1))]. 1027 x \ (y @ x) = (x \ y) @ 1. [para(278(a,1),1025(a,1,2)),flip(a)]. 1028 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). [para(1025(a,1),605(a,1,2,1))]. 1030 x V (y V ((x V y) @ z)) = (x V y) V (x @ z). [para(1019(a,1),17(a,1)),flip(a)]. 1031 ((x V y) V (x @ z)) @ u = (x V y) @ u. [para(1019(a,1),870(a,1,1))]. 1032 (x V (y V (x @ z))) @ u = (x V y) @ u. [para(17(a,1),1031(a,1,1))]. 1033 (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. [para(866(a,1),1032(a,2,1))]. 1034 1 V (x \ (y * z)) = 1 V ((x \ y) * z). [para(1021(a,1),757(a,1)),flip(a)]. 1035 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. [para(1034(a,1),625(a,1))]. 1036 x V (y V ((x V y) @ z)) = x V (y V (x @ z)). [para(1030(a,2),17(a,1))]. 1037 (x @ y) V ((1 \ x) @ y) = (x / (x @ y)) \ x. [para(1007(a,1),994(a,1,2))]. 1038 (x @ y) V (x @ y) = (x / (x @ y)) \ x. [para(119(a,1),1037(a,1,2,1))]. 1039 (x / (x @ y)) \ x = x @ y. [para(1038(a,1),19(a,1))]. 1040 ((x @ y) / x) \ y = y @ x. [para(998(a,2),1039(a,1,1))]. 1041 (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). [para(1020(a,1),876(a,1,2))]. 1042 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. [para(1035(a,1),1028(a,1)),flip(a)]. 1043 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. [para(1042(a,1),756(a,1)),flip(a)]. 1045 (x ^ (y V ((z @ 1) * x))) \ x = 1. [para(1043(a,1),130(a,1)),flip(a)]. 1046 ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(722(a,1),1045(a,1,1,2))]. 1047 ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(119(a,1),1046(a,1,1,1,2,2))]. 1048 (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. [para(20(a,1),1047(a,1,1))]. 1049 (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. [para(305(a,1),1048(a,1,1))]. 1050 (x ^ y) \ (y ^ (1 @ x)) = 1. [para(119(a,1),1049(a,1,2,2,2))]. 1051 x * ((y ^ z) \ (z ^ (1 @ y))) = x. [para(1050(a,2),30(a,1,2))]. 1052 x @ (y ^ (1 @ z)) = x @ (z ^ y). [para(1051(a,1),560(a,1,2)),flip(a)]. 1053 x @ (y ^ (1 @ z)) = x @ (y ^ z). [para(20(a,1),1052(a,2,2))]. 1054 x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). [para(539(a,1),1052(a,1,2,2)),flip(a)]. 1055 x @ ((y @ z) ^ u) = x @ (z ^ u). [para(1054(a,2),1052(a,1))]. 1056 x @ ((y @ z) ^ u) = x @ (u ^ z). [para(1054(a,2),1053(a,1))]. 1057 x @ (y ^ (z @ u)) = x @ (u ^ y). [para(20(a,1),1055(a,1,2))]. 1058 x @ (y ^ (z @ u)) = x @ (y ^ u). [para(20(a,1),1056(a,1,2))]. 1060 (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). [para(1057(a,1),697(a,1))]. 1061 (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). [para(1058(a,1),697(a,1))]. 1062 (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. [para(1058(a,1),1001(a,1,1))]. 1063 (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). [para(1061(a,1),256(a,1,2))]. 1064 x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). [para(1063(a,1),60(a,2))]. 1065 (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). [para(889(a,1),1064(a,1,2)),flip(a)]. 1066 x ^ (1 @ (x ^ y)) = x ^ (1 @ y). [para(1065(a,1),1061(a,1))]. 1067 (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. [para(1066(a,1),895(a,1,2))]. 1068 (x \ (x ^ (1 @ y))) @ 1 = 1. [para(1067(a,1),1062(a,1)),flip(a)]. 1069 x \ ((x ^ (1 @ y)) @ x) = 1. [para(1068(a,1),1027(a,2))]. 1070 ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. [para(1069(a,1),99(a,1,2,2))]. 1071 ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. [para(30(a,1),1070(a,1,2))]. 1072 x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. [para(1071(a,1),21(a,1)),flip(a)]. 1073 (x ^ (1 @ y)) @ x = x. [para(1072(a,1),844(a,1))]. 1074 (x ^ (1 @ y)) @ z = x @ z. [para(1073(a,1),750(a,1,1)),flip(a)]. 1075 x ^ (1 @ y) = x @ (y ^ x). [para(1074(a,1),1060(a,1)),flip(a)]. 1076 x ^ (1 @ y) = x @ (x ^ y). [para(1074(a,1),1061(a,1)),flip(a)]. 1077 (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). [para(1075(a,2),750(a,1))]. 1079 (x @ y) ^ (1 @ z) = x @ (z ^ y). [para(1077(a,2),1058(a,1))]. 1082 x ^ (y @ (x ^ z)) = x ^ (y @ z). [para(1079(a,1),305(a,1,2))]. 1084 x @ (x ^ y) = x ^ (x @ y). [para(1082(a,1),738(a,1)),flip(a)]. 1086 x ^ (1 @ y) = x ^ (x @ y). [para(1084(a,1),1076(a,2))]. 1087 x ^ (y ^ (y @ x)) = x ^ y. [para(1086(a,1),305(a,1,2))]. 1088 x ^ ((x @ y) ^ y) = y ^ x. [para(1087(a,1),60(a,2))]. 1089 x ^ (y ^ (x @ y)) = y ^ x. [para(20(a,1),1088(a,1,2))]. 1090 x ^ (y ^ (x @ y)) = x ^ y. [para(1089(a,2),20(a,1))]. 1091 (x ^ y) ^ (x @ y) = x ^ y. [para(1090(a,1),16(a,2))]. 1092 (x ^ y) V (x @ y) = x @ y. [para(1091(a,1),65(a,1,1))]. 1093 (x ^ y) V (y @ x) = y @ x. [para(20(a,1),1092(a,1,1))]. 1094 (x * y) V (((z V (x * (y * u))) / u) @ (x * y)) = ((z V (x * (y * u))) / u) @ (x * y). [para(163(a,1),1093(a,1,1))]. 1109 x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). [para(1041(a,1),17(a,1)),flip(a)]. 1110 (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). [para(1109(a,1),1036(a,1))]. 1111 x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). [para(1110(a,1),17(a,1)),flip(a)]. 1112 x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). [para(885(a,1),1111(a,1,2)),flip(a)]. 1113 (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. [para(1112(a,1),1033(a,1,1))]. 1114 (x V (y @ z)) @ u = (x V (y @ x)) @ u. [para(65(a,1),1113(a,1,1,2,2))]. 1115 (x V (y @ x)) @ z = (x V y) @ z. [para(697(a,1),1114(a,1,1,2)),flip(a)]. 1118 (x V (y @ z)) @ u = (x V (y @ w)) @ u. [para(1114(a,2),1114(a,2))]. 1120 (x V y) @ (x V x) = x V (y @ x). [para(1115(a,1),785(a,1))]. 1122 (x V y) @ x = x V (y @ x). [para(19(a,1),1120(a,1,2))]. 1123 (x V y) @ y = y V (x @ y). [para(21(a,1),1122(a,1,1))]. 1124 (x V (y \ x)) @ (y \ x) = y \ x. [para(1123(a,2),984(a,1))]. 1125 (x V (y \ x)) @ z = (y \ x) @ z. [para(1124(a,1),750(a,1,1)),flip(a)]. 1127 (x V (y @ z)) @ (x V u) = x V (y @ u). [para(1118(a,1),785(a,1))]. 1128 (x V (y @ z)) @ u = (y V x) @ u. [para(860(a,1),1118(a,1,1)),flip(a)]. 1133 ((x @ y) V z) @ u = (x V z) @ u. [para(21(a,1),1128(a,1,1))]. 1135 (x V y) @ (x V z) = x V (y @ z). [para(697(a,1),1127(a,1,1,2))]. 1136 (x V y) @ (z V x) = x V (y @ z). [para(21(a,1),1135(a,1,2))]. 1137 ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). [para(1135(a,1),784(a,1)),flip(a)]. 1138 (x \ y) @ (y V z) = y V ((x \ y) @ z). [para(1135(a,1),1125(a,1)),flip(a)]. 1139 (x V y) \ (x V ((z \ x) @ y)) = ((x V y) \ (z \ x)) @ 1. [para(1138(a,1),1027(a,1,2))]. 1140 (x V y) @ (z V u) = (x @ z) V (y @ u). [para(1137(a,1),1133(a,1)),flip(a)]. 1144 (x @ y) V (z @ x) = x V (z @ y). [para(1140(a,1),1136(a,1))]. 1146 (x @ y) V (y @ z) = y V (x @ z). [para(1144(a,1),21(a,1)),flip(a)]. 1159 (x V y) \ (x V ((x / y) \ x)) = ((x V y) \ ((x / y) \ x)) @ 1. [para(700(a,1),1139(a,1,2,2))]. 1166 ((x V y) \ ((x / y) \ x)) @ 1 = 1. [para(1159(a,1),912(a,1))]. 1168 ((x V y) @ ((x / y) \ x)) \ ((x / y) \ x) = 1. [para(1166(a,1),1001(a,2))]. 1169 ((x V y) @ y) \ ((x / y) \ x) = 1. [para(556(a,1),1168(a,1,1))]. 1171 (x V (y @ x)) \ ((y / x) \ y) = 1. [para(1123(a,1),1169(a,1,1))]. 1172 (x / y) ^ (x / ((y V (x @ y)) * 1)) = x / y. [para(1171(a,1),179(a,1,2,2,2))]. 1173 (x / y) ^ (x / (y V (x @ y))) = x / y. [para(30(a,1),1172(a,1,2,2))]. 1174 x / (y V (x @ y)) = x / y. [para(1173(a,1),235(a,1)),flip(a)]. 1175 x / ((y @ z) V (x @ z)) = x / (y @ z). [para(539(a,1),1174(a,1,2,2))]. 1177 x / (x V (y @ x)) = x / (y @ x). [para(1146(a,1),1175(a,1,2))]. 1179 (x / (y @ x)) \ x = x V (y @ x). [para(1177(a,1),34(a,1,1))]. 1180 ((x @ y) / (y @ (x @ y))) \ (x @ y) = y V (x @ (x @ y)). [para(1179(a,2),1146(a,1))]. 1181 ((x @ y) / y) \ (x @ y) = y V (x @ (x @ y)). [para(699(a,1),1180(a,1,1,2))]. 1182 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (y * ((y \ x) / z))). [para(196(a,1),1094(a,1,2,1,1)),flip(a)]. 1183 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / z. [para(607(a,1),669(a,2,2))]. 1184 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (x / z)). [para(793(a,1),1182(a,2,2))]. 1185 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V (x / z). [para(697(a,1),1184(a,2,2))]. 1187 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (x / z) V (y * ((y \ x) / z)). [para(1185(a,2),21(a,1))]. 1188 (x / y) @ (z * ((z \ x) / y)) = (x / y) V (z * ((z \ x) / y)). [para(196(a,1),1187(a,1,1,1))]. 1189 (x / y) V (z * ((z \ x) / y)) = (x / y) @ (x / y). [para(1188(a,1),793(a,1))]. 1190 (x / y) V (z * ((z \ x) / y)) = x / y. [para(1189(a,2),697(a,1))]. 1192 ((x \ y) / z) \ (x \ (y / z)) = 1. [para(1190(a,1),938(a,1,2,2))]. 1193 (x \ (y / z)) / 1 = (x \ y) / z. [para(1192(a,1),1183(a,1,2))]. 1195 (x \ y) / z = x \ (y / z). [para(1193(a,1),146(a,1))]. 1197 (((x @ y) / x) \ y) / x = (y / x) @ 1. [para(1195(a,2),993(a,1))]. 1199 (x @ y) / y = (x / y) @ 1. [para(1040(a,1),1197(a,1,1))]. 1200 x V (((y / x) @ 1) \ (y @ x)) = ((y @ x) / x) \ (y @ x). [para(1199(a,1),157(a,1,2,1))]. 1208 x V (((y / x) \ y) @ (1 \ x)) = ((y @ x) / x) \ (y @ x). [para(1023(a,2),1200(a,1,2))]. 1209 ((x @ y) / y) \ (x @ y) = y V (((x / y) \ x) @ y). [para(119(a,1),1208(a,1,2,2)),flip(a)]. 1210 ((x @ y) / y) \ (x @ y) = (x / y) \ x. [para(1209(a,2),741(a,1))]. 1211 x V (y @ (y @ x)) = (y / x) \ y. [para(1210(a,1),1181(a,1)),flip(a)]. 1212 x V (y @ x) = (y / x) \ y. [para(539(a,1),1211(a,1,2))]. 1216 (x / y) \ x = (x V y) @ y. [para(1212(a,1),1123(a,2)),flip(a)]. 1219 ((x / y) \ x) @ (x V y) = x V y # label("Goal 13"). [para(1216(a,2),752(a,1,1))]. 1220 $F # answer("Goal 13"). [resolve(1219,a,55,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 54x 55 58x 59 given #1038 (H,wt=13): 1213 ((x / y) \ x) @ z = (x V y) @ z. [para(1211(a,1),1128(a,1,1))]. given #1039 (H,wt=13): 1219 ((x / y) \ x) @ (x V y) = x V y # label("Goal 13"). [para(1216(a,2),752(a,1,1))]. given #1040 (H,wt=15): 1217 (x / y) \ x = (x V y) @ ((z / y) \ z). [para(1216(a,2),556(a,2)),flip(a)]. given #1041 (H,wt=15): 1221 (x @ (x / y)) / ((x V y) @ 1) = x / y. [para(1213(a,1),1029(a,1,2))]. given #1042 (H,wt=13): 1223 (x @ (x / y)) / (y @ 1) = x / y. [para(1221(a,1),1178(a,1)),flip(a)]. -------- Proof 13 -------- "Goal 14". ============================== PROOF ================================= % Proof 13 at 10.46 (+ 0.23) seconds: "Goal 14". % Length of proof is 1002. % Level of proof is 171. % Maximum clause weight is 41. % Given clauses 1042. 14 (x V y) @ (x / (y \ x)) = x / (y \ x) # label("Goal 14") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 56 c32 / (c33 \ c32) != (c32 V c33) @ (c32 / (c33 \ c32)) # label("Goal 14") # answer("Goal 14"). [deny(14)]. 57 (c32 V c33) @ (c32 / (c33 \ c32)) != c32 / (c33 \ c32) # answer("Goal 14"). [copy(56),flip(a)]. 60 x ^ (y ^ z) = z ^ (x ^ y). [para(20(a,1),16(a,1)),flip(a)]. 61 (x ^ y) ^ z = x ^ (z ^ y). [para(20(a,1),16(a,2,2))]. 62 (x V y) V z = y V (x V z). [para(21(a,1),17(a,1,1))]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 93 x ^ (y ^ (x V z)) = y ^ x. [para(23(a,1),60(a,1,2)),flip(a)]. 94 x V (y V z) = y V (x V z). [para(62(a,1),17(a,1))]. 95 x V (y V (x ^ z)) = y V x. [para(22(a,1),63(a,1,2)),flip(a)]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 123 1 ^ (x \ (y V x)) = 1. [para(21(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 129 1 ^ ((x * (x \ y)) \ y) = 1. [para(99(a,1),123(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 157 x V ((y / x) \ y) = (y / x) \ y. [para(155(a,1),65(a,1,1))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 162 1 V ((x * (x \ y)) \ y) = (x * (x \ y)) \ y. [para(129(a,1),65(a,1,1))]. 163 (x * y) ^ ((z V (x * (y * u))) / u) = x * y. [para(28(a,1),134(a,1,2,1,2))]. 164 x V (y / (z ^ (x \ y))) = y / (z ^ (x \ y)). [para(137(a,1),65(a,1,1))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 169 x ^ ((1 ^ y) \ x) = x. [para(20(a,1),167(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 171 (x * (x \ y)) / ((y V (x * (x \ y))) \ (x * (x \ y))) = y. [para(151(a,2),99(a,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 179 x ^ (y / (z * (z \ (x \ y)))) = x. [para(102(a,1),135(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 196 x V (y * (((y \ x) / z) * z)) = x. [para(108(a,1),70(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 201 x V (((y ^ x) / z) * z) = x. [para(20(a,1),194(a,1,2,1,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 204 x ^ (((y ^ z) / x) \ z) = x. [para(201(a,1),111(a,1,2,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 256 (1 @ x) ^ (y @ x) = y @ x. [para(255(a,1),68(a,1,1))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 266 (x \ y) V ((x \ 1) * y) = x \ y. [para(183(a,1),66(a,1,2))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 302 (1 @ x) ^ x = x. [para(301(a,1),20(a,1)),flip(a)]. 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 305 x ^ (y ^ (1 @ x)) = x ^ y. [para(301(a,1),61(a,1,1)),flip(a)]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 307 x ^ ((y / x) \ (1 @ y)) = x. [para(301(a,1),204(a,1,2,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 313 x ^ (1 @ ((y / x) \ y)) = x. [para(158(a,1),310(a,1,2,2))]. 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 318 1 V (x \ (y ^ x)) = 1. [para(65(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 322 (x \ (y ^ x)) V 1 = 1. [para(318(a,1),21(a,1)),flip(a)]. 323 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. [para(93(a,1),318(a,1,2,2))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 325 1 V (x @ 1) = 1. [para(299(a,1),182(a,1,2))]. 326 1 ^ (x @ 1) = x @ 1. [para(325(a,1),68(a,1,1))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 329 x V ((y @ 1) * x) = x. [para(324(a,1),180(a,1,2,1))]. 330 x ^ ((y @ 1) \ x) = x. [para(326(a,1),169(a,1,2,1))]. 331 x ^ ((y @ 1) * x) = (y @ 1) * x. [para(329(a,1),68(a,1,1))]. 332 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. [para(305(a,1),126(a,1,2,1))]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 353 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. [para(331(a,1),323(a,1,2,2))]. 354 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. [para(67(a,1),332(a,1,2,1))]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 359 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. [para(22(a,1),354(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 393 x ^ ((y \ 1) \ (1 @ (y \ x))) = x. [para(390(a,1),307(a,1,2,1))]. 394 x / ((y / (z ^ (x \ y))) \ x) = x V (y / (z ^ (x \ y))). [para(164(a,1),32(a,1,2,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 422 (x * y) \ x = 1 / y. [para(419(a,2),336(a,2)),flip(a)]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 453 1 / (x @ y) = 1 / y. [para(447(a,2),336(a,2)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 456 (x @ y) \ (1 @ y) = 1. [para(454(a,1),432(a,1,2))]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 466 1 / (x * y) = y \ (1 / x). [para(379(a,1),462(a,1,2)),flip(a)]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 472 1 / ((x @ y) * z) = z \ (1 / y). [para(453(a,1),466(a,2,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 481 ((x @ y) \ z) \ 1 = z \ (1 @ y). [para(454(a,1),476(a,2,2))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 483 1 / ((x @ y) * z) = 1 / (y * z). [para(472(a,2),466(a,2)),flip(a)]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 493 x ^ (1 @ (y * (y \ x))) = x. [para(480(a,1),393(a,1,2))]. 494 ((x @ y) \ y) \ 1 = 1. [para(481(a,2),432(a,1))]. 495 ((x @ y) \ y) @ 1 = ((x @ y) \ y) * 1. [para(494(a,1),324(a,1,2)),flip(a)]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 515 1 V (x @ (y \ (z ^ y))) = 1. [para(512(a,2),322(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. 524 1 @ ((x @ y) \ y) = 1. [para(520(a,2),456(a,1))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 526 1 ^ ((x @ y) \ y) = (x @ y) \ y. [para(524(a,1),302(a,1,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 529 1 / ((x @ (y \ z)) * u) = u \ (1 @ (z \ y)). [para(527(a,2),472(a,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 552 x @ (y / (1 / z)) = x @ (y * z). [para(422(a,1),551(a,1,2,2))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 567 x @ (y * (y \ (z * (1 / u)))) = x @ (z / u). [para(561(a,1),560(a,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 574 ((x @ y) \ y) @ 1 = (x @ y) \ y. [para(495(a,2),30(a,1))]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 588 ((x @ y) * z) V 1 = 1 V (u @ (y * z)). [para(587(a,1),512(a,1,2)),flip(a)]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 595 1 V (x @ (y \ (z ^ (u @ y)))) = 1. [para(594(a,1),515(a,1,2))]. 596 (x \ (y ^ (z @ x))) V 1 = 1. [para(595(a,1),512(a,1)),flip(a)]. 597 1 V (x \ (y ^ (z @ x))) = 1. [para(595(a,1),514(a,1)),flip(a)]. 598 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). [para(596(a,1),23(a,1,1))]. 599 1 V ((x * (x \ y)) \ y) = 1. [para(493(a,1),597(a,1,2,2))]. 600 (x * (x \ y)) \ y = 1. [para(599(a,1),162(a,1)),flip(a)]. 604 ((x @ y) * z) V 1 = 1 V (y * z). [para(588(a,2),514(a,1))]. 605 1 V ((x @ y) * z) = 1 V (y * z). [para(604(a,1),21(a,1)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 607 x ^ ((y \ (z / x)) \ (y \ z)) = x. [para(606(a,1),181(a,1,2))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 613 1 / ((x @ (y \ z)) * u) = 1 @ (u \ (z \ y)). [para(529(a,2),520(a,2)),flip(a)]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 617 x @ (y / (z \ u)) = x @ (y * (u \ z)). [para(615(a,1),566(a,1)),flip(a)]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 622 (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. [para(598(a,1),359(a,1))]. 623 1 @ (x \ (y \ z)) = 1 / ((z \ y) * x). [para(613(a,1),483(a,1))]. 624 x @ (y / (1 / ((z \ u) * w))) = x @ (y / (w \ (u \ z))). [para(623(a,1),591(a,1,2,2))]. 625 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. [para(21(a,1),353(a,1,2,1,2))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 633 (x * (x \ y)) / (y \ (x * (x \ y))) = y. [para(99(a,1),171(a,1,2,1))]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 668 x / ((y / (z ^ (x \ y))) \ x) = y / (z ^ (x \ y)). [para(394(a,2),164(a,1))]. 669 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / (z ^ ((x \ (y / z)) \ (x \ y))). [para(607(a,1),668(a,1,2,1,2))]. 670 x @ (y / (z \ (u \ w))) = x @ (y * ((w \ u) * z)). [para(624(a,1),552(a,1))]. 671 x @ (y / ((z * u) \ w)) = x @ (y * ((w \ z) * u)). [para(606(a,2),670(a,1,2,2))]. 672 x @ (y * (z \ (u * w))) = x @ (y * ((z \ u) * w)). [para(671(a,1),617(a,1)),flip(a)]. 673 x @ (y * ((y \ z) * (1 / u))) = x @ (z / u). [para(672(a,1),567(a,1))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 693 (x \ 1) @ 1 = 1. [para(689(a,1),324(a,1)),flip(a)]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 699 x @ (y @ x) = x. [para(697(a,1),539(a,2))]. 700 ((x / y) \ x) @ y = (x / y) \ x. [para(697(a,1),556(a,1)),flip(a)]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 711 (x \ (x / y)) @ 1 = 1. [para(463(a,2),693(a,1,1))]. 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 716 (x \ (y V x)) @ 1 = 1. [para(77(a,1),711(a,1,1,2))]. 717 ((x V (y ^ z)) \ (x V y)) @ 1 = 1. [para(95(a,1),716(a,1,1,2))]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 721 x \ (x @ (x V y)) = x \ (x V y). [para(709(a,2),645(a,1))]. 722 x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. [para(710(a,1),98(a,1,2,2,2))]. 723 x ^ ((y @ (y / x)) \ y) = x. [para(710(a,2),313(a,1,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 728 (x @ (x / y)) ^ (x / y) = x @ (x / y). [para(723(a,1),137(a,1,2,2))]. 729 1 ^ ((x @ (x / y)) \ (x / y)) = 1. [para(723(a,1),165(a,1,2,2,2))]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. 738 x ^ (x @ (x ^ y)) = x @ (x ^ y). [para(736(a,1),68(a,1,1))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 741 x V (((y / x) \ y) @ x) = (y / x) \ y. [para(700(a,2),157(a,1,2))]. 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. 743 ((1 @ x) * y) / y = 1 @ x. [para(742(a,1),382(a,1,2))]. 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. 747 ((x V (y ^ z)) \ (x V z)) @ 1 = 1. [para(20(a,1),717(a,1,1,1,2))]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 749 (x @ (x / y)) \ (x / y) = 1. [para(729(a,1),526(a,1)),flip(a)]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. 754 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. [para(738(a,1),622(a,1,1))]. 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. 756 1 V ((x \ 1) * y) = 1 V (x \ y). [para(703(a,1),514(a,1,2))]. 757 1 V ((1 / x) * y) = 1 V (x \ y). [para(336(a,2),756(a,1,2,1))]. 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 764 ((x V (y @ (y / z))) \ (x V (y / z))) @ 1 = 1. [para(728(a,1),747(a,1,1,1,2))]. 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 793 x @ (y * ((y \ z) / u)) = x @ (z / u). [para(762(a,1),673(a,1))]. 795 (x * ((x \ y) / z)) @ (y / z) = x * ((x \ y) / z). [para(793(a,1),697(a,1))]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 806 (1 @ x) \ ((1 @ x) * y) = y. [para(800(a,2),742(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 842 (x @ 1) \ (x / y) = (1 @ (x / y)) * 1. [para(749(a,1),813(a,1,2)),flip(a)]. 843 x V ((y ^ x) @ x) = x. [para(821(a,1),99(a,1,2))]. 844 x V ((x ^ y) @ x) = x. [para(20(a,1),843(a,1,2,1))]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 856 (x @ 1) \ (x / y) = 1 @ (x / y). [para(842(a,2),30(a,1))]. 857 ((x @ 1) * (1 @ (x / y))) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. [para(856(a,1),633(a,1,1,2))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. 866 x V (y @ (x ^ z)) = x V (y @ x). [para(66(a,1),864(a,1,2,2)),flip(a)]. 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. 868 x @ (x V (y @ z)) = x V (x @ z). [para(539(a,1),867(a,2,2))]. 869 x \ (x V (x @ y)) = x \ (x V y). [para(867(a,1),721(a,1,2))]. 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. 871 x * (x \ (x V y)) = x V (x @ y). [para(867(a,1),820(a,2))]. 872 x V (y V (y @ x)) = y V x. [para(867(a,1),860(a,1,2))]. 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. 875 x V (y V (x @ y)) = x V y. [para(872(a,1),94(a,1)),flip(a)]. 876 x V ((y @ z) V (x @ z)) = x V (y @ z). [para(539(a,1),875(a,1,2,2))]. 877 x \ (x V (y @ z)) = x \ (x V (x @ z)). [para(868(a,1),721(a,1,2)),flip(a)]. 878 (x V (x @ y)) \ (x V y) = 1. [para(871(a,1),600(a,1,1))]. 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. 882 x \ (x V (y @ z)) = x \ (x V z). [para(877(a,2),869(a,1))]. 884 (x V y) \ (x V (y V (z @ u))) = (x V y) \ ((x V y) V u). [para(17(a,1),882(a,1,2))]. 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. 886 (x @ y) ^ (x @ (y V z)) = x @ y. [para(885(a,1),67(a,1,2))]. 888 (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). [para(65(a,1),886(a,1,2,2))]. 889 (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). [para(888(a,1),20(a,1)),flip(a)]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 894 (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. [para(738(a,1),754(a,1,2,2,2))]. 895 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. [para(539(a,1),894(a,1,2,2))]. 897 ((x * ((x \ y) / z)) V (x * ((x \ y) / z))) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(795(a,1),878(a,1,1,2))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 899 (x V y) \ (x V (y V (z @ u))) = (x V y) \ (x V (y V u)). [para(17(a,1),884(a,2,2))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 909 (x V y) \ (x V (y V y)) = (x V y) \ (x V ((z / y) \ z)). [para(741(a,1),899(a,1,2,2)),flip(a)]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 911 (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V y). [para(19(a,1),909(a,1,2,2)),flip(a)]. 912 (x V y) \ (x V ((z / y) \ z)) = 1. [para(911(a,2),278(a,1))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 933 (x @ (x / y)) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. [para(351(a,1),857(a,1,1))]. 934 (x @ (x / y)) / ((x / y) \ ((x @ 1) * (1 @ (x / y)))) = x / y. [para(856(a,1),933(a,1,2,2,2))]. 935 (x @ (x / y)) / ((x / y) \ (x @ (x / y))) = x / y. [para(351(a,1),934(a,1,2,2))]. 936 (x * ((x \ y) / z)) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(19(a,1),897(a,1,1))]. 937 (x * ((x \ y) / z)) \ ((y / z) V (x * ((x \ y) / z))) = 1. [para(21(a,1),936(a,1,2))]. 938 ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))) = 1. [para(937(a,1),606(a,1)),flip(a)]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 972 (1 @ x) \ x = x @ 1. [para(970(a,1),806(a,1,2))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 984 (x \ y) V (y @ (x \ y)) = x \ y. [para(982(a,2),266(a,1,2))]. 985 x @ (y \ x) = (1 / y) * x. [para(336(a,2),982(a,2,1))]. 986 x / (x @ (y \ x)) = 1 @ y. [para(982(a,2),382(a,1,2))]. 988 (x @ y) / x = 1 @ (y / x). [para(983(a,1),743(a,1,1))]. 989 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). [para(985(a,1),605(a,1,2,1))]. 990 x / (x @ y) = 1 @ (x / y). [para(556(a,1),986(a,1,2))]. 993 ((x @ y) / x) \ (y / x) = (y / x) @ 1. [para(988(a,2),972(a,1,1))]. 994 (x @ y) V ((1 @ (x / y)) \ x) = (x / (x @ y)) \ x. [para(990(a,1),157(a,1,2,1))]. 998 (x @ y) / x = y / (y @ x). [para(990(a,2),988(a,2))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1001 (x @ y) \ y = (x \ y) @ 1. [para(999(a,1),574(a,1)),flip(a)]. 1002 (x \ y) @ (z \ y) = (x @ z) \ y. [para(999(a,1),707(a,1))]. 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1006 (x @ 1) \ y = (x \ y) @ y. [para(119(a,1),1002(a,1,2)),flip(a)]. 1007 (x @ (y / z)) \ y = (x \ y) @ z. [para(1002(a,1),556(a,1))]. 1009 x ^ ((y \ x) @ x) = x. [para(1006(a,1),330(a,1,2))]. 1010 (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. [para(539(a,1),1009(a,1,2))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1013 (x @ y) ^ ((z \ x) @ y) = x @ y. [para(1012(a,1),1010(a,1,2))]. 1014 (x @ y) ^ ((x V z) @ y) = x @ y. [para(34(a,1),1013(a,1,2,1))]. 1015 (x @ y) ^ ((z V x) @ y) = x @ y. [para(81(a,1),1013(a,1,2,1))]. 1017 (x @ y) V ((x V z) @ y) = (x V z) @ y. [para(1014(a,1),65(a,1,1))]. 1018 (x @ y) V ((z V x) @ y) = (z V x) @ y. [para(1015(a,1),65(a,1,1))]. 1019 (x V y) V ((x V y) @ z) = (x V y) V (x @ z). [para(1017(a,1),876(a,1,2))]. 1020 (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. [para(750(a,1),1018(a,1,1))]. 1021 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). [para(28(a,1),989(a,1,2))]. 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. 1025 (x \ y) @ (x \ z) = x \ (y @ z). [para(697(a,1),1023(a,2,1))]. 1027 x \ (y @ x) = (x \ y) @ 1. [para(278(a,1),1025(a,1,2)),flip(a)]. 1028 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). [para(1025(a,1),605(a,1,2,1))]. 1029 (x @ (x / y)) / (((x / y) \ x) @ 1) = x / y. [para(1027(a,1),935(a,1,2))]. 1030 x V (y V ((x V y) @ z)) = (x V y) V (x @ z). [para(1019(a,1),17(a,1)),flip(a)]. 1031 ((x V y) V (x @ z)) @ u = (x V y) @ u. [para(1019(a,1),870(a,1,1))]. 1032 (x V (y V (x @ z))) @ u = (x V y) @ u. [para(17(a,1),1031(a,1,1))]. 1033 (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. [para(866(a,1),1032(a,2,1))]. 1034 1 V (x \ (y * z)) = 1 V ((x \ y) * z). [para(1021(a,1),757(a,1)),flip(a)]. 1035 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. [para(1034(a,1),625(a,1))]. 1036 x V (y V ((x V y) @ z)) = x V (y V (x @ z)). [para(1030(a,2),17(a,1))]. 1037 (x @ y) V ((1 \ x) @ y) = (x / (x @ y)) \ x. [para(1007(a,1),994(a,1,2))]. 1038 (x @ y) V (x @ y) = (x / (x @ y)) \ x. [para(119(a,1),1037(a,1,2,1))]. 1039 (x / (x @ y)) \ x = x @ y. [para(1038(a,1),19(a,1))]. 1040 ((x @ y) / x) \ y = y @ x. [para(998(a,2),1039(a,1,1))]. 1041 (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). [para(1020(a,1),876(a,1,2))]. 1042 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. [para(1035(a,1),1028(a,1)),flip(a)]. 1043 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. [para(1042(a,1),756(a,1)),flip(a)]. 1045 (x ^ (y V ((z @ 1) * x))) \ x = 1. [para(1043(a,1),130(a,1)),flip(a)]. 1046 ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(722(a,1),1045(a,1,1,2))]. 1047 ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(119(a,1),1046(a,1,1,1,2,2))]. 1048 (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. [para(20(a,1),1047(a,1,1))]. 1049 (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. [para(305(a,1),1048(a,1,1))]. 1050 (x ^ y) \ (y ^ (1 @ x)) = 1. [para(119(a,1),1049(a,1,2,2,2))]. 1051 x * ((y ^ z) \ (z ^ (1 @ y))) = x. [para(1050(a,2),30(a,1,2))]. 1052 x @ (y ^ (1 @ z)) = x @ (z ^ y). [para(1051(a,1),560(a,1,2)),flip(a)]. 1053 x @ (y ^ (1 @ z)) = x @ (y ^ z). [para(20(a,1),1052(a,2,2))]. 1054 x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). [para(539(a,1),1052(a,1,2,2)),flip(a)]. 1055 x @ ((y @ z) ^ u) = x @ (z ^ u). [para(1054(a,2),1052(a,1))]. 1056 x @ ((y @ z) ^ u) = x @ (u ^ z). [para(1054(a,2),1053(a,1))]. 1057 x @ (y ^ (z @ u)) = x @ (u ^ y). [para(20(a,1),1055(a,1,2))]. 1058 x @ (y ^ (z @ u)) = x @ (y ^ u). [para(20(a,1),1056(a,1,2))]. 1060 (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). [para(1057(a,1),697(a,1))]. 1061 (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). [para(1058(a,1),697(a,1))]. 1062 (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. [para(1058(a,1),1001(a,1,1))]. 1063 (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). [para(1061(a,1),256(a,1,2))]. 1064 x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). [para(1063(a,1),60(a,2))]. 1065 (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). [para(889(a,1),1064(a,1,2)),flip(a)]. 1066 x ^ (1 @ (x ^ y)) = x ^ (1 @ y). [para(1065(a,1),1061(a,1))]. 1067 (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. [para(1066(a,1),895(a,1,2))]. 1068 (x \ (x ^ (1 @ y))) @ 1 = 1. [para(1067(a,1),1062(a,1)),flip(a)]. 1069 x \ ((x ^ (1 @ y)) @ x) = 1. [para(1068(a,1),1027(a,2))]. 1070 ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. [para(1069(a,1),99(a,1,2,2))]. 1071 ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. [para(30(a,1),1070(a,1,2))]. 1072 x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. [para(1071(a,1),21(a,1)),flip(a)]. 1073 (x ^ (1 @ y)) @ x = x. [para(1072(a,1),844(a,1))]. 1074 (x ^ (1 @ y)) @ z = x @ z. [para(1073(a,1),750(a,1,1)),flip(a)]. 1075 x ^ (1 @ y) = x @ (y ^ x). [para(1074(a,1),1060(a,1)),flip(a)]. 1076 x ^ (1 @ y) = x @ (x ^ y). [para(1074(a,1),1061(a,1)),flip(a)]. 1077 (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). [para(1075(a,2),750(a,1))]. 1079 (x @ y) ^ (1 @ z) = x @ (z ^ y). [para(1077(a,2),1058(a,1))]. 1082 x ^ (y @ (x ^ z)) = x ^ (y @ z). [para(1079(a,1),305(a,1,2))]. 1084 x @ (x ^ y) = x ^ (x @ y). [para(1082(a,1),738(a,1)),flip(a)]. 1086 x ^ (1 @ y) = x ^ (x @ y). [para(1084(a,1),1076(a,2))]. 1087 x ^ (y ^ (y @ x)) = x ^ y. [para(1086(a,1),305(a,1,2))]. 1088 x ^ ((x @ y) ^ y) = y ^ x. [para(1087(a,1),60(a,2))]. 1089 x ^ (y ^ (x @ y)) = y ^ x. [para(20(a,1),1088(a,1,2))]. 1090 x ^ (y ^ (x @ y)) = x ^ y. [para(1089(a,2),20(a,1))]. 1091 (x ^ y) ^ (x @ y) = x ^ y. [para(1090(a,1),16(a,2))]. 1092 (x ^ y) V (x @ y) = x @ y. [para(1091(a,1),65(a,1,1))]. 1093 (x ^ y) V (y @ x) = y @ x. [para(20(a,1),1092(a,1,1))]. 1094 (x * y) V (((z V (x * (y * u))) / u) @ (x * y)) = ((z V (x * (y * u))) / u) @ (x * y). [para(163(a,1),1093(a,1,1))]. 1109 x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). [para(1041(a,1),17(a,1)),flip(a)]. 1110 (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). [para(1109(a,1),1036(a,1))]. 1111 x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). [para(1110(a,1),17(a,1)),flip(a)]. 1112 x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). [para(885(a,1),1111(a,1,2)),flip(a)]. 1113 (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. [para(1112(a,1),1033(a,1,1))]. 1114 (x V (y @ z)) @ u = (x V (y @ x)) @ u. [para(65(a,1),1113(a,1,1,2,2))]. 1115 (x V (y @ x)) @ z = (x V y) @ z. [para(697(a,1),1114(a,1,1,2)),flip(a)]. 1118 (x V (y @ z)) @ u = (x V (y @ w)) @ u. [para(1114(a,2),1114(a,2))]. 1120 (x V y) @ (x V x) = x V (y @ x). [para(1115(a,1),785(a,1))]. 1122 (x V y) @ x = x V (y @ x). [para(19(a,1),1120(a,1,2))]. 1123 (x V y) @ y = y V (x @ y). [para(21(a,1),1122(a,1,1))]. 1124 (x V (y \ x)) @ (y \ x) = y \ x. [para(1123(a,2),984(a,1))]. 1125 (x V (y \ x)) @ z = (y \ x) @ z. [para(1124(a,1),750(a,1,1)),flip(a)]. 1127 (x V (y @ z)) @ (x V u) = x V (y @ u). [para(1118(a,1),785(a,1))]. 1128 (x V (y @ z)) @ u = (y V x) @ u. [para(860(a,1),1118(a,1,1)),flip(a)]. 1133 ((x @ y) V z) @ u = (x V z) @ u. [para(21(a,1),1128(a,1,1))]. 1134 ((x V (y @ z)) \ u) @ w = (((y V x) @ v5) \ u) @ w. [para(1128(a,1),999(a,1,1,1)),flip(a)]. 1135 (x V y) @ (x V z) = x V (y @ z). [para(697(a,1),1127(a,1,1,2))]. 1136 (x V y) @ (z V x) = x V (y @ z). [para(21(a,1),1135(a,1,2))]. 1137 ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). [para(1135(a,1),784(a,1)),flip(a)]. 1138 (x \ y) @ (y V z) = y V ((x \ y) @ z). [para(1135(a,1),1125(a,1)),flip(a)]. 1139 (x V y) \ (x V ((z \ x) @ y)) = ((x V y) \ (z \ x)) @ 1. [para(1138(a,1),1027(a,1,2))]. 1140 (x V y) @ (z V u) = (x @ z) V (y @ u). [para(1137(a,1),1133(a,1)),flip(a)]. 1142 (x @ y) V (z @ y) = (x V z) @ y. [para(19(a,1),1140(a,1,2)),flip(a)]. 1144 (x @ y) V (z @ x) = x V (z @ y). [para(1140(a,1),1136(a,1))]. 1145 (x @ y) / (((x V z) @ y) \ (x @ y)) = (x @ y) V (z @ y). [para(1142(a,1),32(a,1,2,1))]. 1146 (x @ y) V (y @ z) = y V (x @ z). [para(1144(a,1),21(a,1)),flip(a)]. 1147 ((x V (y @ z)) \ u) @ w = ((y V x) \ u) @ w. [para(697(a,1),1134(a,2,1,1))]. 1148 ((x V y) \ (y V (x / z))) @ 1 = 1. [para(1147(a,1),764(a,1))]. 1149 ((x V y) \ (x / (y \ x))) @ 1 = 1. [para(150(a,1),1148(a,1,1,2))]. 1151 ((x V y) @ (x / (y \ x))) \ (x / (y \ x)) = 1. [para(1149(a,1),1001(a,2))]. 1152 ((x V y) @ y) \ (x / (y \ x)) = 1. [para(551(a,1),1151(a,1,1))]. 1154 (x V (y @ x)) \ (y / (x \ y)) = 1. [para(1123(a,1),1152(a,1,1))]. 1155 (x \ y) ^ (((x V (y @ x)) * 1) \ y) = x \ y. [para(1154(a,1),181(a,1,2,1,2))]. 1156 (x \ y) ^ ((x V (y @ x)) \ y) = x \ y. [para(30(a,1),1155(a,1,2,1))]. 1157 (x V (y @ x)) \ y = x \ y. [para(1156(a,1),188(a,1)),flip(a)]. 1158 ((x V y) \ x) @ z = (y \ x) @ z. [para(1157(a,1),1147(a,1,1)),flip(a)]. 1159 (x V y) \ (x V ((x / y) \ x)) = ((x V y) \ ((x / y) \ x)) @ 1. [para(700(a,1),1139(a,1,2,2))]. 1160 (x @ y) / (((x V z) \ x) @ (y \ y)) = (x @ y) V (z @ y). [para(1023(a,2),1145(a,1,2))]. 1161 (x @ y) / (((x V z) \ x) @ (y \ y)) = (x V z) @ y. [para(1160(a,2),1142(a,1))]. 1162 (x @ y) / (((x V z) \ x) @ 1) = (x V z) @ y. [para(278(a,1),1161(a,1,2,2))]. 1164 (x @ y) / ((z \ x) @ 1) = (x V z) @ y. [para(1158(a,1),1162(a,1,2))]. 1166 ((x V y) \ ((x / y) \ x)) @ 1 = 1. [para(1159(a,1),912(a,1))]. 1168 ((x V y) @ ((x / y) \ x)) \ ((x / y) \ x) = 1. [para(1166(a,1),1001(a,2))]. 1169 ((x V y) @ y) \ ((x / y) \ x) = 1. [para(556(a,1),1168(a,1,1))]. 1171 (x V (y @ x)) \ ((y / x) \ y) = 1. [para(1123(a,1),1169(a,1,1))]. 1172 (x / y) ^ (x / ((y V (x @ y)) * 1)) = x / y. [para(1171(a,1),179(a,1,2,2,2))]. 1173 (x / y) ^ (x / (y V (x @ y))) = x / y. [para(30(a,1),1172(a,1,2,2))]. 1174 x / (y V (x @ y)) = x / y. [para(1173(a,1),235(a,1)),flip(a)]. 1175 x / ((y @ z) V (x @ z)) = x / (y @ z). [para(539(a,1),1174(a,1,2,2))]. 1176 x / ((y V x) @ z) = x / (y @ z). [para(1142(a,1),1175(a,1,2))]. 1177 x / (x V (y @ x)) = x / (y @ x). [para(1146(a,1),1175(a,1,2))]. 1178 (x @ y) / ((x V z) @ u) = (x @ y) / (z @ u). [para(1128(a,1),1176(a,1,2))]. 1179 (x / (y @ x)) \ x = x V (y @ x). [para(1177(a,1),34(a,1,1))]. 1180 ((x @ y) / (y @ (x @ y))) \ (x @ y) = y V (x @ (x @ y)). [para(1179(a,2),1146(a,1))]. 1181 ((x @ y) / y) \ (x @ y) = y V (x @ (x @ y)). [para(699(a,1),1180(a,1,1,2))]. 1182 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (y * ((y \ x) / z))). [para(196(a,1),1094(a,1,2,1,1)),flip(a)]. 1183 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / z. [para(607(a,1),669(a,2,2))]. 1184 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (x / z)). [para(793(a,1),1182(a,2,2))]. 1185 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V (x / z). [para(697(a,1),1184(a,2,2))]. 1187 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (x / z) V (y * ((y \ x) / z)). [para(1185(a,2),21(a,1))]. 1188 (x / y) @ (z * ((z \ x) / y)) = (x / y) V (z * ((z \ x) / y)). [para(196(a,1),1187(a,1,1,1))]. 1189 (x / y) V (z * ((z \ x) / y)) = (x / y) @ (x / y). [para(1188(a,1),793(a,1))]. 1190 (x / y) V (z * ((z \ x) / y)) = x / y. [para(1189(a,2),697(a,1))]. 1192 ((x \ y) / z) \ (x \ (y / z)) = 1. [para(1190(a,1),938(a,1,2,2))]. 1193 (x \ (y / z)) / 1 = (x \ y) / z. [para(1192(a,1),1183(a,1,2))]. 1195 (x \ y) / z = x \ (y / z). [para(1193(a,1),146(a,1))]. 1197 (((x @ y) / x) \ y) / x = (y / x) @ 1. [para(1195(a,2),993(a,1))]. 1199 (x @ y) / y = (x / y) @ 1. [para(1040(a,1),1197(a,1,1))]. 1200 x V (((y / x) @ 1) \ (y @ x)) = ((y @ x) / x) \ (y @ x). [para(1199(a,1),157(a,1,2,1))]. 1208 x V (((y / x) \ y) @ (1 \ x)) = ((y @ x) / x) \ (y @ x). [para(1023(a,2),1200(a,1,2))]. 1209 ((x @ y) / y) \ (x @ y) = y V (((x / y) \ x) @ y). [para(119(a,1),1208(a,1,2,2)),flip(a)]. 1210 ((x @ y) / y) \ (x @ y) = (x / y) \ x. [para(1209(a,2),741(a,1))]. 1211 x V (y @ (y @ x)) = (y / x) \ y. [para(1210(a,1),1181(a,1)),flip(a)]. 1213 ((x / y) \ x) @ z = (x V y) @ z. [para(1211(a,1),1128(a,1,1))]. 1221 (x @ (x / y)) / ((x V y) @ 1) = x / y. [para(1213(a,1),1029(a,1,2))]. 1223 (x @ (x / y)) / (y @ 1) = x / y. [para(1221(a,1),1178(a,1)),flip(a)]. 1224 (x V y) @ (x / (y \ x)) = x / (y \ x). [para(1223(a,1),1164(a,1)),flip(a)]. 1225 $F # answer("Goal 14"). [resolve(1224,a,57,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 56x 57 given #1043 (H,wt=13): 1226 (x / (y @ 1)) @ z = (x / y) @ z. [para(1223(a,1),1206(a,1,1)),flip(a)]. given #1044 (H,wt=15): 1222 (x @ y) / ((x V z) @ 1) = (x / z) @ y. [para(1213(a,1),1165(a,1,2))]. given #1045 (H,wt=13): 1228 (x @ y) / (z @ 1) = (x / z) @ y. [para(1222(a,1),1178(a,1)),flip(a)]. given #1046 (H,wt=13): 1229 (x / (y \ x)) @ z = (x V y) @ z. [para(1228(a,1),1164(a,1))]. -------- Proof 14 -------- "Goal 12". ============================== PROOF ================================= % Proof 14 at 10.53 (+ 0.23) seconds: "Goal 12". % Length of proof is 992. % Level of proof is 172. % Maximum clause weight is 41. % Given clauses 1046. 12 (x / (y \ x)) @ (x V y) = x V y # label("Goal 12") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 52 c28 V c29 != (c28 / (c29 \ c28)) @ (c28 V c29) # label("Goal 12") # answer("Goal 12"). [deny(12)]. 53 (c28 / (c29 \ c28)) @ (c28 V c29) != c28 V c29 # answer("Goal 12") # label("Goal 12"). [copy(52),flip(a)]. 60 x ^ (y ^ z) = z ^ (x ^ y). [para(20(a,1),16(a,1)),flip(a)]. 61 (x ^ y) ^ z = x ^ (z ^ y). [para(20(a,1),16(a,2,2))]. 62 (x V y) V z = y V (x V z). [para(21(a,1),17(a,1,1))]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 93 x ^ (y ^ (x V z)) = y ^ x. [para(23(a,1),60(a,1,2)),flip(a)]. 94 x V (y V z) = y V (x V z). [para(62(a,1),17(a,1))]. 95 x V (y V (x ^ z)) = y V x. [para(22(a,1),63(a,1,2)),flip(a)]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 123 1 ^ (x \ (y V x)) = 1. [para(21(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 129 1 ^ ((x * (x \ y)) \ y) = 1. [para(99(a,1),123(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 157 x V ((y / x) \ y) = (y / x) \ y. [para(155(a,1),65(a,1,1))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 162 1 V ((x * (x \ y)) \ y) = (x * (x \ y)) \ y. [para(129(a,1),65(a,1,1))]. 163 (x * y) ^ ((z V (x * (y * u))) / u) = x * y. [para(28(a,1),134(a,1,2,1,2))]. 164 x V (y / (z ^ (x \ y))) = y / (z ^ (x \ y)). [para(137(a,1),65(a,1,1))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 169 x ^ ((1 ^ y) \ x) = x. [para(20(a,1),167(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 179 x ^ (y / (z * (z \ (x \ y)))) = x. [para(102(a,1),135(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 196 x V (y * (((y \ x) / z) * z)) = x. [para(108(a,1),70(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 201 x V (((y ^ x) / z) * z) = x. [para(20(a,1),194(a,1,2,1,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 204 x ^ (((y ^ z) / x) \ z) = x. [para(201(a,1),111(a,1,2,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 256 (1 @ x) ^ (y @ x) = y @ x. [para(255(a,1),68(a,1,1))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 266 (x \ y) V ((x \ 1) * y) = x \ y. [para(183(a,1),66(a,1,2))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 305 x ^ (y ^ (1 @ x)) = x ^ y. [para(301(a,1),61(a,1,1)),flip(a)]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 307 x ^ ((y / x) \ (1 @ y)) = x. [para(301(a,1),204(a,1,2,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 313 x ^ (1 @ ((y / x) \ y)) = x. [para(158(a,1),310(a,1,2,2))]. 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 318 1 V (x \ (y ^ x)) = 1. [para(65(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 322 (x \ (y ^ x)) V 1 = 1. [para(318(a,1),21(a,1)),flip(a)]. 323 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. [para(93(a,1),318(a,1,2,2))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 325 1 V (x @ 1) = 1. [para(299(a,1),182(a,1,2))]. 326 1 ^ (x @ 1) = x @ 1. [para(325(a,1),68(a,1,1))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 329 x V ((y @ 1) * x) = x. [para(324(a,1),180(a,1,2,1))]. 330 x ^ ((y @ 1) \ x) = x. [para(326(a,1),169(a,1,2,1))]. 331 x ^ ((y @ 1) * x) = (y @ 1) * x. [para(329(a,1),68(a,1,1))]. 332 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. [para(305(a,1),126(a,1,2,1))]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 353 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. [para(331(a,1),323(a,1,2,2))]. 354 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. [para(67(a,1),332(a,1,2,1))]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 359 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. [para(22(a,1),354(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 393 x ^ ((y \ 1) \ (1 @ (y \ x))) = x. [para(390(a,1),307(a,1,2,1))]. 394 x / ((y / (z ^ (x \ y))) \ x) = x V (y / (z ^ (x \ y))). [para(164(a,1),32(a,1,2,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 422 (x * y) \ x = 1 / y. [para(419(a,2),336(a,2)),flip(a)]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 453 1 / (x @ y) = 1 / y. [para(447(a,2),336(a,2)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 466 1 / (x * y) = y \ (1 / x). [para(379(a,1),462(a,1,2)),flip(a)]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 472 1 / ((x @ y) * z) = z \ (1 / y). [para(453(a,1),466(a,2,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 481 ((x @ y) \ z) \ 1 = z \ (1 @ y). [para(454(a,1),476(a,2,2))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 483 1 / ((x @ y) * z) = 1 / (y * z). [para(472(a,2),466(a,2)),flip(a)]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 493 x ^ (1 @ (y * (y \ x))) = x. [para(480(a,1),393(a,1,2))]. 494 ((x @ y) \ y) \ 1 = 1. [para(481(a,2),432(a,1))]. 495 ((x @ y) \ y) @ 1 = ((x @ y) \ y) * 1. [para(494(a,1),324(a,1,2)),flip(a)]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 515 1 V (x @ (y \ (z ^ y))) = 1. [para(512(a,2),322(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 529 1 / ((x @ (y \ z)) * u) = u \ (1 @ (z \ y)). [para(527(a,2),472(a,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 552 x @ (y / (1 / z)) = x @ (y * z). [para(422(a,1),551(a,1,2,2))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 567 x @ (y * (y \ (z * (1 / u)))) = x @ (z / u). [para(561(a,1),560(a,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 574 ((x @ y) \ y) @ 1 = (x @ y) \ y. [para(495(a,2),30(a,1))]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 588 ((x @ y) * z) V 1 = 1 V (u @ (y * z)). [para(587(a,1),512(a,1,2)),flip(a)]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 595 1 V (x @ (y \ (z ^ (u @ y)))) = 1. [para(594(a,1),515(a,1,2))]. 596 (x \ (y ^ (z @ x))) V 1 = 1. [para(595(a,1),512(a,1)),flip(a)]. 597 1 V (x \ (y ^ (z @ x))) = 1. [para(595(a,1),514(a,1)),flip(a)]. 598 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). [para(596(a,1),23(a,1,1))]. 599 1 V ((x * (x \ y)) \ y) = 1. [para(493(a,1),597(a,1,2,2))]. 600 (x * (x \ y)) \ y = 1. [para(599(a,1),162(a,1)),flip(a)]. 604 ((x @ y) * z) V 1 = 1 V (y * z). [para(588(a,2),514(a,1))]. 605 1 V ((x @ y) * z) = 1 V (y * z). [para(604(a,1),21(a,1)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 607 x ^ ((y \ (z / x)) \ (y \ z)) = x. [para(606(a,1),181(a,1,2))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 613 1 / ((x @ (y \ z)) * u) = 1 @ (u \ (z \ y)). [para(529(a,2),520(a,2)),flip(a)]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 617 x @ (y / (z \ u)) = x @ (y * (u \ z)). [para(615(a,1),566(a,1)),flip(a)]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 622 (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. [para(598(a,1),359(a,1))]. 623 1 @ (x \ (y \ z)) = 1 / ((z \ y) * x). [para(613(a,1),483(a,1))]. 624 x @ (y / (1 / ((z \ u) * w))) = x @ (y / (w \ (u \ z))). [para(623(a,1),591(a,1,2,2))]. 625 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. [para(21(a,1),353(a,1,2,1,2))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 668 x / ((y / (z ^ (x \ y))) \ x) = y / (z ^ (x \ y)). [para(394(a,2),164(a,1))]. 669 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / (z ^ ((x \ (y / z)) \ (x \ y))). [para(607(a,1),668(a,1,2,1,2))]. 670 x @ (y / (z \ (u \ w))) = x @ (y * ((w \ u) * z)). [para(624(a,1),552(a,1))]. 671 x @ (y / ((z * u) \ w)) = x @ (y * ((w \ z) * u)). [para(606(a,2),670(a,1,2,2))]. 672 x @ (y * (z \ (u * w))) = x @ (y * ((z \ u) * w)). [para(671(a,1),617(a,1)),flip(a)]. 673 x @ (y * ((y \ z) * (1 / u))) = x @ (z / u). [para(672(a,1),567(a,1))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 693 (x \ 1) @ 1 = 1. [para(689(a,1),324(a,1)),flip(a)]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 699 x @ (y @ x) = x. [para(697(a,1),539(a,2))]. 700 ((x / y) \ x) @ y = (x / y) \ x. [para(697(a,1),556(a,1)),flip(a)]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 711 (x \ (x / y)) @ 1 = 1. [para(463(a,2),693(a,1,1))]. 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 716 (x \ (y V x)) @ 1 = 1. [para(77(a,1),711(a,1,1,2))]. 717 ((x V (y ^ z)) \ (x V y)) @ 1 = 1. [para(95(a,1),716(a,1,1,2))]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 721 x \ (x @ (x V y)) = x \ (x V y). [para(709(a,2),645(a,1))]. 722 x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. [para(710(a,1),98(a,1,2,2,2))]. 723 x ^ ((y @ (y / x)) \ y) = x. [para(710(a,2),313(a,1,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 728 (x @ (x / y)) ^ (x / y) = x @ (x / y). [para(723(a,1),137(a,1,2,2))]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. 738 x ^ (x @ (x ^ y)) = x @ (x ^ y). [para(736(a,1),68(a,1,1))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 741 x V (((y / x) \ y) @ x) = (y / x) \ y. [para(700(a,2),157(a,1,2))]. 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. 743 ((1 @ x) * y) / y = 1 @ x. [para(742(a,1),382(a,1,2))]. 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. 747 ((x V (y ^ z)) \ (x V z)) @ 1 = 1. [para(20(a,1),717(a,1,1,1,2))]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. 754 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. [para(738(a,1),622(a,1,1))]. 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. 756 1 V ((x \ 1) * y) = 1 V (x \ y). [para(703(a,1),514(a,1,2))]. 757 1 V ((1 / x) * y) = 1 V (x \ y). [para(336(a,2),756(a,1,2,1))]. 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 763 (x / y) V (x @ (x / y)) = x / y. [para(728(a,1),85(a,1,2))]. 764 ((x V (y @ (y / z))) \ (x V (y / z))) @ 1 = 1. [para(728(a,1),747(a,1,1,1,2))]. 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 793 x @ (y * ((y \ z) / u)) = x @ (z / u). [para(762(a,1),673(a,1))]. 795 (x * ((x \ y) / z)) @ (y / z) = x * ((x \ y) / z). [para(793(a,1),697(a,1))]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 806 (1 @ x) \ ((1 @ x) * y) = y. [para(800(a,2),742(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 843 x V ((y ^ x) @ x) = x. [para(821(a,1),99(a,1,2))]. 844 x V ((x ^ y) @ x) = x. [para(20(a,1),843(a,1,2,1))]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. 866 x V (y @ (x ^ z)) = x V (y @ x). [para(66(a,1),864(a,1,2,2)),flip(a)]. 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. 868 x @ (x V (y @ z)) = x V (x @ z). [para(539(a,1),867(a,2,2))]. 869 x \ (x V (x @ y)) = x \ (x V y). [para(867(a,1),721(a,1,2))]. 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. 871 x * (x \ (x V y)) = x V (x @ y). [para(867(a,1),820(a,2))]. 872 x V (y V (y @ x)) = y V x. [para(867(a,1),860(a,1,2))]. 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. 875 x V (y V (x @ y)) = x V y. [para(872(a,1),94(a,1)),flip(a)]. 876 x V ((y @ z) V (x @ z)) = x V (y @ z). [para(539(a,1),875(a,1,2,2))]. 877 x \ (x V (y @ z)) = x \ (x V (x @ z)). [para(868(a,1),721(a,1,2)),flip(a)]. 878 (x V (x @ y)) \ (x V y) = 1. [para(871(a,1),600(a,1,1))]. 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. 882 x \ (x V (y @ z)) = x \ (x V z). [para(877(a,2),869(a,1))]. 884 (x V y) \ (x V (y V (z @ u))) = (x V y) \ ((x V y) V u). [para(17(a,1),882(a,1,2))]. 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. 886 (x @ y) ^ (x @ (y V z)) = x @ y. [para(885(a,1),67(a,1,2))]. 888 (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). [para(65(a,1),886(a,1,2,2))]. 889 (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). [para(888(a,1),20(a,1)),flip(a)]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 894 (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. [para(738(a,1),754(a,1,2,2,2))]. 895 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. [para(539(a,1),894(a,1,2,2))]. 897 ((x * ((x \ y) / z)) V (x * ((x \ y) / z))) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(795(a,1),878(a,1,1,2))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 899 (x V y) \ (x V (y V (z @ u))) = (x V y) \ (x V (y V u)). [para(17(a,1),884(a,2,2))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 909 (x V y) \ (x V (y V y)) = (x V y) \ (x V ((z / y) \ z)). [para(741(a,1),899(a,1,2,2)),flip(a)]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 911 (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V y). [para(19(a,1),909(a,1,2,2)),flip(a)]. 912 (x V y) \ (x V ((z / y) \ z)) = 1. [para(911(a,2),278(a,1))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 936 (x * ((x \ y) / z)) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(19(a,1),897(a,1,1))]. 937 (x * ((x \ y) / z)) \ ((y / z) V (x * ((x \ y) / z))) = 1. [para(21(a,1),936(a,1,2))]. 938 ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))) = 1. [para(937(a,1),606(a,1)),flip(a)]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 972 (1 @ x) \ x = x @ 1. [para(970(a,1),806(a,1,2))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 984 (x \ y) V (y @ (x \ y)) = x \ y. [para(982(a,2),266(a,1,2))]. 985 x @ (y \ x) = (1 / y) * x. [para(336(a,2),982(a,2,1))]. 986 x / (x @ (y \ x)) = 1 @ y. [para(982(a,2),382(a,1,2))]. 988 (x @ y) / x = 1 @ (y / x). [para(983(a,1),743(a,1,1))]. 989 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). [para(985(a,1),605(a,1,2,1))]. 990 x / (x @ y) = 1 @ (x / y). [para(556(a,1),986(a,1,2))]. 993 ((x @ y) / x) \ (y / x) = (y / x) @ 1. [para(988(a,2),972(a,1,1))]. 994 (x @ y) V ((1 @ (x / y)) \ x) = (x / (x @ y)) \ x. [para(990(a,1),157(a,1,2,1))]. 998 (x @ y) / x = y / (y @ x). [para(990(a,2),988(a,2))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1001 (x @ y) \ y = (x \ y) @ 1. [para(999(a,1),574(a,1)),flip(a)]. 1002 (x \ y) @ (z \ y) = (x @ z) \ y. [para(999(a,1),707(a,1))]. 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1006 (x @ 1) \ y = (x \ y) @ y. [para(119(a,1),1002(a,1,2)),flip(a)]. 1007 (x @ (y / z)) \ y = (x \ y) @ z. [para(1002(a,1),556(a,1))]. 1009 x ^ ((y \ x) @ x) = x. [para(1006(a,1),330(a,1,2))]. 1010 (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. [para(539(a,1),1009(a,1,2))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1013 (x @ y) ^ ((z \ x) @ y) = x @ y. [para(1012(a,1),1010(a,1,2))]. 1014 (x @ y) ^ ((x V z) @ y) = x @ y. [para(34(a,1),1013(a,1,2,1))]. 1015 (x @ y) ^ ((z V x) @ y) = x @ y. [para(81(a,1),1013(a,1,2,1))]. 1017 (x @ y) V ((x V z) @ y) = (x V z) @ y. [para(1014(a,1),65(a,1,1))]. 1018 (x @ y) V ((z V x) @ y) = (z V x) @ y. [para(1015(a,1),65(a,1,1))]. 1019 (x V y) V ((x V y) @ z) = (x V y) V (x @ z). [para(1017(a,1),876(a,1,2))]. 1020 (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. [para(750(a,1),1018(a,1,1))]. 1021 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). [para(28(a,1),989(a,1,2))]. 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. 1025 (x \ y) @ (x \ z) = x \ (y @ z). [para(697(a,1),1023(a,2,1))]. 1027 x \ (y @ x) = (x \ y) @ 1. [para(278(a,1),1025(a,1,2)),flip(a)]. 1028 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). [para(1025(a,1),605(a,1,2,1))]. 1030 x V (y V ((x V y) @ z)) = (x V y) V (x @ z). [para(1019(a,1),17(a,1)),flip(a)]. 1031 ((x V y) V (x @ z)) @ u = (x V y) @ u. [para(1019(a,1),870(a,1,1))]. 1032 (x V (y V (x @ z))) @ u = (x V y) @ u. [para(17(a,1),1031(a,1,1))]. 1033 (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. [para(866(a,1),1032(a,2,1))]. 1034 1 V (x \ (y * z)) = 1 V ((x \ y) * z). [para(1021(a,1),757(a,1)),flip(a)]. 1035 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. [para(1034(a,1),625(a,1))]. 1036 x V (y V ((x V y) @ z)) = x V (y V (x @ z)). [para(1030(a,2),17(a,1))]. 1037 (x @ y) V ((1 \ x) @ y) = (x / (x @ y)) \ x. [para(1007(a,1),994(a,1,2))]. 1038 (x @ y) V (x @ y) = (x / (x @ y)) \ x. [para(119(a,1),1037(a,1,2,1))]. 1039 (x / (x @ y)) \ x = x @ y. [para(1038(a,1),19(a,1))]. 1040 ((x @ y) / x) \ y = y @ x. [para(998(a,2),1039(a,1,1))]. 1041 (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). [para(1020(a,1),876(a,1,2))]. 1042 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. [para(1035(a,1),1028(a,1)),flip(a)]. 1043 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. [para(1042(a,1),756(a,1)),flip(a)]. 1045 (x ^ (y V ((z @ 1) * x))) \ x = 1. [para(1043(a,1),130(a,1)),flip(a)]. 1046 ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(722(a,1),1045(a,1,1,2))]. 1047 ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(119(a,1),1046(a,1,1,1,2,2))]. 1048 (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. [para(20(a,1),1047(a,1,1))]. 1049 (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. [para(305(a,1),1048(a,1,1))]. 1050 (x ^ y) \ (y ^ (1 @ x)) = 1. [para(119(a,1),1049(a,1,2,2,2))]. 1051 x * ((y ^ z) \ (z ^ (1 @ y))) = x. [para(1050(a,2),30(a,1,2))]. 1052 x @ (y ^ (1 @ z)) = x @ (z ^ y). [para(1051(a,1),560(a,1,2)),flip(a)]. 1053 x @ (y ^ (1 @ z)) = x @ (y ^ z). [para(20(a,1),1052(a,2,2))]. 1054 x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). [para(539(a,1),1052(a,1,2,2)),flip(a)]. 1055 x @ ((y @ z) ^ u) = x @ (z ^ u). [para(1054(a,2),1052(a,1))]. 1056 x @ ((y @ z) ^ u) = x @ (u ^ z). [para(1054(a,2),1053(a,1))]. 1057 x @ (y ^ (z @ u)) = x @ (u ^ y). [para(20(a,1),1055(a,1,2))]. 1058 x @ (y ^ (z @ u)) = x @ (y ^ u). [para(20(a,1),1056(a,1,2))]. 1060 (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). [para(1057(a,1),697(a,1))]. 1061 (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). [para(1058(a,1),697(a,1))]. 1062 (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. [para(1058(a,1),1001(a,1,1))]. 1063 (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). [para(1061(a,1),256(a,1,2))]. 1064 x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). [para(1063(a,1),60(a,2))]. 1065 (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). [para(889(a,1),1064(a,1,2)),flip(a)]. 1066 x ^ (1 @ (x ^ y)) = x ^ (1 @ y). [para(1065(a,1),1061(a,1))]. 1067 (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. [para(1066(a,1),895(a,1,2))]. 1068 (x \ (x ^ (1 @ y))) @ 1 = 1. [para(1067(a,1),1062(a,1)),flip(a)]. 1069 x \ ((x ^ (1 @ y)) @ x) = 1. [para(1068(a,1),1027(a,2))]. 1070 ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. [para(1069(a,1),99(a,1,2,2))]. 1071 ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. [para(30(a,1),1070(a,1,2))]. 1072 x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. [para(1071(a,1),21(a,1)),flip(a)]. 1073 (x ^ (1 @ y)) @ x = x. [para(1072(a,1),844(a,1))]. 1074 (x ^ (1 @ y)) @ z = x @ z. [para(1073(a,1),750(a,1,1)),flip(a)]. 1075 x ^ (1 @ y) = x @ (y ^ x). [para(1074(a,1),1060(a,1)),flip(a)]. 1076 x ^ (1 @ y) = x @ (x ^ y). [para(1074(a,1),1061(a,1)),flip(a)]. 1077 (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). [para(1075(a,2),750(a,1))]. 1079 (x @ y) ^ (1 @ z) = x @ (z ^ y). [para(1077(a,2),1058(a,1))]. 1082 x ^ (y @ (x ^ z)) = x ^ (y @ z). [para(1079(a,1),305(a,1,2))]. 1084 x @ (x ^ y) = x ^ (x @ y). [para(1082(a,1),738(a,1)),flip(a)]. 1086 x ^ (1 @ y) = x ^ (x @ y). [para(1084(a,1),1076(a,2))]. 1087 x ^ (y ^ (y @ x)) = x ^ y. [para(1086(a,1),305(a,1,2))]. 1088 x ^ ((x @ y) ^ y) = y ^ x. [para(1087(a,1),60(a,2))]. 1089 x ^ (y ^ (x @ y)) = y ^ x. [para(20(a,1),1088(a,1,2))]. 1090 x ^ (y ^ (x @ y)) = x ^ y. [para(1089(a,2),20(a,1))]. 1091 (x ^ y) ^ (x @ y) = x ^ y. [para(1090(a,1),16(a,2))]. 1092 (x ^ y) V (x @ y) = x @ y. [para(1091(a,1),65(a,1,1))]. 1093 (x ^ y) V (y @ x) = y @ x. [para(20(a,1),1092(a,1,1))]. 1094 (x * y) V (((z V (x * (y * u))) / u) @ (x * y)) = ((z V (x * (y * u))) / u) @ (x * y). [para(163(a,1),1093(a,1,1))]. 1109 x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). [para(1041(a,1),17(a,1)),flip(a)]. 1110 (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). [para(1109(a,1),1036(a,1))]. 1111 x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). [para(1110(a,1),17(a,1)),flip(a)]. 1112 x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). [para(885(a,1),1111(a,1,2)),flip(a)]. 1113 (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. [para(1112(a,1),1033(a,1,1))]. 1114 (x V (y @ z)) @ u = (x V (y @ x)) @ u. [para(65(a,1),1113(a,1,1,2,2))]. 1115 (x V (y @ x)) @ z = (x V y) @ z. [para(697(a,1),1114(a,1,1,2)),flip(a)]. 1118 (x V (y @ z)) @ u = (x V (y @ w)) @ u. [para(1114(a,2),1114(a,2))]. 1119 ((x / y) V x) @ z = (x / y) @ z. [para(763(a,1),1115(a,1,1)),flip(a)]. 1120 (x V y) @ (x V x) = x V (y @ x). [para(1115(a,1),785(a,1))]. 1121 (x V (x / y)) @ z = (x / y) @ z. [para(21(a,1),1119(a,1,1))]. 1122 (x V y) @ x = x V (y @ x). [para(19(a,1),1120(a,1,2))]. 1123 (x V y) @ y = y V (x @ y). [para(21(a,1),1122(a,1,1))]. 1124 (x V (y \ x)) @ (y \ x) = y \ x. [para(1123(a,2),984(a,1))]. 1125 (x V (y \ x)) @ z = (y \ x) @ z. [para(1124(a,1),750(a,1,1)),flip(a)]. 1127 (x V (y @ z)) @ (x V u) = x V (y @ u). [para(1118(a,1),785(a,1))]. 1128 (x V (y @ z)) @ u = (y V x) @ u. [para(860(a,1),1118(a,1,1)),flip(a)]. 1133 ((x @ y) V z) @ u = (x V z) @ u. [para(21(a,1),1128(a,1,1))]. 1134 ((x V (y @ z)) \ u) @ w = (((y V x) @ v5) \ u) @ w. [para(1128(a,1),999(a,1,1,1)),flip(a)]. 1135 (x V y) @ (x V z) = x V (y @ z). [para(697(a,1),1127(a,1,1,2))]. 1136 (x V y) @ (z V x) = x V (y @ z). [para(21(a,1),1135(a,1,2))]. 1137 ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). [para(1135(a,1),784(a,1)),flip(a)]. 1138 (x \ y) @ (y V z) = y V ((x \ y) @ z). [para(1135(a,1),1125(a,1)),flip(a)]. 1139 (x V y) \ (x V ((z \ x) @ y)) = ((x V y) \ (z \ x)) @ 1. [para(1138(a,1),1027(a,1,2))]. 1140 (x V y) @ (z V u) = (x @ z) V (y @ u). [para(1137(a,1),1133(a,1)),flip(a)]. 1142 (x @ y) V (z @ y) = (x V z) @ y. [para(19(a,1),1140(a,1,2)),flip(a)]. 1144 (x @ y) V (z @ x) = x V (z @ y). [para(1140(a,1),1136(a,1))]. 1145 (x @ y) / (((x V z) @ y) \ (x @ y)) = (x @ y) V (z @ y). [para(1142(a,1),32(a,1,2,1))]. 1146 (x @ y) V (y @ z) = y V (x @ z). [para(1144(a,1),21(a,1)),flip(a)]. 1147 ((x V (y @ z)) \ u) @ w = ((y V x) \ u) @ w. [para(697(a,1),1134(a,2,1,1))]. 1148 ((x V y) \ (y V (x / z))) @ 1 = 1. [para(1147(a,1),764(a,1))]. 1149 ((x V y) \ (x / (y \ x))) @ 1 = 1. [para(150(a,1),1148(a,1,1,2))]. 1151 ((x V y) @ (x / (y \ x))) \ (x / (y \ x)) = 1. [para(1149(a,1),1001(a,2))]. 1152 ((x V y) @ y) \ (x / (y \ x)) = 1. [para(551(a,1),1151(a,1,1))]. 1154 (x V (y @ x)) \ (y / (x \ y)) = 1. [para(1123(a,1),1152(a,1,1))]. 1155 (x \ y) ^ (((x V (y @ x)) * 1) \ y) = x \ y. [para(1154(a,1),181(a,1,2,1,2))]. 1156 (x \ y) ^ ((x V (y @ x)) \ y) = x \ y. [para(30(a,1),1155(a,1,2,1))]. 1157 (x V (y @ x)) \ y = x \ y. [para(1156(a,1),188(a,1)),flip(a)]. 1158 ((x V y) \ x) @ z = (y \ x) @ z. [para(1157(a,1),1147(a,1,1)),flip(a)]. 1159 (x V y) \ (x V ((x / y) \ x)) = ((x V y) \ ((x / y) \ x)) @ 1. [para(700(a,1),1139(a,1,2,2))]. 1160 (x @ y) / (((x V z) \ x) @ (y \ y)) = (x @ y) V (z @ y). [para(1023(a,2),1145(a,1,2))]. 1161 (x @ y) / (((x V z) \ x) @ (y \ y)) = (x V z) @ y. [para(1160(a,2),1142(a,1))]. 1162 (x @ y) / (((x V z) \ x) @ 1) = (x V z) @ y. [para(278(a,1),1161(a,1,2,2))]. 1164 (x @ y) / ((z \ x) @ 1) = (x V z) @ y. [para(1158(a,1),1162(a,1,2))]. 1165 (x @ y) / (((x / z) \ x) @ 1) = (x / z) @ y. [para(1164(a,2),1121(a,1))]. 1166 ((x V y) \ ((x / y) \ x)) @ 1 = 1. [para(1159(a,1),912(a,1))]. 1168 ((x V y) @ ((x / y) \ x)) \ ((x / y) \ x) = 1. [para(1166(a,1),1001(a,2))]. 1169 ((x V y) @ y) \ ((x / y) \ x) = 1. [para(556(a,1),1168(a,1,1))]. 1171 (x V (y @ x)) \ ((y / x) \ y) = 1. [para(1123(a,1),1169(a,1,1))]. 1172 (x / y) ^ (x / ((y V (x @ y)) * 1)) = x / y. [para(1171(a,1),179(a,1,2,2,2))]. 1173 (x / y) ^ (x / (y V (x @ y))) = x / y. [para(30(a,1),1172(a,1,2,2))]. 1174 x / (y V (x @ y)) = x / y. [para(1173(a,1),235(a,1)),flip(a)]. 1175 x / ((y @ z) V (x @ z)) = x / (y @ z). [para(539(a,1),1174(a,1,2,2))]. 1176 x / ((y V x) @ z) = x / (y @ z). [para(1142(a,1),1175(a,1,2))]. 1177 x / (x V (y @ x)) = x / (y @ x). [para(1146(a,1),1175(a,1,2))]. 1178 (x @ y) / ((x V z) @ u) = (x @ y) / (z @ u). [para(1128(a,1),1176(a,1,2))]. 1179 (x / (y @ x)) \ x = x V (y @ x). [para(1177(a,1),34(a,1,1))]. 1180 ((x @ y) / (y @ (x @ y))) \ (x @ y) = y V (x @ (x @ y)). [para(1179(a,2),1146(a,1))]. 1181 ((x @ y) / y) \ (x @ y) = y V (x @ (x @ y)). [para(699(a,1),1180(a,1,1,2))]. 1182 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (y * ((y \ x) / z))). [para(196(a,1),1094(a,1,2,1,1)),flip(a)]. 1183 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / z. [para(607(a,1),669(a,2,2))]. 1184 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (x / z)). [para(793(a,1),1182(a,2,2))]. 1185 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V (x / z). [para(697(a,1),1184(a,2,2))]. 1187 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (x / z) V (y * ((y \ x) / z)). [para(1185(a,2),21(a,1))]. 1188 (x / y) @ (z * ((z \ x) / y)) = (x / y) V (z * ((z \ x) / y)). [para(196(a,1),1187(a,1,1,1))]. 1189 (x / y) V (z * ((z \ x) / y)) = (x / y) @ (x / y). [para(1188(a,1),793(a,1))]. 1190 (x / y) V (z * ((z \ x) / y)) = x / y. [para(1189(a,2),697(a,1))]. 1192 ((x \ y) / z) \ (x \ (y / z)) = 1. [para(1190(a,1),938(a,1,2,2))]. 1193 (x \ (y / z)) / 1 = (x \ y) / z. [para(1192(a,1),1183(a,1,2))]. 1195 (x \ y) / z = x \ (y / z). [para(1193(a,1),146(a,1))]. 1197 (((x @ y) / x) \ y) / x = (y / x) @ 1. [para(1195(a,2),993(a,1))]. 1199 (x @ y) / y = (x / y) @ 1. [para(1040(a,1),1197(a,1,1))]. 1200 x V (((y / x) @ 1) \ (y @ x)) = ((y @ x) / x) \ (y @ x). [para(1199(a,1),157(a,1,2,1))]. 1208 x V (((y / x) \ y) @ (1 \ x)) = ((y @ x) / x) \ (y @ x). [para(1023(a,2),1200(a,1,2))]. 1209 ((x @ y) / y) \ (x @ y) = y V (((x / y) \ x) @ y). [para(119(a,1),1208(a,1,2,2)),flip(a)]. 1210 ((x @ y) / y) \ (x @ y) = (x / y) \ x. [para(1209(a,2),741(a,1))]. 1211 x V (y @ (y @ x)) = (y / x) \ y. [para(1210(a,1),1181(a,1)),flip(a)]. 1213 ((x / y) \ x) @ z = (x V y) @ z. [para(1211(a,1),1128(a,1,1))]. 1222 (x @ y) / ((x V z) @ 1) = (x / z) @ y. [para(1213(a,1),1165(a,1,2))]. 1228 (x @ y) / (z @ 1) = (x / z) @ y. [para(1222(a,1),1178(a,1)),flip(a)]. 1229 (x / (y \ x)) @ z = (x V y) @ z. [para(1228(a,1),1164(a,1))]. 1230 (x / (y \ x)) @ (x V y) = x V y # label("Goal 12"). [para(1229(a,2),697(a,1))]. 1231 $F # answer("Goal 12"). [resolve(1230,a,53,a)]. ============================== end of proof ========================== % Disable descendants (x means already disabled): 52x 53 given #1047 (H,wt=13): 1230 (x / (y \ x)) @ (x V y) = x V y # label("Goal 12"). [para(1229(a,2),697(a,1))]. given #1048 (H,wt=15): 1224 (x V y) @ (x / (y \ x)) = x / (y \ x). [para(1223(a,1),1164(a,1)),flip(a)]. given #1049 (H,wt=15): 1227 (x / (y @ 1)) @ z = (x / (y @ u)) @ z. [para(750(a,1),1226(a,1,1,2))]. given #1050 (H,wt=13): 1232 (x / (y @ z)) @ u = (x / y) @ u. [para(1227(a,1),1226(a,1))]. given #1051 (H,wt=17): 1234 ((x @ y) / z) @ (y / u) = (x @ y) / (z @ u). [para(1232(a,1),758(a,1))]. -------- Proof 15 -------- "Goal 7". ============================== PROOF ================================= % Proof 15 at 10.64 (+ 0.24) seconds: "Goal 7". % Length of proof is 1001. % Level of proof is 175. % Maximum clause weight is 41. % Given clauses 1051. 7 (x / y) @ (z / u) = (x @ z) / (y @ u) # label("Goal 7") # label(non_clause) # label(goal). [goal]. 16 (x ^ y) ^ z = x ^ (y ^ z) # label("Associativity-^"). [assumption]. 17 (x V y) V z = x V (y V z) # label("Associativity-V"). [assumption]. 18 x ^ x = x # label("Idempotence-^"). [assumption]. 19 x V x = x # label("Idempotence-V"). [assumption]. 20 x ^ y = y ^ x # label("Commutativity-^"). [assumption]. 21 x V y = y V x # label("Commutativity-V"). [assumption]. 22 (x ^ y) V x = x # label("Absorption a"). [assumption]. 23 (x V y) ^ x = x # label("Absorption b"). [assumption]. 24 (x * ((x \ y) ^ z)) V y = y # label("Residual a"). [assumption]. 25 ((x ^ (y / z)) * z) V y = y # label("Residual b"). [assumption]. 26 (x \ ((x * y) V z)) ^ y = y # label("Residual c"). [assumption]. 27 (((x * y) V z) / y) ^ x = x # label("Residual d"). [assumption]. 28 (x * y) * z = x * (y * z) # label("Associativity-* (fusion)"). [assumption]. 29 1 * x = x # label("Left monoid unit"). [assumption]. 30 x * 1 = x # label("Right monoid unit"). [assumption]. 31 x V y = x / ((x V y) \ x) # label("GMV a"). [assumption]. 32 x / ((x V y) \ x) = x V y. [copy(31),flip(a)]. 33 x V y = (x / (x V y)) \ x # label("GMV b"). [assumption]. 34 (x / (x V y)) \ x = x V y. [copy(33),flip(a)]. 35 x @ y = (x * (x \ 1)) * ((y \ 1) \ 1) # label("Definition-@"). [assumption]. 45 (c20 @ c22) / (c21 @ c23) != (c20 / c21) @ (c22 / c23) # label("Goal 7") # answer("Goal 7"). [deny(7)]. 46 (c20 / c21) @ (c22 / c23) != (c20 @ c22) / (c21 @ c23) # answer("Goal 7"). [copy(45),flip(a)]. 60 x ^ (y ^ z) = z ^ (x ^ y). [para(20(a,1),16(a,1)),flip(a)]. 61 (x ^ y) ^ z = x ^ (z ^ y). [para(20(a,1),16(a,2,2))]. 62 (x V y) V z = y V (x V z). [para(21(a,1),17(a,1,1))]. 63 x V (y V z) = z V (x V y). [para(21(a,1),17(a,1)),flip(a)]. 64 (x V y) V z = x V (z V y). [para(21(a,1),17(a,2,2))]. 65 (x ^ y) V y = y. [para(20(a,1),22(a,1,1))]. 66 x V (x ^ y) = x. [para(22(a,1),21(a,1)),flip(a)]. 67 x ^ (x V y) = x. [para(23(a,1),20(a,1)),flip(a)]. 68 (x V y) ^ y = y. [para(21(a,1),23(a,1,1))]. 69 (x * (y ^ (x \ z))) V z = z. [para(20(a,1),24(a,1,1,2))]. 70 x V (y * ((y \ x) ^ z)) = x. [para(24(a,1),21(a,1)),flip(a)]. 71 x V ((y ^ (x / z)) * z) = x. [para(25(a,1),21(a,1)),flip(a)]. 72 x ^ (y \ ((y * x) V z)) = x. [para(26(a,1),20(a,1)),flip(a)]. 73 (x \ (y V (x * z))) ^ z = z. [para(21(a,1),26(a,1,1,2))]. 74 x ^ (((x * y) V z) / y) = x. [para(27(a,1),20(a,1)),flip(a)]. 75 ((x V (y * z)) / z) ^ y = y. [para(21(a,1),27(a,1,1,1))]. 77 x / ((x V y) \ x) = y V x. [para(32(a,2),21(a,1))]. 78 x / ((y V x) \ x) = x V y. [para(21(a,1),32(a,1,2,1))]. 79 (x ^ y) / (x \ (x ^ y)) = (x ^ y) V x. [para(22(a,1),32(a,1,2,1))]. 80 (x / (x V x)) \ x = x. [para(34(a,2),19(a,1))]. 81 (x / (x V y)) \ x = y V x. [para(34(a,2),21(a,1))]. 82 ((x ^ y) / x) \ (x ^ y) = (x ^ y) V x. [para(22(a,1),34(a,1,1,2))]. 83 x @ y = x * ((x \ 1) * ((y \ 1) \ 1)). [para(35(a,2),28(a,1))]. 84 (1 \ 1) * ((x \ 1) \ 1) = 1 @ x. [para(29(a,1),35(a,2,1)),flip(a)]. 85 x V (y ^ x) = x. [para(65(a,1),21(a,1)),flip(a)]. 86 x V ((x ^ y) ^ z) = x. [para(16(a,2),66(a,1,2))]. 87 (x V y) V (y ^ z) = x V y. [para(66(a,1),17(a,2,2))]. 88 x ^ (y V x) = x. [para(21(a,1),67(a,1,2))]. 89 x ^ (y V (z V x)) = x. [para(17(a,1),88(a,1,2))]. 90 (x / x) \ x = x. [para(19(a,1),80(a,1,1,2))]. 91 x V (y ^ (x ^ z)) = x. [para(20(a,1),86(a,1,2))]. 92 x ^ (y V (x V z)) = x. [para(21(a,1),89(a,1,2,2))]. 93 x ^ (y ^ (x V z)) = y ^ x. [para(23(a,1),60(a,1,2)),flip(a)]. 94 x V (y V z) = y V (x V z). [para(62(a,1),17(a,1))]. 96 (x ^ y) V (z V x) = z V x. [para(66(a,1),63(a,1,2)),flip(a)]. 97 (x ^ y) V (z V y) = z V y. [para(85(a,1),63(a,1,2)),flip(a)]. 98 x V (y * (z ^ (y \ x))) = x. [para(69(a,1),21(a,1)),flip(a)]. 99 x V (y * (y \ x)) = x. [para(18(a,1),70(a,1,2,2))]. 100 x V (y * (z * ((y * z) \ x))) = x. [para(28(a,1),99(a,1,2))]. 101 x V (1 \ x) = x. [para(29(a,1),99(a,1,2))]. 102 x ^ (y * (y \ x)) = y * (y \ x). [para(99(a,1),68(a,1,1))]. 103 x ^ (1 \ x) = 1 \ x. [para(101(a,1),68(a,1,1))]. 104 x V ((y ^ (z ^ (x / u))) * u) = x. [para(16(a,1),71(a,1,2,1))]. 105 x V ((x / y) * y) = x. [para(18(a,1),71(a,1,2,1))]. 106 x V (((x / y) ^ z) * y) = x. [para(20(a,1),71(a,1,2,1))]. 107 x V (x / 1) = x. [para(30(a,1),105(a,1,2))]. 108 x ^ ((x / y) * y) = (x / y) * y. [para(105(a,1),68(a,1,1))]. 109 x ^ (x / 1) = x / 1. [para(107(a,1),68(a,1,1))]. 110 x ^ (y \ (y * x)) = x. [para(19(a,1),72(a,1,2,2))]. 111 x ^ (y \ (z V (y * x))) = x. [para(21(a,1),72(a,1,2,2))]. 112 x ^ ((y ^ (z / x)) \ z) = x. [para(25(a,1),72(a,1,2,2))]. 113 1 ^ (x \ (x V y)) = 1. [para(30(a,1),72(a,1,2,2,1))]. 114 x ^ ((y * z) \ (y * (z * x))) = x. [para(28(a,1),110(a,1,2,2))]. 115 x ^ (1 \ x) = x. [para(29(a,1),110(a,1,2,2))]. 116 1 ^ (x \ x) = 1. [para(30(a,1),110(a,1,2,2))]. 117 x V (y \ (y * x)) = y \ (y * x). [para(110(a,1),65(a,1,1))]. 118 (x \ (x * y)) V y = x \ (x * y). [para(110(a,1),85(a,1,2))]. 119 1 \ x = x. [para(115(a,1),103(a,1)),flip(a)]. 120 (x ^ 1) ^ (y \ y) = x ^ 1. [para(116(a,1),16(a,2,2))]. 121 1 V (x \ x) = x \ x. [para(116(a,1),65(a,1,1))]. 122 1 ^ ((x V y) \ (x V (y V z))) = 1. [para(17(a,1),113(a,1,2,2))]. 123 1 ^ (x \ (y V x)) = 1. [para(21(a,1),113(a,1,2,2))]. 124 1 ^ ((x ^ y) \ x) = 1. [para(22(a,1),113(a,1,2,2))]. 125 1 V (x \ (x V y)) = x \ (x V y). [para(113(a,1),65(a,1,1))]. 126 1 ^ ((x ^ y) \ y) = 1. [para(65(a,1),113(a,1,2,2))]. 127 (x \ (x V y)) V 1 = x \ (x V y). [para(113(a,1),85(a,1,2))]. 128 1 ^ (x V (y \ y)) = 1. [para(121(a,1),92(a,1,2,2))]. 129 1 ^ ((x * (x \ y)) \ y) = 1. [para(99(a,1),123(a,1,2,2))]. 130 1 V ((x ^ y) \ x) = (x ^ y) \ x. [para(124(a,1),65(a,1,1))]. 131 1 ^ ((x \ x) V y) = 1. [para(21(a,1),128(a,1,2))]. 132 1 ^ (x \ (y \ (z V (y * x)))) = 1. [para(73(a,1),124(a,1,2,1))]. 133 x ^ ((x * y) / y) = x. [para(19(a,1),74(a,1,2,1))]. 134 x ^ ((y V (x * z)) / z) = x. [para(21(a,1),74(a,1,2,1))]. 135 x ^ (y / ((x \ y) ^ z)) = x. [para(24(a,1),74(a,1,2,1))]. 136 1 ^ ((x V y) / x) = 1. [para(29(a,1),74(a,1,2,1,1))]. 137 x ^ (y / (z ^ (x \ y))) = x. [para(69(a,1),74(a,1,2,1))]. 138 (x * y) ^ ((x * (y * z)) / z) = x * y. [para(28(a,1),133(a,1,2,1))]. 139 1 ^ (x / x) = 1. [para(29(a,1),133(a,1,2,1))]. 140 x ^ (x / 1) = x. [para(30(a,1),133(a,1,2,1))]. 141 x V ((x * y) / y) = (x * y) / y. [para(133(a,1),65(a,1,1))]. 142 (x ^ 1) ^ (y / y) = x ^ 1. [para(139(a,1),16(a,2,2))]. 143 1 V (x / x) = x / x. [para(139(a,1),65(a,1,1))]. 144 (x / x) V 1 = x / x. [para(139(a,1),85(a,1,2))]. 146 x / 1 = x. [para(140(a,1),109(a,1)),flip(a)]. 147 1 ^ (x / (x ^ y)) = 1. [para(22(a,1),136(a,1,2,1))]. 148 (x / (y \ x)) ^ y = y. [para(99(a,1),75(a,1,1,1))]. 149 x ^ (y / (x \ y)) = x. [para(148(a,1),20(a,1)),flip(a)]. 150 x V (y / (x \ y)) = y / (x \ y). [para(148(a,1),22(a,1,1))]. 151 x / ((y V x) \ x) = y V x. [para(21(a,1),77(a,1,2,1))]. 152 (x / (y V x)) \ x = y V x. [para(21(a,1),81(a,1,1,2))]. 153 x V (y * (z * (u ^ ((y * z) \ x)))) = x. [para(28(a,1),98(a,1,2))]. 154 x ^ ((y * z) \ (u V (y * (z * x)))) = x. [para(28(a,1),111(a,1,2,2,2))]. 155 x ^ ((y / x) \ y) = x. [para(105(a,1),111(a,1,2,2))]. 156 x ^ (((y / x) ^ z) \ y) = x. [para(106(a,1),111(a,1,2,2))]. 157 x V ((y / x) \ y) = (y / x) \ y. [para(155(a,1),65(a,1,1))]. 158 ((x / y) \ x) V y = (x / y) \ x. [para(155(a,1),85(a,1,2))]. 159 1 ^ (x \ ((y / x) \ y)) = 1. [para(155(a,1),126(a,1,2,1))]. 160 x V (x * (y ^ 1)) = x. [para(120(a,1),98(a,1,2,2))]. 161 x V (x * (1 ^ y)) = x. [para(20(a,1),160(a,1,2,2))]. 162 1 V ((x * (x \ y)) \ y) = (x * (x \ y)) \ y. [para(129(a,1),65(a,1,1))]. 163 (x * y) ^ ((z V (x * (y * u))) / u) = x * y. [para(28(a,1),134(a,1,2,1,2))]. 164 x V (y / (z ^ (x \ y))) = y / (z ^ (x \ y)). [para(137(a,1),65(a,1,1))]. 165 1 ^ (x \ (y / (z ^ (x \ y)))) = 1. [para(137(a,1),126(a,1,2,1))]. 166 x V ((y ^ 1) * x) = x. [para(142(a,1),71(a,1,2,1))]. 167 x ^ ((y ^ 1) \ x) = x. [para(142(a,1),112(a,1,2,1))]. 168 x V ((1 ^ y) * x) = x. [para(20(a,1),166(a,1,2,1))]. 169 x ^ ((1 ^ y) \ x) = x. [para(20(a,1),167(a,1,2,1))]. 170 x V ((y ^ 1) \ x) = (y ^ 1) \ x. [para(167(a,1),65(a,1,1))]. 171 (x * (x \ y)) / ((y V (x * (x \ y))) \ (x * (x \ y))) = y. [para(151(a,2),99(a,1))]. 172 ((x * (x \ y)) / (y V (x * (x \ y)))) \ (x * (x \ y)) = y. [para(152(a,2),99(a,1))]. 173 1 V (x \ ((y / x) \ y)) = x \ ((y / x) \ y). [para(159(a,1),65(a,1,1))]. 174 1 * ((x \ 1) \ 1) = 1 @ x. [para(119(a,1),84(a,1,1))]. 175 (x \ 1) \ 1 = 1 @ x. [para(174(a,1),29(a,1)),flip(a)]. 176 (x \ y) ^ (z * ((x * z) \ y)) = z * ((x * z) \ y). [para(100(a,1),73(a,1,1,2))]. 177 x V (y * (z * (z \ (y \ x)))) = x. [para(102(a,1),70(a,1,2,2))]. 178 x V (x * (y * (y \ 1))) = x. [para(102(a,1),161(a,1,2,2))]. 179 x ^ (y / (z * (z \ (x \ y)))) = x. [para(102(a,1),135(a,1,2,2))]. 180 x V ((y * (y \ 1)) * x) = x. [para(102(a,1),168(a,1,2,1))]. 181 x ^ ((y * (y \ (z / x))) \ z) = x. [para(102(a,1),156(a,1,2,1))]. 182 x V (y * ((y \ 1) * x)) = x. [para(28(a,1),180(a,1,2))]. 183 (x \ y) ^ ((x \ 1) * y) = (x \ 1) * y. [para(182(a,1),73(a,1,1,2))]. 184 x V ((y ^ z) * (z \ x)) = x. [para(149(a,1),104(a,1,2,1,2))]. 185 x V (y * ((y V z) \ x)) = x. [para(67(a,1),184(a,1,2,1))]. 186 x V (y * ((z V y) \ x)) = x. [para(88(a,1),184(a,1,2,1))]. 187 x V ((1 V y) \ x) = x. [para(29(a,1),185(a,1,2))]. 188 (x \ y) ^ ((x V z) \ y) = (x V z) \ y. [para(185(a,1),73(a,1,1,2))]. 189 x V ((y V 1) \ x) = x. [para(21(a,1),187(a,1,2,1))]. 190 x V ((y / y) \ x) = x. [para(143(a,1),187(a,1,2,1))]. 191 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y V ((1 V x) \ y). [para(187(a,1),152(a,1,1,2))]. 192 x ^ ((y V 1) \ x) = (y V 1) \ x. [para(189(a,1),68(a,1,1))]. 193 x V (((y / z) * z) * (y \ x)) = x. [para(105(a,1),186(a,1,2,2,1))]. 194 x V (((x ^ y) / z) * z) = x. [para(108(a,1),86(a,1,2))]. 195 x V (y ^ ((x / z) * z)) = x. [para(108(a,1),91(a,1,2,2))]. 196 x V (y * (((y \ x) / z) * z)) = x. [para(108(a,1),70(a,1,2,2))]. 197 x V ((((x / y) / z) * z) * y) = x. [para(108(a,1),106(a,1,2,1))]. 198 x V (x * ((1 / y) * y)) = x. [para(108(a,1),161(a,1,2,2))]. 199 x ^ (y / (((x \ y) / z) * z)) = x. [para(108(a,1),135(a,1,2,2))]. 200 x V (((1 / y) * y) * x) = x. [para(108(a,1),168(a,1,2,1))]. 201 x V (((y ^ x) / z) * z) = x. [para(20(a,1),194(a,1,2,1,1))]. 202 x ^ (((y ^ z) / x) \ y) = x. [para(194(a,1),111(a,1,2,2))]. 203 x V ((1 / y) * (y * x)) = x. [para(28(a,1),200(a,1,2))]. 204 x ^ (((y ^ z) / x) \ z) = x. [para(201(a,1),111(a,1,2,2))]. 205 x ^ ((y / x) \ (y V z)) = x. [para(23(a,1),202(a,1,2,1,1))]. 206 (1 / x) ^ (y / (x * y)) = 1 / x. [para(203(a,1),134(a,1,2,1))]. 207 (x / y) ^ ((x V z) / y) = x / y. [para(205(a,1),137(a,1,2,2))]. 208 x ^ (y V (z \ (z * x))) = x. [para(117(a,1),92(a,1,2,2))]. 209 1 ^ (x \ (x * (y \ y))) = 1. [para(117(a,1),131(a,1,2))]. 210 x ^ ((1 V y) * x) = x. [para(187(a,1),208(a,1,2))]. 211 x ^ ((y V 1) * x) = x. [para(189(a,1),208(a,1,2))]. 212 x ^ ((y / y) * x) = x. [para(190(a,1),208(a,1,2))]. 213 x V ((y V 1) * x) = (y V 1) * x. [para(211(a,1),65(a,1,1))]. 214 (x / x) * x = x. [para(212(a,1),108(a,1)),flip(a)]. 215 x ^ ((x * (y \ y)) / 1) = x. [para(209(a,1),137(a,1,2,2))]. 216 x ^ (x * (y \ y)) = x. [para(146(a,1),215(a,1,2))]. 217 x * (x \ x) = x. [para(216(a,1),102(a,1)),flip(a)]. 218 (x \ x) ^ ((y * x) \ (y * x)) = x \ x. [para(217(a,1),114(a,1,2,2,2))]. 219 x ^ (y \ (y * (z V x))) = x. [para(118(a,1),89(a,1,2))]. 220 x ^ ((x * (y V z)) / z) = x. [para(219(a,1),137(a,1,2,2))]. 221 x ^ ((x * y) / (z * (z \ y))) = x. [para(99(a,1),220(a,1,2,1,2))]. 222 x ^ (x * (y V 1)) = x. [para(146(a,1),220(a,1,2))]. 223 x ^ (x * (1 V y)) = x. [para(21(a,1),222(a,1,2,2))]. 224 x V (x / (y V 1)) = x. [para(222(a,1),195(a,1,2))]. 225 x ^ (x / (y V 1)) = x / (y V 1). [para(224(a,1),68(a,1,1))]. 226 x V (x / (y / y)) = x. [para(144(a,1),224(a,1,2,2))]. 227 (x / (y V 1)) / (x \ (x / (y V 1))) = x V (x / (y V 1)). [para(224(a,1),151(a,1,2,1))]. 228 x ^ (x / (y / y)) = x / (y / y). [para(226(a,1),68(a,1,1))]. 229 x ^ (x * (y \ (y V z))) = x. [para(125(a,1),223(a,1,2,2))]. 230 1 ^ ((x V 1) \ (x V ((y ^ z) \ y))) = 1. [para(130(a,1),122(a,1,2,2,2))]. 231 1 ^ (((x V y) \ z) \ (x \ z)) = 1. [para(185(a,1),132(a,1,2,2,2))]. 232 x ^ ((((x * y) * z) / z) / y) = x. [para(141(a,1),74(a,1,2,1))]. 233 x ^ ((y / (x V z)) \ y) = x. [para(158(a,1),92(a,1,2))]. 234 x V ((x / (y V z)) * y) = x. [para(233(a,1),98(a,1,2,2))]. 235 (x / y) ^ (x / (y V z)) = x / (y V z). [para(234(a,1),75(a,1,1,1))]. 236 x V ((y / z) * (z * (y \ x))) = x. [para(28(a,1),193(a,1,2))]. 237 x V (((x / y) / z) * (z * y)) = x. [para(28(a,1),197(a,1,2))]. 238 x ^ (((x * (y * z)) / z) / y) = x. [para(28(a,1),232(a,1,2,1,1))]. 239 x ^ (((x * y) / y) / (y / y)) = x. [para(214(a,1),238(a,1,2,1,1,2))]. 240 (x / x) ^ ((x / x) / (x / x)) = x / x. [para(214(a,1),239(a,1,2,1,1))]. 241 (x ^ y) / (x \ (x ^ y)) = x. [para(79(a,2),22(a,1))]. 242 (x ^ y) / (y \ (y ^ x)) = y. [para(20(a,1),241(a,1,1))]. 243 (x ^ y) / (y \ (x ^ y)) = y. [para(20(a,1),242(a,1,2,2))]. 244 1 / ((x \ (x V y)) \ (1 ^ (x \ (x V y)))) = x \ (x V y). [para(113(a,1),243(a,1,1))]. 245 1 / (((x ^ y) \ y) \ (1 ^ ((x ^ y) \ y))) = (x ^ y) \ y. [para(126(a,1),243(a,1,1))]. 246 x / ((x * (y V 1)) \ (x ^ (x * (y V 1)))) = x * (y V 1). [para(222(a,1),243(a,1,1))]. 247 ((x ^ y) / x) \ (x ^ y) = x. [para(82(a,2),22(a,1))]. 248 ((x ^ y) / y) \ (y ^ x) = y. [para(20(a,1),247(a,1,1,1))]. 249 (1 ^ x) \ (1 ^ x) = 1. [para(146(a,1),247(a,1,1))]. 250 ((1 / x) * x) \ (1 ^ ((1 / x) * x)) = 1. [para(108(a,1),249(a,1,1))]. 251 ((x ^ y) / y) \ (x ^ y) = y. [para(20(a,1),248(a,1,2))]. 252 (x / ((1 V y) * x)) \ (x ^ ((1 V y) * x)) = (1 V y) * x. [para(210(a,1),251(a,1,1,1))]. 253 x * ((x \ 1) * (1 @ y)) = x @ y. [para(175(a,1),83(a,2,2,2)),flip(a)]. 254 (x \ 1) * ((1 @ x) * (1 @ y)) = (x \ 1) @ y. [para(175(a,1),253(a,1,2,1))]. 255 (1 @ x) V (y @ x) = 1 @ x. [para(253(a,1),182(a,1,2))]. 256 (1 @ x) ^ (y @ x) = y @ x. [para(255(a,1),68(a,1,1))]. 257 (x @ y) ^ (1 @ y) = x @ y. [para(255(a,1),88(a,1,2))]. 258 x ^ (y / (z * (u ^ ((x * z) \ y)))) = x. [para(153(a,1),134(a,1,2,1))]. 259 x ^ (((y ^ (z / (u * x))) * u) \ z) = x. [para(71(a,1),154(a,1,2,2))]. 260 (x \ 1) ^ ((y * x) \ y) = x \ 1. [para(178(a,1),154(a,1,2,2))]. 261 x ^ ((y * (1 / x)) \ y) = x. [para(198(a,1),154(a,1,2,2))]. 262 (x \ (y \ z)) ^ ((y * x) \ z) = x \ (y \ z). [para(177(a,1),154(a,1,2,2))]. 263 1 ^ ((x * (1 / y)) \ (x / y)) = 1. [para(261(a,1),165(a,1,2,2,2))]. 264 (x / y) V ((x * (1 / y)) * 1) = x / y. [para(263(a,1),98(a,1,2,2))]. 265 ((x \ 1) * y) V (x \ y) = x \ y. [para(183(a,1),22(a,1,1))]. 266 (x \ y) V ((x \ 1) * y) = x \ y. [para(183(a,1),66(a,1,2))]. 267 x ^ ((y \ 1) \ (y \ x)) = x. [para(265(a,1),72(a,1,2,2))]. 268 (x \ 1) ^ ((x \ y) / y) = x \ 1. [para(265(a,1),74(a,1,2,1))]. 269 (x \ y) V ((x V z) \ y) = x \ y. [para(188(a,1),66(a,1,2))]. 270 1 ^ ((x \ y) / ((x V z) \ y)) = 1. [para(188(a,1),147(a,1,2,2))]. 271 1 ^ ((x V 1) \ ((x ^ 1) \ x)) = 1. [para(170(a,1),230(a,1,2,2))]. 272 (x V 1) ^ (((x ^ 1) \ x) / 1) = x V 1. [para(271(a,1),137(a,1,2,2))]. 273 (x / x) / (x / x) = x / x. [para(240(a,1),228(a,1)),flip(a)]. 274 (x / x) \ (x / x) = x / x. [para(273(a,1),90(a,1,1))]. 275 ((1 / x) * x) \ ((1 / x) * x) = 1. [para(108(a,1),250(a,1,2))]. 276 (x \ x) ^ 1 = x \ x. [para(275(a,1),218(a,1,2))]. 277 1 V (x \ x) = 1. [para(276(a,1),85(a,1,2))]. 278 x \ x = 1. [para(277(a,1),121(a,1)),flip(a)]. 281 (x \ x) \ y = y. [para(278(a,2),119(a,1,1))]. 283 x ^ ((x \ 1) \ 1) = x. [para(278(a,1),267(a,1,2,2))]. 284 (x \ 1) ^ (1 / x) = x \ 1. [para(278(a,1),268(a,1,2,1))]. 285 1 ^ ((x V y) \ x) = (x V y) \ x. [para(278(a,1),188(a,1,1))]. 286 1 V ((x V y) \ x) = x \ x. [para(278(a,1),269(a,1,1))]. 287 x / x = 1. [para(278(a,1),274(a,1)),flip(a)]. 294 (x / (x V y)) ^ 1 = x / (x V y). [para(287(a,1),207(a,1,2))]. 295 1 ^ (x / (x V y)) = x / (x V y). [para(287(a,1),235(a,1,1))]. 299 x * ((x \ 1) * 1) = x @ 1. [para(281(a,1),83(a,2,2,2)),flip(a)]. 300 1 V ((x \ 1) * x) = 1. [para(283(a,1),98(a,1,2,2))]. 301 x ^ (1 @ x) = x. [para(175(a,1),283(a,1,2))]. 302 (1 @ x) ^ x = x. [para(301(a,1),20(a,1)),flip(a)]. 303 x V (1 @ x) = 1 @ x. [para(301(a,1),65(a,1,1))]. 304 (1 @ x) V x = 1 @ x. [para(301(a,1),85(a,1,2))]. 305 x ^ (y ^ (1 @ x)) = x ^ y. [para(301(a,1),61(a,1,1)),flip(a)]. 306 x V (y V (1 @ x)) = y V (1 @ x). [para(301(a,1),97(a,1,1))]. 307 x ^ ((y / x) \ (1 @ y)) = x. [para(301(a,1),204(a,1,2,1,1))]. 308 (1 / x) ^ (x \ 1) = x \ 1. [para(300(a,1),75(a,1,1,1))]. 309 x V (y V (1 @ x)) = (1 @ x) V y. [para(303(a,1),64(a,1,1)),flip(a)]. 310 x ^ (1 @ (y V x)) = x. [para(304(a,1),89(a,1,2))]. 311 x ^ (1 @ (x V y)) = x. [para(304(a,1),92(a,1,2))]. 312 1 ^ (1 @ (x \ (x V y))) = 1. [para(127(a,1),310(a,1,2,2))]. 313 x ^ (1 @ ((y / x) \ y)) = x. [para(158(a,1),310(a,1,2,2))]. 314 x V (1 @ (x V y)) = 1 @ (x V y). [para(311(a,1),65(a,1,1))]. 315 1 V ((x ^ (y \ 1)) * y) = 1. [para(284(a,1),104(a,1,2,1,2))]. 316 1 V ((x V y) \ x) = 1. [para(286(a,2),278(a,1))]. 317 1 V (x \ (x ^ y)) = 1. [para(22(a,1),316(a,1,2,1))]. 318 1 V (x \ (y ^ x)) = 1. [para(65(a,1),316(a,1,2,1))]. 319 1 V ((x V y) \ (y ^ z)) = 1. [para(96(a,1),316(a,1,2,1))]. 321 1 V ((x \ (x ^ y)) ^ z) = 1 V (x \ (x ^ y)). [para(317(a,1),87(a,1,1))]. 322 (x \ (y ^ x)) V 1 = 1. [para(318(a,1),21(a,1)),flip(a)]. 323 1 V ((x ^ (y V z)) \ (x ^ y)) = 1. [para(93(a,1),318(a,1,2,2))]. 324 x * (x \ 1) = x @ 1. [para(30(a,1),299(a,1,2))]. 325 1 V (x @ 1) = 1. [para(299(a,1),182(a,1,2))]. 326 1 ^ (x @ 1) = x @ 1. [para(325(a,1),68(a,1,1))]. 327 x * ((x \ 1) * y) = (x @ 1) * y. [para(324(a,1),28(a,1,1)),flip(a)]. 328 x * (y * ((x * y) \ 1)) = (x * y) @ 1. [para(324(a,1),28(a,1)),flip(a)]. 329 x V ((y @ 1) * x) = x. [para(324(a,1),180(a,1,2,1))]. 330 x ^ ((y @ 1) \ x) = x. [para(326(a,1),169(a,1,2,1))]. 331 x ^ ((y @ 1) * x) = (y @ 1) * x. [para(329(a,1),68(a,1,1))]. 332 1 ^ ((x ^ y) \ (y ^ (1 @ x))) = 1. [para(305(a,1),126(a,1,2,1))]. 333 1 ^ (1 @ ((x ^ y) \ y)) = 1. [para(65(a,1),312(a,1,2,2,2))]. 334 x ^ ((y ^ (x \ 1)) \ 1) = x. [para(315(a,1),111(a,1,2,2))]. 335 (1 / x) ^ (x \ 1) = 1 / x. [para(155(a,1),334(a,1,2,1))]. 336 1 / x = x \ 1. [para(335(a,1),308(a,1))]. 337 (1 / x) \ 1 = 1 @ x. [para(336(a,2),175(a,1,1))]. 338 1 / (x \ 1) = 1 @ x. [para(336(a,2),175(a,1))]. 339 x V (x * (y * (1 / y))) = x. [para(336(a,2),178(a,1,2,2,2))]. 340 x ^ ((1 / y) \ (y \ x)) = x. [para(336(a,2),267(a,1,2,1))]. 341 1 @ (x V 1) = x V 1. [para(337(a,1),152(a,1))]. 342 1 / (1 / x) = 1 @ x. [para(337(a,1),336(a,2))]. 343 (1 / x) ^ ((y * x) \ y) = 1 / x. [para(339(a,1),154(a,1,2,2))]. 344 (1 / x) ^ ((x \ y) / y) = 1 / x. [para(340(a,1),137(a,1,2,2))]. 345 x V (((y V z) \ y) * x) = x. [para(285(a,1),168(a,1,2,1))]. 346 x V ((y / (y V z)) * x) = x. [para(294(a,1),166(a,1,2,1))]. 347 x V (((y ^ z) / y) * x) = x. [para(22(a,1),346(a,1,2,1,2))]. 348 1 V ((x ^ y) / x) = 1. [para(30(a,1),347(a,1,2))]. 349 1 V ((x ^ y) / y) = 1. [para(20(a,1),348(a,1,2,1))]. 350 1 V (1 \ (((x ^ y) / x) ^ z)) = 1. [para(348(a,1),319(a,1,2,1))]. 351 (x @ 1) * (1 @ y) = x @ y. [para(327(a,1),253(a,1))]. 352 (x @ 1) * ((1 @ y) * z) = (x @ y) * z. [para(351(a,1),28(a,1,1)),flip(a)]. 353 1 V ((x ^ (((y @ 1) * x) V z)) \ ((y @ 1) * x)) = 1. [para(331(a,1),323(a,1,2,2))]. 354 1 ^ (x \ ((x V y) ^ (1 @ x))) = 1. [para(67(a,1),332(a,1,2,1))]. 355 1 ^ ((x * y) \ (x / (1 / y))) = 1. [para(343(a,1),165(a,1,2,2,2))]. 356 1 V (((x ^ y) / x) ^ z) = 1. [para(119(a,1),350(a,1,2))]. 357 1 V (x ^ ((y ^ z) / y)) = 1. [para(20(a,1),356(a,1,2))]. 358 1 V (x ^ (((y / z) * z) / y)) = 1. [para(108(a,1),357(a,1,2,2,1))]. 359 1 ^ ((x ^ y) \ (x ^ (1 @ (x ^ y)))) = 1. [para(22(a,1),354(a,1,2,2,1))]. 360 1 V ((x / (y * x)) * y) = 1. [para(138(a,1),358(a,1,2))]. 361 (1 / x) ^ (y / (x * y)) = y / (x * y). [para(360(a,1),75(a,1,1,1))]. 362 x ^ ((x / (1 / y)) / (y * 1)) = x. [para(355(a,1),258(a,1,2,2,2))]. 363 x ^ ((x / (1 / y)) / y) = x. [para(30(a,1),362(a,1,2,2))]. 364 x ^ ((x / (1 @ y)) / (y \ 1)) = x. [para(338(a,1),363(a,1,2,1,2))]. 365 x V ((x / (1 @ y)) / (y \ 1)) = (x / (1 @ y)) / (y \ 1). [para(364(a,1),65(a,1,1))]. 366 (x \ y) ^ ((z * x) \ (z * y)) = x \ y. [para(221(a,1),259(a,1,2,1,1))]. 367 (x / y) V (x * (1 / y)) = x / y. [para(30(a,1),264(a,1,2))]. 368 (x \ (x / y)) ^ (1 / y) = 1 / y. [para(367(a,1),73(a,1,1,2))]. 369 (x / y) V (x * (y \ 1)) = x / y. [para(336(a,1),367(a,1,2,2))]. 370 (x \ 1) ^ (y \ (y / x)) = x \ 1. [para(369(a,1),111(a,1,2,2))]. 371 x ^ ((x / y) / (y \ 1)) = x. [para(369(a,1),134(a,1,2,1))]. 372 1 V ((x \ 1) / (y \ (y / x))) = 1. [para(370(a,1),349(a,1,2,1))]. 373 (x V 1) ^ ((x ^ 1) \ x) = x V 1. [para(146(a,1),272(a,1,2))]. 374 (x ^ 1) ^ (x / (x V 1)) = x ^ 1. [para(373(a,1),137(a,1,2,2))]. 375 x ^ (1 ^ (x / (x V 1))) = x ^ 1. [para(374(a,1),16(a,1)),flip(a)]. 376 x ^ (x / (x V 1)) = x ^ 1. [para(295(a,1),375(a,1,2))]. 377 x / (x V 1) = x ^ 1. [para(376(a,1),225(a,1)),flip(a)]. 378 x / (1 V x) = x ^ 1. [para(21(a,1),377(a,1,2))]. 379 x / (y * x) = 1 / y. [para(361(a,1),206(a,1))]. 381 x / (y * x) = y \ 1. [para(379(a,2),336(a,1))]. 382 x / ((y \ 1) * x) = 1 @ y. [para(379(a,2),338(a,1))]. 383 x ^ (((x \ y) / y) \ 1) = x. [para(381(a,1),199(a,1,2))]. 384 ((x * y) \ 1) ^ (y \ (x \ 1)) = (x * y) \ 1. [para(381(a,1),370(a,1,2,2))]. 385 (x @ y) ^ (z / ((y \ 1) * z)) = x @ y. [para(382(a,2),257(a,1,2))]. 386 (((x \ y) / y) * x) V 1 = 1. [para(383(a,1),69(a,1,1,2))]. 387 (1 / x) ^ ((x \ y) / y) = (x \ y) / y. [para(386(a,1),27(a,1,1,1))]. 388 ((x \ 1) * y) ^ ((z @ x) \ y) = (x \ 1) * y. [para(385(a,1),112(a,1,2,1))]. 389 (x \ y) / y = 1 / x. [para(387(a,1),344(a,1))]. 390 (x \ y) / y = x \ 1. [para(389(a,2),336(a,1))]. 391 ((x \ y) / y) \ 1 = 1 @ x. [para(389(a,2),337(a,1,1))]. 393 x ^ ((y \ 1) \ (1 @ (y \ x))) = x. [para(390(a,1),307(a,1,2,1))]. 394 x / ((y / (z ^ (x \ y))) \ x) = x V (y / (z ^ (x \ y))). [para(164(a,1),32(a,1,2,1))]. 395 1 V ((x \ (x ^ y)) ^ z) = 1. [para(321(a,2),317(a,1))]. 397 (x \ (y \ 1)) ^ ((y * x) \ 1) = (y * x) \ 1. [para(384(a,1),20(a,1)),flip(a)]. 398 (x * y) \ 1 = y \ (x \ 1). [para(397(a,1),262(a,1))]. 399 (x \ (y \ 1)) \ 1 = 1 @ (y * x). [para(398(a,1),175(a,1,1))]. 400 ((x \ 1) * y) \ 1 = y \ (1 @ x). [para(175(a,1),398(a,2,2))]. 401 (x * (x \ 1)) \ 1 = 1. [para(398(a,2),278(a,1))]. 404 x * (y * (y \ (x \ 1))) = (x * y) @ 1. [para(398(a,1),328(a,1,2,2))]. 405 (x @ 1) \ 1 = 1. [para(324(a,1),401(a,1,1))]. 406 ((x @ 1) * y) \ 1 = y \ 1. [para(405(a,1),398(a,2,2))]. 409 1 @ ((x \ 1) * y) = (y \ (1 @ x)) \ 1. [para(175(a,1),399(a,1,1,2)),flip(a)]. 410 ((x \ 1) * y) * (y \ (1 @ x)) = ((x \ 1) * y) @ 1. [para(400(a,1),324(a,1,2))]. 413 (x \ y) V (z * ((x * z) \ y)) = x \ y. [para(176(a,1),66(a,1,2))]. 414 1 V (x * ((y * x) \ (y ^ z))) = 1. [para(176(a,1),395(a,1,2))]. 415 1 V (x * ((y * x) \ y)) = 1. [para(18(a,1),414(a,1,2,2,2))]. 416 (x \ 1) ^ ((y * x) \ y) = (y * x) \ y. [para(415(a,1),73(a,1,1,2))]. 417 (x \ (y \ z)) ^ ((y * x) \ z) = (y * x) \ z. [para(413(a,1),73(a,1,1,2))]. 419 (x * y) \ x = y \ 1. [para(416(a,1),260(a,1))]. 420 x ^ ((y \ (y / x)) \ 1) = x. [para(419(a,1),181(a,1,2))]. 422 (x * y) \ x = 1 / y. [para(419(a,2),336(a,2)),flip(a)]. 423 (x \ 1) / y = (y * x) \ 1. [para(419(a,1),390(a,1,1))]. 424 ((x \ 1) / y) \ 1 = 1 @ (y * x). [para(419(a,1),391(a,1,1,1))]. 425 (1 / x) / y = 1 / (y * x). [para(422(a,1),389(a,1,1))]. 426 (x \ (x / y)) ^ (1 / y) = x \ (x / y). [para(420(a,1),137(a,1,2,2))]. 427 (x \ 1) / y = x \ (y \ 1). [para(423(a,2),398(a,1))]. 428 (x \ 1) / (y \ 1) = x \ (1 @ y). [para(423(a,2),400(a,1))]. 429 (x \ 1) \ (y \ 1) = (1 @ x) / y. [para(175(a,1),427(a,1,1)),flip(a)]. 430 x \ ((x \ 1) \ 1) = 1. [para(427(a,1),287(a,1))]. 431 1 V (x \ ((y \ (y / x)) \ 1)) = 1. [para(427(a,1),372(a,1,2))]. 432 x \ (1 @ x) = 1. [para(175(a,1),430(a,1,2))]. 433 1 / ((x \ 1) \ 1) = x \ 1. [para(430(a,1),390(a,1,1))]. 434 1 ^ (((x V y) \ (1 @ x)) \ 1) = 1. [para(432(a,1),231(a,1,2,2))]. 435 1 / (1 @ x) = x \ 1. [para(432(a,1),390(a,1,1))]. 436 (1 @ x) \ 1 = x \ 1. [para(435(a,1),336(a,1)),flip(a)]. 437 ((x \ 1) \ 1) \ 1 = x \ 1. [para(175(a,2),436(a,1,1))]. 438 ((x @ 1) * (1 @ y)) \ 1 = y \ 1. [para(436(a,1),406(a,2))]. 439 1 @ (x \ 1) = x \ 1. [para(433(a,1),338(a,1)),flip(a)]. 441 1 @ (1 / x) = x \ 1. [para(336(a,2),439(a,1,2))]. 442 (x @ 1) * (y \ 1) = x @ (y \ 1). [para(439(a,1),351(a,1,2))]. 443 1 @ ((x * y) \ x) = y \ 1. [para(419(a,2),439(a,1,2))]. 444 (x @ y) \ 1 = y \ 1. [para(351(a,1),438(a,1,1))]. 445 (x \ 1) \ 1 = 1 @ (y @ x). [para(444(a,1),175(a,1,1))]. 446 (x @ y) * (y \ 1) = (x @ y) @ 1. [para(444(a,1),324(a,1,2))]. 447 1 / (x @ y) = y \ 1. [para(444(a,1),336(a,2))]. 450 (x \ (y \ 1)) \ 1 = 1 @ ((z @ y) * x). [para(444(a,1),399(a,1,1,2))]. 451 (x @ y) * (z * (z \ (y \ 1))) = ((x @ y) * z) @ 1. [para(444(a,1),404(a,1,2,2,2))]. 452 (x @ y) \ (z \ 1) = (y \ 1) / z. [para(444(a,1),427(a,1,1)),flip(a)]. 453 1 / (x @ y) = 1 / y. [para(447(a,2),336(a,2)),flip(a)]. 454 1 @ (x @ y) = 1 @ y. [para(445(a,1),175(a,1))]. 455 x * ((x \ 1) * (1 @ (y @ z))) = x @ z. [para(445(a,1),83(a,2,2,2)),flip(a)]. 456 (x @ y) \ (1 @ y) = 1. [para(454(a,1),432(a,1,2))]. 457 (x @ y) \ (z \ 1) = (z * y) \ 1. [para(452(a,2),423(a,1))]. 458 (x @ y) \ (z \ 1) = y \ (z \ 1). [para(452(a,2),427(a,1))]. 459 (x @ 1) \ (y \ 1) = y \ 1. [para(30(a,1),457(a,2,1))]. 460 (x @ y) \ (1 @ z) = ((z \ 1) * y) \ 1. [para(175(a,1),457(a,1,2))]. 461 x * ((y @ 1) * (x \ 1)) = (x * (y @ 1)) @ 1. [para(459(a,1),404(a,1,2,2))]. 462 x \ (x / y) = 1 / y. [para(426(a,1),368(a,1))]. 463 x \ (x / y) = y \ 1. [para(462(a,2),336(a,1))]. 464 x \ (x / (y \ 1)) = 1 @ y. [para(462(a,2),338(a,1))]. 466 1 / (x * y) = y \ (1 / x). [para(379(a,1),462(a,1,2)),flip(a)]. 467 1 @ (x \ (x / y)) = y \ 1. [para(462(a,2),441(a,1,2))]. 468 (x V 1) \ 1 = x \ (x ^ 1). [para(377(a,1),463(a,1,2)),flip(a)]. 469 (x \ 1) / (y / x) = y \ 1. [para(463(a,1),390(a,1,1))]. 470 (x \ y) \ (x \ 1) = y \ 1. [para(390(a,1),463(a,1,2))]. 471 (x \ (y \ 1)) \ (x \ 1) = 1 @ y. [para(390(a,1),464(a,1,2))]. 472 1 / ((x @ y) * z) = z \ (1 / y). [para(453(a,1),466(a,2,2))]. 473 (x \ 1) / (y \ 1) = (y \ x) \ 1. [para(390(a,1),469(a,1,2))]. 474 x \ ((y / x) \ 1) = y \ 1. [para(469(a,1),427(a,1)),flip(a)]. 475 (x \ 1) \ (y \ 1) = (y / x) \ 1. [para(469(a,1),463(a,1,2))]. 476 (x \ y) \ 1 = y \ (1 @ x). [para(391(a,1),474(a,1,2)),flip(a)]. 477 x \ (1 @ (x * y)) = (y \ 1) \ 1. [para(424(a,1),474(a,1,2))]. 478 (x \ (1 @ y)) \ 1 = 1 @ (y \ x). [para(476(a,1),175(a,1,1))]. 479 ((x V 1) \ y) \ 1 = y \ (x V 1). [para(341(a,1),476(a,2,2))]. 480 (x \ 1) \ (1 @ y) = 1 @ (x * y). [para(476(a,1),399(a,1))]. 481 ((x @ y) \ z) \ 1 = z \ (1 @ y). [para(454(a,1),476(a,2,2))]. 482 (x \ 1) \ (y \ 1) = 1 @ (x / y). [para(474(a,1),471(a,1,1))]. 483 1 / ((x @ y) * z) = 1 / (y * z). [para(472(a,2),466(a,2)),flip(a)]. 484 (1 @ x) / y = (y / x) \ 1. [para(475(a,1),429(a,1)),flip(a)]. 485 (1 @ x) / y = 1 / (y / x). [para(484(a,2),336(a,2)),flip(a)]. 486 (1 @ x) / (y \ x) = 1 @ y. [para(484(a,2),391(a,1))]. 487 x \ ((1 @ x) / y) = y \ 1. [para(484(a,2),474(a,1,2))]. 488 (x \ 1) / (y \ (x \ 1)) = 1 @ y. [para(439(a,1),486(a,1,1))]. 489 x \ (1 @ (x * y)) = 1 @ y. [para(477(a,2),175(a,1))]. 490 1 @ ((x \ 1) * y) = 1 @ (x \ y). [para(478(a,1),409(a,2))]. 491 1 ^ ((1 @ x) \ (x V 1)) = 1. [para(479(a,1),434(a,1,2))]. 492 (x V 1) V ((1 @ x) * 1) = x V 1. [para(491(a,1),98(a,1,2,2))]. 493 x ^ (1 @ (y * (y \ x))) = x. [para(480(a,1),393(a,1,2))]. 494 ((x @ y) \ y) \ 1 = 1. [para(481(a,2),432(a,1))]. 495 ((x @ y) \ y) @ 1 = ((x @ y) \ y) * 1. [para(494(a,1),324(a,1,2)),flip(a)]. 496 1 @ (x / y) = (1 @ x) / y. [para(482(a,1),429(a,1))]. 497 1 @ (x / y) = (y / x) \ 1. [para(482(a,1),475(a,1))]. 498 1 @ ((x @ y) / z) = (1 @ y) / z. [para(454(a,1),496(a,2,1))]. 499 1 @ (x / y) = 1 / (y / x). [para(496(a,2),485(a,1))]. 500 1 @ (x / (y \ x)) = 1 @ y. [para(496(a,2),486(a,1))]. 501 x \ (1 @ (x / y)) = y \ 1. [para(496(a,2),487(a,1,2))]. 502 ((x / y) \ z) / z = 1 @ (y / x). [para(497(a,2),390(a,2))]. 503 1 @ ((x * y) / y) = 1 / (1 / x). [para(379(a,1),499(a,2,2))]. 504 1 @ ((x / y) \ x) = (y \ 1) \ 1. [para(501(a,1),478(a,1,1)),flip(a)]. 505 (x \ 1) / (y \ 1) = 1 @ (x \ y). [para(470(a,1),488(a,1,2))]. 506 (x \ 1) \ (1 @ (x \ y)) = 1 @ y. [para(490(a,1),489(a,1,2))]. 507 (x V 1) V (1 @ x) = x V 1. [para(30(a,1),492(a,1,2))]. 508 x V (1 V (1 @ x)) = x V 1. [para(507(a,1),17(a,1)),flip(a)]. 509 1 V (1 @ x) = x V 1. [para(508(a,1),306(a,1)),flip(a)]. 510 (x @ y) V 1 = 1 V (1 @ y). [para(454(a,1),509(a,1,2)),flip(a)]. 511 (x @ y) V 1 = y V 1. [para(510(a,2),509(a,1))]. 512 1 V (x @ y) = y V 1. [para(511(a,1),21(a,1)),flip(a)]. 514 1 V (x @ y) = 1 V y. [para(512(a,2),21(a,1))]. 515 1 V (x @ (y \ (z ^ y))) = 1. [para(512(a,2),322(a,1))]. 517 1 @ ((x V y) / y) = (x V y) / y. [para(152(a,1),502(a,1,1)),flip(a)]. 518 1 @ ((x * y) / y) = 1 @ x. [para(503(a,2),342(a,1))]. 519 1 @ ((x / y) \ x) = 1 @ y. [para(504(a,2),175(a,1))]. 520 1 @ (x \ y) = x \ (1 @ y). [para(505(a,1),428(a,1))]. 521 1 @ (x \ y) = (y \ x) \ 1. [para(505(a,1),473(a,1))]. 522 (1 @ x) / (1 @ ((y V (1 @ x)) \ x)) = (1 @ x) V y. [para(520(a,2),78(a,1,2))]. 523 1 @ (x \ (y @ z)) = x \ (1 @ z). [para(454(a,1),520(a,2,2))]. 524 1 @ ((x @ y) \ y) = 1. [para(520(a,2),456(a,1))]. 525 1 @ (x \ (x * y)) = 1 @ y. [para(520(a,2),489(a,1))]. 526 1 ^ ((x @ y) \ y) = (x @ y) \ y. [para(524(a,1),302(a,1,1))]. 527 1 @ (x \ y) = 1 / (y \ x). [para(521(a,2),336(a,2)),flip(a)]. 528 1 V (x \ (1 @ ((y / x) \ y))) = 1. [para(521(a,2),431(a,1,2,2))]. 529 1 / ((x @ (y \ z)) * u) = u \ (1 @ (z \ y)). [para(527(a,2),472(a,2,2))]. 530 1 @ (x * (x \ y)) = 1 @ y. [para(506(a,1),480(a,1)),flip(a)]. 531 1 @ (x \ (y @ z)) = 1 @ (x \ z). [para(523(a,2),520(a,2)),flip(a)]. 532 1 V (1 @ (x \ ((y / x) \ y))) = 1. [para(520(a,2),528(a,1,2))]. 533 1 V (x \ ((y / x) \ y)) = 1. [para(532(a,1),514(a,1)),flip(a)]. 534 x \ ((y / x) \ y) = 1. [para(533(a,1),173(a,1)),flip(a)]. 537 1 @ ((x @ y) * z) = 1 @ (y * z). [para(450(a,1),399(a,1))]. 538 (x @ y) \ (1 @ (y * z)) = 1 @ z. [para(537(a,1),489(a,1,2))]. 539 x @ (y @ z) = x @ z. [para(455(a,1),253(a,1)),flip(a)]. 541 x @ ((1 @ y) / z) = x @ (y / z). [para(496(a,1),539(a,1,2))]. 542 x @ (y / (z \ y)) = x @ (1 @ z). [para(500(a,1),539(a,1,2)),flip(a)]. 543 x @ (1 @ (y \ z)) = x @ ((y \ 1) * z). [para(490(a,1),539(a,1,2))]. 544 x @ ((1 @ y) / z) = x @ ((u @ y) / z). [para(498(a,1),539(a,1,2))]. 545 x @ ((y * z) / z) = x @ (1 @ y). [para(518(a,1),539(a,1,2)),flip(a)]. 546 x @ ((y / z) \ y) = x @ (1 @ z). [para(519(a,1),539(a,1,2)),flip(a)]. 547 x @ (y \ (y * z)) = x @ (1 @ z). [para(525(a,1),539(a,1,2)),flip(a)]. 548 x @ (y * (y \ z)) = x @ (1 @ z). [para(530(a,1),539(a,1,2)),flip(a)]. 549 x @ (y \ (z @ u)) = x @ (1 @ (y \ u)). [para(531(a,1),539(a,1,2)),flip(a)]. 550 x @ (1 @ (y * z)) = x @ ((u @ y) * z). [para(537(a,1),539(a,1,2))]. 551 x @ (y / (z \ y)) = x @ z. [para(542(a,2),539(a,1))]. 552 x @ (y / (1 / z)) = x @ (y * z). [para(422(a,1),551(a,1,2,2))]. 554 x @ ((y * z) / z) = x @ y. [para(545(a,2),539(a,1))]. 556 x @ ((y / z) \ y) = x @ z. [para(546(a,2),539(a,1))]. 558 x @ (y \ (y * z)) = x @ z. [para(547(a,2),539(a,1))]. 560 x @ (y * (y \ z)) = x @ z. [para(548(a,2),539(a,1))]. 561 x @ (y * (1 / z)) = x @ (y / z). [para(462(a,1),560(a,1,2,2))]. 562 x @ (y * (z \ 1)) = x @ (y / z). [para(463(a,1),560(a,1,2,2))]. 564 x @ (y * (z * (1 / u))) = x @ ((y * z) / u). [para(28(a,1),561(a,1,2))]. 565 x @ (y / (z @ u)) = x @ (y * (u \ 1)). [para(447(a,1),561(a,1,2,2)),flip(a)]. 566 x @ (y * (1 @ (z \ u))) = x @ (y / (u \ z)). [para(527(a,2),561(a,1,2,2))]. 567 x @ (y * (y \ (z * (1 / u)))) = x @ (z / u). [para(561(a,1),560(a,2))]. 568 (x @ y) \ (1 @ z) = y \ (1 @ z). [para(460(a,2),400(a,1))]. 569 1 @ ((x @ y) \ z) = y \ (1 @ z). [para(568(a,1),520(a,2))]. 570 1 @ ((x @ y) \ z) = 1 @ (y \ z). [para(569(a,2),520(a,2)),flip(a)]. 571 (x \ (y @ z)) \ 1 = 1 @ (z \ x). [para(570(a,1),521(a,1)),flip(a)]. 574 ((x @ y) \ y) @ 1 = (x @ y) \ y. [para(495(a,2),30(a,1))]. 575 x @ ((y \ 1) * z) = x @ (y \ z). [para(543(a,1),539(a,1))]. 576 x @ ((1 / y) * z) = x @ (y \ z). [para(336(a,2),575(a,1,2,1))]. 577 x @ ((y @ z) \ u) = x @ ((z \ 1) * u). [para(444(a,1),575(a,1,2,1)),flip(a)]. 579 x @ (((y / z) \ 1) * y) = x @ z. [para(575(a,2),556(a,1))]. 580 x @ ((1 @ (y / z)) * z) = x @ y. [para(497(a,2),579(a,1,2,1))]. 581 x @ ((1 @ y) * z) = x @ (y * z). [para(554(a,1),580(a,1,2,1))]. 582 x @ ((y / z) * z) = x @ y. [para(581(a,1),580(a,1))]. 583 x @ ((y @ z) / u) = x @ (z / u). [para(544(a,1),541(a,1))]. 585 x @ (y \ (z @ u)) = x @ (y \ u). [para(549(a,2),539(a,1))]. 587 x @ ((y @ z) * u) = x @ (z * u). [para(550(a,1),539(a,1))]. 588 ((x @ y) * z) V 1 = 1 V (u @ (y * z)). [para(587(a,1),512(a,1,2)),flip(a)]. 591 x @ (y / (z @ u)) = x @ (y / u). [para(565(a,2),562(a,1))]. 592 (1 @ x) / (y @ z) = 1 @ (x / z). [para(591(a,1),496(a,1)),flip(a)]. 593 (1 @ x) / (y @ (z \ x)) = 1 @ z. [para(592(a,2),551(a,1))]. 594 x @ ((y @ z) \ u) = x @ (z \ u). [para(577(a,2),575(a,1))]. 595 1 V (x @ (y \ (z ^ (u @ y)))) = 1. [para(594(a,1),515(a,1,2))]. 596 (x \ (y ^ (z @ x))) V 1 = 1. [para(595(a,1),512(a,1)),flip(a)]. 597 1 V (x \ (y ^ (z @ x))) = 1. [para(595(a,1),514(a,1)),flip(a)]. 598 1 ^ (x \ (y ^ (z @ x))) = x \ (y ^ (z @ x)). [para(596(a,1),23(a,1,1))]. 599 1 V ((x * (x \ y)) \ y) = 1. [para(493(a,1),597(a,1,2,2))]. 600 (x * (x \ y)) \ y = 1. [para(599(a,1),162(a,1)),flip(a)]. 604 ((x @ y) * z) V 1 = 1 V (y * z). [para(588(a,2),514(a,1))]. 605 1 V ((x @ y) * z) = 1 V (y * z). [para(604(a,1),21(a,1)),flip(a)]. 606 (x * y) \ z = y \ (x \ z). [para(417(a,1),262(a,1))]. 607 x ^ ((y \ (z / x)) \ (y \ z)) = x. [para(606(a,1),181(a,1,2))]. 608 (x \ 1) \ (x \ y) = (x @ 1) \ y. [para(324(a,1),606(a,1,1)),flip(a)]. 609 (1 @ x) \ ((y @ 1) \ z) = (y @ x) \ z. [para(351(a,1),606(a,1,1)),flip(a)]. 610 (x * (y @ 1)) @ 1 = x * (y @ (x \ 1)). [para(442(a,1),461(a,1,2)),flip(a)]. 611 (x * (y @ 1)) @ 1 = x * (y @ (1 / x)). [para(336(a,2),610(a,2,2,2))]. 612 (x * (y @ (1 / x))) * (1 @ z) = (x * (y @ 1)) @ z. [para(611(a,1),351(a,1,1))]. 613 1 / ((x @ (y \ z)) * u) = 1 @ (u \ (z \ y)). [para(529(a,2),520(a,2)),flip(a)]. 614 x @ (y * (1 @ (y \ z))) = x @ z. [para(566(a,2),551(a,1))]. 615 x @ (y * (1 @ z)) = x @ (y * z). [para(558(a,1),614(a,1,2,2))]. 616 x @ (y * (1 @ z)) = x @ (y * (u @ z)). [para(539(a,1),615(a,1,2,2))]. 617 x @ (y / (z \ u)) = x @ (y * (u \ z)). [para(615(a,1),566(a,1)),flip(a)]. 620 x @ (y * (z @ u)) = x @ (y * u). [para(616(a,1),615(a,1))]. 622 (x ^ y) \ (x ^ (1 @ (x ^ y))) = 1. [para(598(a,1),359(a,1))]. 623 1 @ (x \ (y \ z)) = 1 / ((z \ y) * x). [para(613(a,1),483(a,1))]. 624 x @ (y / (1 / ((z \ u) * w))) = x @ (y / (w \ (u \ z))). [para(623(a,1),591(a,1,2,2))]. 625 1 V ((x ^ (y V ((z @ 1) * x))) \ ((z @ 1) * x)) = 1. [para(21(a,1),353(a,1,2,1,2))]. 626 ((x \ 1) * x) @ 1 = ((x \ 1) * x) * 1. [para(432(a,1),410(a,1,2)),flip(a)]. 627 ((x \ 1) * x) @ 1 = (x \ 1) * x. [para(626(a,2),30(a,1))]. 628 1 @ (x V (1 @ y)) = (1 @ y) V x. [para(522(a,1),593(a,1)),flip(a)]. 629 1 @ (x V (1 @ y)) = x V (1 @ y). [para(628(a,2),21(a,1))]. 630 (1 @ x) V (y \ (y * (1 @ x))) = 1 @ (y \ (y * (1 @ x))). [para(118(a,1),628(a,1,2)),flip(a)]. 632 x * ((y @ (1 / x)) * (1 @ z)) = (x * (y @ 1)) @ z. [para(612(a,1),28(a,1)),flip(a)]. 633 (x * (x \ y)) / (y \ (x * (x \ y))) = y. [para(99(a,1),171(a,1,2,1))]. 634 ((x * (x \ y)) / y) \ (x * (x \ y)) = y. [para(99(a,1),172(a,1,1,2))]. 635 1 @ (x / (y * (y \ x))) = x / (y * (y \ x)). [para(634(a,1),502(a,1,1)),flip(a)]. 636 (((1 V x) \ y) / y) \ ((1 V x) \ y) = y. [para(191(a,2),187(a,1))]. 637 ((1 V x) \ 1) \ ((1 V x) \ y) = y. [para(390(a,1),636(a,1,1))]. 639 (x / (y V 1)) / (x \ (x / (y V 1))) = x. [para(227(a,2),224(a,1))]. 640 (x / (1 V y)) / (x \ (x / (y V 1))) = x. [para(21(a,1),639(a,1,1,2))]. 641 (x / (1 V y)) / ((y V 1) \ 1) = x. [para(463(a,1),640(a,1,2))]. 643 1 / ((x \ (x V y)) \ 1) = x \ (x V y). [para(113(a,1),244(a,1,2,2))]. 645 1 @ (x \ (x V y)) = x \ (x V y). [para(643(a,1),338(a,1)),flip(a)]. 646 1 / (((x ^ y) \ y) \ 1) = (x ^ y) \ y. [para(126(a,1),245(a,1,2,2))]. 648 1 @ ((x ^ y) \ y) = (x ^ y) \ y. [para(646(a,1),338(a,1)),flip(a)]. 649 x / ((x * (y V 1)) \ x) = x * (y V 1). [para(222(a,1),246(a,1,2,2))]. 650 x / ((y V 1) \ 1) = x * (y V 1). [para(419(a,1),649(a,1,2))]. 651 1 ^ ((x \ 1) * (x V 1)) = 1. [para(650(a,1),270(a,1,2))]. 653 (x / (1 V y)) * (y V 1) = x. [para(650(a,1),641(a,1))]. 654 1 ^ ((x \ 1) * (1 V x)) = 1. [para(21(a,1),651(a,1,2,2))]. 655 (x ^ 1) * (x V 1) = x. [para(378(a,1),653(a,1,1))]. 656 (x ^ 1) * ((x V 1) * y) = x * y. [para(655(a,1),28(a,1,1)),flip(a)]. 657 ((x \ 1) * (1 V x)) ^ 1 = 1. [para(654(a,1),20(a,1)),flip(a)]. 658 (x / ((1 V y) * x)) \ x = (1 V y) * x. [para(210(a,1),252(a,1,2))]. 659 ((1 V x) \ 1) \ y = (1 V x) * y. [para(381(a,1),658(a,1,1))]. 661 (1 V x) * ((1 V x) \ y) = y. [para(659(a,1),637(a,1))]. 662 (1 V x) * ((x V 1) \ y) = y. [para(21(a,1),661(a,1,2,1))]. 663 (x V 1) * ((x V 1) \ y) = y. [para(21(a,1),662(a,1,1))]. 664 (x V 1) * (x \ (x ^ 1)) = 1. [para(468(a,1),663(a,1,2))]. 665 x * (x \ (x ^ 1)) = (x ^ 1) * 1. [para(664(a,1),656(a,1,2)),flip(a)]. 666 x * (x \ (x ^ 1)) = x ^ 1. [para(665(a,2),30(a,1))]. 667 ((x \ 1) * (1 V x)) * (((x \ 1) * (1 V x)) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(657(a,1),666(a,1,2,2))]. 668 x / ((y / (z ^ (x \ y))) \ x) = y / (z ^ (x \ y)). [para(394(a,2),164(a,1))]. 669 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / (z ^ ((x \ (y / z)) \ (x \ y))). [para(607(a,1),668(a,1,2,1,2))]. 670 x @ (y / (z \ (u \ w))) = x @ (y * ((w \ u) * z)). [para(624(a,1),552(a,1))]. 671 x @ (y / ((z * u) \ w)) = x @ (y * ((w \ z) * u)). [para(606(a,2),670(a,1,2,2))]. 672 x @ (y * (z \ (u * w))) = x @ (y * ((z \ u) * w)). [para(671(a,1),617(a,1)),flip(a)]. 673 x @ (y * ((y \ z) * (1 / u))) = x @ (z / u). [para(672(a,1),567(a,1))]. 674 x @ ((y * (y \ z)) / u) = x @ (z / u). [para(673(a,1),564(a,1)),flip(a)]. 675 x @ ((y * (1 / z)) / u) = x @ ((y / z) / u). [para(462(a,1),674(a,1,2,1,2))]. 676 1 @ (x \ (x * (1 @ y))) = x \ (x * (1 @ y)). [para(630(a,1),117(a,1))]. 677 x \ (x * (1 @ y)) = 1 @ (1 @ y). [para(676(a,1),558(a,1))]. 678 x \ (x * (1 @ y)) = 1 @ y. [para(677(a,2),539(a,1))]. 680 (x @ 1) \ (x @ y) = 1 @ y. [para(351(a,1),678(a,1,2))]. 682 (1 @ x) \ (1 @ y) = (z @ x) \ (z @ y). [para(680(a,1),609(a,1,2))]. 683 (x @ y) \ (x @ (y * z)) = 1 @ z. [para(682(a,1),538(a,1))]. 685 (x @ y) \ (x @ z) = 1 @ (y \ z). [para(560(a,1),683(a,1,2))]. 686 ((x \ 1) * (1 V x)) * ((1 V x) \ ((x \ 1) \ 1)) = ((x \ 1) * (1 V x)) ^ 1. [para(606(a,1),667(a,1,2))]. 687 (x \ 1) * ((1 V x) * ((1 V x) \ ((x \ 1) \ 1))) = ((x \ 1) * (1 V x)) ^ 1. [para(686(a,1),28(a,1)),flip(a)]. 688 (x \ 1) * ((x \ 1) \ 1) = ((x \ 1) * (1 V x)) ^ 1. [para(661(a,1),687(a,1,2))]. 689 (x \ 1) * ((x \ 1) \ 1) = 1. [para(688(a,2),657(a,1))]. 691 (x \ 1) * (1 @ x) = 1. [para(175(a,1),689(a,1,2))]. 692 x @ x = x * 1. [para(689(a,1),83(a,2,2))]. 695 ((x \ 1) \ 1) * (x \ 1) = 1. [para(437(a,1),689(a,1,2))]. 697 x @ x = x # label("Goal 1"). [para(692(a,2),30(a,1))]. 699 x @ (y @ x) = x. [para(697(a,1),539(a,2))]. 700 ((x / y) \ x) @ y = (x / y) \ x. [para(697(a,1),556(a,1)),flip(a)]. 702 x @ (y * (y \ x)) = x. [para(697(a,1),560(a,2))]. 703 ((x \ 1) * y) @ (x \ y) = (x \ 1) * y. [para(697(a,1),575(a,1)),flip(a)]. 704 ((x @ y) / z) @ (y / z) = (x @ y) / z. [para(697(a,1),583(a,1)),flip(a)]. 705 ((x @ y) * z) @ (y * z) = (x @ y) * z. [para(697(a,1),587(a,1)),flip(a)]. 706 (x / (y @ z)) @ (x / z) = x / (y @ z). [para(697(a,1),591(a,1)),flip(a)]. 707 ((x @ y) \ z) @ (y \ z) = (x @ y) \ z. [para(697(a,1),594(a,1)),flip(a)]. 708 x \ (x @ (x * y)) = 1 @ y. [para(697(a,1),683(a,1,1))]. 709 x \ (x @ y) = 1 @ (x \ y). [para(697(a,1),685(a,1,1))]. 710 (x @ y) \ x = 1 @ (y \ x). [para(697(a,1),685(a,1,2))]. 712 (x \ 1) * ((1 @ x) * y) = 1 * y. [para(691(a,1),28(a,1,1)),flip(a)]. 713 x V (((x / (1 @ y)) / (y \ 1)) * 1) = x. [para(691(a,1),237(a,1,2,2))]. 714 ((1 @ x) \ y) ^ (1 \ ((x \ 1) * y)) = (1 @ x) \ y. [para(691(a,1),366(a,1,2,1))]. 715 x @ (y * (z * (z \ u))) = x @ (y * u). [para(702(a,1),620(a,1,2,2)),flip(a)]. 718 (1 @ x) * (x \ 1) = 1. [para(175(a,1),695(a,1,1))]. 719 x * ((1 @ y) * (y \ 1)) = x. [para(718(a,2),30(a,1,2))]. 720 x \ (x @ y) = 1 @ (x \ (z @ y)). [para(539(a,1),709(a,1,2))]. 721 x \ (x @ (x V y)) = x \ (x V y). [para(709(a,2),645(a,1))]. 722 x V ((x @ y) * (z ^ (1 @ (y \ x)))) = x. [para(710(a,1),98(a,1,2,2,2))]. 723 x ^ ((y @ (y / x)) \ y) = x. [para(710(a,2),313(a,1,2))]. 724 1 ^ ((x @ (y ^ x)) \ x) = 1. [para(710(a,2),333(a,1,2))]. 725 (x \ (y @ z)) \ 1 = (x @ z) \ x. [para(710(a,2),571(a,2))]. 726 (x @ y) \ x = y \ (y @ x). [para(710(a,2),709(a,2)),flip(a)]. 727 (x @ y) * (y \ 1) = x @ 1. [para(719(a,1),352(a,1)),flip(a)]. 729 1 ^ ((x @ (x / y)) \ (x / y)) = 1. [para(723(a,1),165(a,1,2,2,2))]. 730 1 ^ ((x @ (x ^ y)) \ x) = 1. [para(20(a,1),724(a,1,2,1,2))]. 731 (x @ y) @ 1 = x @ 1. [para(727(a,1),446(a,1)),flip(a)]. 732 (x \ 1) \ ((y @ x) \ z) = (y @ 1) \ z. [para(727(a,1),606(a,1,1)),flip(a)]. 733 (x @ y) @ z = (x @ 1) * (1 @ z). [para(731(a,1),351(a,1,1)),flip(a)]. 735 x V ((x @ (x ^ y)) * 1) = x. [para(730(a,1),98(a,1,2,2))]. 736 x V (x @ (x ^ y)) = x. [para(30(a,1),735(a,1,2))]. 737 (x V y) V ((x V y) @ x) = x V y. [para(23(a,1),736(a,1,2,2))]. 738 x ^ (x @ (x ^ y)) = x @ (x ^ y). [para(736(a,1),68(a,1,1))]. 739 x V (x @ ((y V 1) \ x)) = x. [para(192(a,1),736(a,1,2,2))]. 740 (x V (x @ ((y V 1) \ x))) / (x @ ((y V 1) \ x)) = 1 @ (x / (x @ ((y V 1) \ x))). [para(739(a,1),517(a,1,2,1)),flip(a)]. 741 x V (((y / x) \ y) @ x) = (y / x) \ y. [para(700(a,2),157(a,1,2))]. 742 (x \ 1) * ((1 @ x) * y) = y. [para(712(a,2),29(a,1))]. 743 ((1 @ x) * y) / y = 1 @ x. [para(742(a,1),382(a,1,2))]. 744 (x \ 1) @ y = 1 @ y. [para(742(a,1),254(a,1)),flip(a)]. 745 (1 @ x) @ y = 1 @ y. [para(175(a,1),744(a,1,1))]. 746 (x V (1 @ y)) @ z = 1 @ z. [para(629(a,1),745(a,1,1))]. 748 (x \ (x @ y)) * ((x \ (z @ y)) \ 1) = 1 @ 1. [para(720(a,2),727(a,1,1))]. 749 (x @ (x / y)) \ (x / y) = 1. [para(729(a,1),526(a,1)),flip(a)]. 750 (x @ y) @ z = x @ z # label("Goal 2"). [para(733(a,2),351(a,1))]. 753 x V (y V ((x V y) @ x)) = x V y. [para(737(a,1),17(a,1)),flip(a)]. 754 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ (x @ (x ^ y))))) = 1. [para(738(a,1),622(a,1,1))]. 755 x V ((1 @ y) V (1 @ x)) = x V (1 @ y). [para(746(a,1),753(a,1,2,2))]. 756 1 V ((x \ 1) * y) = 1 V (x \ y). [para(703(a,1),514(a,1,2))]. 757 1 V ((1 / x) * y) = 1 V (x \ y). [para(336(a,2),756(a,1,2,1))]. 758 ((x @ y) / (z @ u)) @ (y / u) = (x @ y) / (z @ u). [para(704(a,1),591(a,1)),flip(a)]. 759 ((x @ y) \ (z @ u)) @ (y \ u) = (x @ y) \ (z @ u). [para(707(a,1),585(a,1)),flip(a)]. 760 x V ((x / (1 @ y)) / (y \ 1)) = x. [para(30(a,1),713(a,1,2))]. 761 (x / (1 @ y)) / (y \ 1) = x. [para(760(a,1),365(a,1)),flip(a)]. 762 x @ (y * (z * (1 / u))) = x @ (y * (z / u)). [para(462(a,1),715(a,1,2,2,2))]. 765 (1 @ x) V (1 @ y) = y V (1 @ x). [para(755(a,1),306(a,1)),flip(a)]. 766 (1 @ x) V (1 @ y) = x V (1 @ y). [para(755(a,1),309(a,1)),flip(a)]. 767 x V (1 @ y) = y V (1 @ x). [para(766(a,1),765(a,1))]. 768 (x V y) V (1 @ x) = 1 @ (x V y). [para(767(a,1),314(a,1))]. 769 (x @ y) V (1 @ z) = z V (1 @ y). [para(539(a,1),767(a,1,2)),flip(a)]. 770 x V (y V (1 @ x)) = 1 @ (x V y). [para(768(a,1),17(a,1)),flip(a)]. 774 1 @ (x V y) = y V (1 @ x). [para(770(a,1),306(a,1))]. 775 1 @ (x V y) = x V (1 @ y). [para(21(a,1),774(a,1,2))]. 776 x @ (y V (1 @ z)) = x @ (z V y). [para(774(a,1),539(a,1,2))]. 778 x @ (y V (1 @ z)) = x @ (y V z). [para(775(a,1),539(a,1,2))]. 779 x @ (y V (1 @ z)) = x @ (y V (u @ z)). [para(769(a,1),776(a,1,2))]. 781 x @ (y V (z @ u)) = x @ (u V y). [para(779(a,1),776(a,1))]. 782 x @ (y V (z @ u)) = x @ (y V u). [para(779(a,1),778(a,1))]. 784 x @ ((y @ z) V u) = x @ (z V u). [para(21(a,1),781(a,1,2))]. 785 (x V (y @ z)) @ (x V z) = x V (y @ z). [para(782(a,1),697(a,1))]. 786 (x \ (x @ y)) * ((x @ y) \ x) = 1 @ 1. [para(725(a,1),748(a,1,2))]. 787 (x \ (x @ y)) * ((x @ y) \ x) = 1. [para(786(a,2),697(a,1))]. 788 x V (((x @ y) / (x \ (x @ y))) * 1) = x. [para(787(a,1),236(a,1,2,2))]. 789 x V ((x @ y) / (x \ (x @ y))) = x. [para(30(a,1),788(a,1,2))]. 790 (x @ y) / (x \ (x @ y)) = x. [para(789(a,1),150(a,1)),flip(a)]. 791 (x @ (x * y)) / (1 @ y) = x. [para(708(a,1),790(a,1,2))]. 792 x @ ((y * z) / u) = x @ (y * (z / u)). [para(762(a,1),564(a,1)),flip(a)]. 793 x @ (y * ((y \ z) / u)) = x @ (z / u). [para(762(a,1),673(a,1))]. 794 x @ (y * ((1 / z) / u)) = x @ ((y / z) / u). [para(792(a,1),675(a,1))]. 795 (x * ((x \ y) / z)) @ (y / z) = x * ((x \ y) / z). [para(793(a,1),697(a,1))]. 796 x @ (y * (1 / (z * u))) = x @ ((y / u) / z). [para(425(a,1),794(a,1,2,2))]. 797 x @ ((y / z) / u) = x @ (y / (u * z)). [para(796(a,1),561(a,1))]. 798 ((1 @ x) \ y) ^ ((x \ 1) * y) = (1 @ x) \ y. [para(119(a,1),714(a,1,2))]. 799 ((x \ 1) * y) ^ ((1 @ x) \ y) = (1 @ x) \ y. [para(798(a,1),20(a,1)),flip(a)]. 800 (1 @ x) \ y = (x \ 1) * y. [para(799(a,1),388(a,1))]. 801 ((x V 1) \ 1) * y = (x V 1) \ y. [para(341(a,1),800(a,1,1)),flip(a)]. 803 ((x \ 1) \ 1) * y = (x \ 1) \ y. [para(439(a,1),800(a,1,1)),flip(a)]. 804 (1 @ (x / y)) \ z = (1 @ (y / x)) * z. [para(497(a,2),800(a,2,1))]. 805 (x \ 1) * (1 @ (x * y)) = 1 @ y. [para(800(a,1),538(a,1))]. 806 (1 @ x) \ ((1 @ x) * y) = y. [para(800(a,2),742(a,1))]. 807 (x \ (x @ y)) \ z = ((x \ (u @ y)) \ 1) * z # label(flip_matches_hint). [para(720(a,2),800(a,1,1))]. 808 ((x V 1) \ 1) * (y * z) = ((x V 1) \ y) * z. [para(801(a,1),28(a,1,1)),flip(a)]. 809 (x \ 1) \ y = (1 @ x) * y. [para(175(a,1),803(a,1,1)),flip(a)]. 810 ((x \ 1) * y) \ z = y \ ((1 @ x) * z). [para(809(a,1),606(a,2,2))]. 811 (1 @ x) * (x \ y) = (x @ 1) \ y. [para(809(a,1),608(a,1))]. 813 (1 @ x) * ((y @ x) \ z) = (y @ 1) \ z. [para(809(a,1),732(a,1))]. 814 x @ (x * y) = x * (1 @ y). [para(805(a,1),253(a,1,2)),flip(a)]. 815 x * (1 @ (x \ y)) = x @ y. [para(814(a,1),560(a,1))]. 816 (x * (1 @ y)) @ z = x @ z. [para(814(a,1),750(a,1,1))]. 817 (x * (1 @ y)) / (1 @ y) = x. [para(814(a,1),791(a,1,1))]. 818 (x * y) @ x = (x * y) * (y \ 1). [para(443(a,1),815(a,1,2)),flip(a)]. 819 x @ (x / y) = x * (y \ 1). [para(467(a,1),815(a,1,2)),flip(a)]. 820 x * (x \ (x V y)) = x @ (x V y). [para(645(a,1),815(a,1,2))]. 821 (x ^ y) * ((x ^ y) \ y) = (x ^ y) @ y. [para(648(a,1),815(a,1,2))]. 822 x * ((y @ x) \ y) = x @ y. [para(710(a,2),815(a,1,2))]. 823 x / (y \ 1) = x * (1 @ y). [para(817(a,1),761(a,1,1))]. 824 x * (((y @ x) \ y) * z) = (x @ y) * z. [para(822(a,1),28(a,1,1)),flip(a)]. 825 (x @ y) * ((z @ (x @ y)) \ z) = x @ z. [para(822(a,2),750(a,1))]. 826 x ^ ((x / y) * (1 @ y)) = x. [para(823(a,1),371(a,1,2))]. 827 (x / (y \ 1)) @ z = x @ z. [para(823(a,2),816(a,1,1))]. 828 x ^ ((x / y) * (1 @ ((z / y) \ z))) = x. [para(556(a,2),826(a,1,2,2))]. 829 (x / (1 @ y)) @ z = x @ z. [para(175(a,1),827(a,1,1,2))]. 830 x / (1 @ y) = x @ (x / y). [para(829(a,1),706(a,1)),flip(a)]. 831 x / (1 @ y) = x * (y \ 1). [para(830(a,2),819(a,1))]. 832 (x * y) @ x = x * (y * (y \ 1)). [para(818(a,2),28(a,1))]. 833 (x V y) V (x @ (x V y)) = x V y. [para(820(a,1),99(a,1,2))]. 834 x ^ (x @ (x V y)) = x. [para(820(a,1),229(a,1,2))]. 835 x V (x @ (x V y)) = x @ (x V y). [para(834(a,1),65(a,1,1))]. 836 x V (y V (x @ (x V z))) = y V (x @ (x V z)). [para(834(a,1),97(a,1,1))]. 837 (x * y) @ x = x * (y @ 1). [para(324(a,1),832(a,2,2))]. 838 ((x @ y) * z) @ y = (x @ y) * (z @ 1). [para(837(a,1),539(a,1)),flip(a)]. 839 (x * (y @ 1)) @ z = (x * y) @ z. [para(837(a,1),750(a,1,1))]. 840 x V (y V (x @ (x V y))) = x V y. [para(833(a,1),17(a,1)),flip(a)]. 841 (1 @ ((x * (x \ y)) / y)) * z = (y / (x * (x \ y))) \ z. [para(635(a,1),804(a,1,1)),flip(a)]. 842 (x @ 1) \ (x / y) = (1 @ (x / y)) * 1. [para(749(a,1),813(a,1,2)),flip(a)]. 843 x V ((y ^ x) @ x) = x. [para(821(a,1),99(a,1,2))]. 844 x V ((x ^ y) @ x) = x. [para(20(a,1),843(a,1,2,1))]. 845 (x @ y) * (z * (((u @ z) \ u) * ((z @ u) \ (y \ 1)))) = ((x @ y) * (z @ u)) @ 1. [para(824(a,2),451(a,1,2))]. 846 (x @ y) * ((z @ y) \ z) = x @ z. [para(539(a,1),825(a,1,2,1))]. 848 (x @ y) * (1 @ (y \ z)) = x @ z. [para(710(a,1),846(a,1,2))]. 849 (x @ y) * (y \ (y @ z)) = x @ z. [para(726(a,1),846(a,1,2))]. 850 (x @ y) * (1 @ z) = x @ (y * z). [para(558(a,1),848(a,1,2))]. 851 (x \ (x @ y)) \ ((z @ x) \ u) = (z @ y) \ u. [para(849(a,1),606(a,1,1)),flip(a)]. 852 x * (y @ ((1 / x) * z)) = (x * (y @ 1)) @ z. [para(850(a,1),632(a,1,2))]. 853 x ^ ((x / y) @ x) = x. [para(815(a,1),828(a,1,2))]. 854 x ^ ((x V y) @ x) = x. [para(32(a,1),853(a,1,2,1))]. 855 x ^ (((y V 1) * x) @ x) = x. [para(213(a,1),854(a,1,2,1))]. 856 (x @ 1) \ (x / y) = 1 @ (x / y). [para(842(a,2),30(a,1))]. 857 ((x @ 1) * (1 @ (x / y))) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. [para(856(a,1),633(a,1,1,2))]. 858 (x \ (x @ y)) \ z = ((x @ y) \ x) * z. [para(725(a,1),807(a,2,1))]. 859 (x V 1) \ (y * z) = ((x V 1) \ y) * z. [para(808(a,1),801(a,1)),flip(a)]. 860 x V (y @ (y V x)) = y V x. [para(836(a,1),840(a,1))]. 861 x V (y @ ((y @ z) V x)) = (y @ z) V x. [para(750(a,1),860(a,1,2))]. 862 x V (y @ (x V (y @ z))) = (y @ z) V x. [para(21(a,1),861(a,1,2,2))]. 863 x V (y @ (x V z)) = (y @ z) V x. [para(782(a,1),862(a,1,2))]. 864 x V (y @ (x V z)) = x V (y @ z). [para(863(a,2),21(a,1))]. 866 x V (y @ (x ^ z)) = x V (y @ x). [para(66(a,1),864(a,1,2,2)),flip(a)]. 867 x @ (x V y) = x V (x @ y). [para(864(a,1),835(a,1)),flip(a)]. 868 x @ (x V (y @ z)) = x V (x @ z). [para(539(a,1),867(a,2,2))]. 869 x \ (x V (x @ y)) = x \ (x V y). [para(867(a,1),721(a,1,2))]. 870 (x V (x @ y)) @ z = x @ z. [para(867(a,1),750(a,1,1))]. 871 x * (x \ (x V y)) = x V (x @ y). [para(867(a,1),820(a,2))]. 872 x V (y V (y @ x)) = y V x. [para(867(a,1),860(a,1,2))]. 873 ((x @ y) V ((x @ y) @ z)) @ u = x @ u. [para(870(a,2),750(a,1))]. 875 x V (y V (x @ y)) = x V y. [para(872(a,1),94(a,1)),flip(a)]. 876 x V ((y @ z) V (x @ z)) = x V (y @ z). [para(539(a,1),875(a,1,2,2))]. 877 x \ (x V (y @ z)) = x \ (x V (x @ z)). [para(868(a,1),721(a,1,2)),flip(a)]. 878 (x V (x @ y)) \ (x V y) = 1. [para(871(a,1),600(a,1,1))]. 879 ((x @ y) V (x @ z)) @ u = x @ u. [para(750(a,1),873(a,1,1,2))]. 881 x @ ((x @ y) V z) = (x @ y) V (x @ z). [para(879(a,1),785(a,1))]. 882 x \ (x V (y @ z)) = x \ (x V z). [para(877(a,2),869(a,1))]. 884 (x V y) \ (x V (y V (z @ u))) = (x V y) \ ((x V y) V u). [para(17(a,1),882(a,1,2))]. 885 (x @ y) V (x @ z) = x @ (y V z). [para(881(a,1),784(a,1))]. 886 (x @ y) ^ (x @ (y V z)) = x @ y. [para(885(a,1),67(a,1,2))]. 888 (x @ (y ^ z)) ^ (x @ z) = x @ (y ^ z). [para(65(a,1),886(a,1,2,2))]. 889 (x @ y) ^ (x @ (z ^ y)) = x @ (z ^ y). [para(888(a,1),20(a,1)),flip(a)]. 890 ((x @ y) \ x) * ((z @ x) \ u) = (z @ y) \ u. [para(851(a,1),858(a,1)),flip(a)]. 891 (x * (y @ 1)) @ z = x * (y @ (x \ z)). [para(576(a,1),852(a,1,2)),flip(a)]. 892 x * (y @ (x \ z)) = (x * y) @ z. [para(891(a,1),839(a,1))]. 894 (x @ (x ^ y)) \ (x ^ (1 @ (x @ (x ^ y)))) = 1. [para(738(a,1),754(a,1,2,2,2))]. 895 (x @ (x ^ y)) \ (x ^ (1 @ (x ^ y))) = 1. [para(539(a,1),894(a,1,2,2))]. 897 ((x * ((x \ y) / z)) V (x * ((x \ y) / z))) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(795(a,1),878(a,1,1,2))]. 898 (1 @ (x * ((x \ y) / y))) * z = (y / (x * (x \ y))) \ z. [para(792(a,1),841(a,1,1))]. 899 (x V y) \ (x V (y V (z @ u))) = (x V y) \ (x V (y V u)). [para(17(a,1),884(a,2,2))]. 900 (x / (y * (y \ x))) \ z = (1 @ (y * (y \ 1))) * z. [para(390(a,1),898(a,1,1,2,2)),flip(a)]. 902 (x / (y * (y \ x))) \ z = 1 * z. [para(702(a,1),900(a,2,1))]. 903 (x / (y * (y \ x))) \ z = z. [para(902(a,2),29(a,1))]. 905 x / (y * (y \ x)) = 1. [para(903(a,1),278(a,1))]. 906 x / ((y ^ x) @ x) = 1. [para(821(a,1),905(a,1,2))]. 907 (x V y) / (y @ (x V y)) = 1. [para(88(a,1),906(a,1,2,1))]. 908 x / ((((y V z) \ y) * x) @ (x V (((y V z) \ y) * x))) = 1. [para(345(a,1),907(a,1,1))]. 909 (x V y) \ (x V (y V y)) = (x V y) \ (x V ((z / y) \ z)). [para(741(a,1),899(a,1,2,2)),flip(a)]. 910 x / ((((y V z) \ y) * x) @ x) = 1. [para(345(a,1),908(a,1,2,2))]. 911 (x V y) \ (x V ((z / y) \ z)) = (x V y) \ (x V y). [para(19(a,1),909(a,1,2,2)),flip(a)]. 912 (x V y) \ (x V ((z / y) \ z)) = 1. [para(911(a,2),278(a,1))]. 914 1 @ (x / (x @ ((y V 1) \ x))) = x / (x @ ((y V 1) \ x)). [para(739(a,1),740(a,1,1)),flip(a)]. 915 x / (x @ ((y V 1) \ x)) = 1 @ (x * (((y V 1) \ x) \ 1)). [para(914(a,1),565(a,1))]. 916 x / (x @ ((y V 1) \ x)) = 1 @ (x * (x \ (y V 1))). [para(479(a,1),915(a,2,2,2))]. 918 x / (x @ ((y V 1) \ x)) = 1 @ (y V 1). [para(916(a,2),560(a,1))]. 919 x / (x @ ((y V 1) \ x)) = y V 1. [para(918(a,2),341(a,1))]. 921 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ ((y V 1) * (x @ ((y V 1) \ x))). [para(919(a,1),108(a,1,2,1)),flip(a)]. 923 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * ((z @ z) \ (y \ 1))). [para(890(a,1),845(a,1,2,2)),flip(a)]. 924 ((x @ y) * (z @ u)) @ 1 = (x @ y) * (z * (z \ (y \ 1))). [para(458(a,1),923(a,2,2,2))]. 927 ((x @ y) * (z @ u)) @ 1 = ((x @ y) * z) @ 1. [para(924(a,2),451(a,1))]. 928 ((x @ x) * y) @ 1 = (x * (y @ z)) @ 1. [para(697(a,1),927(a,1,1,1)),flip(a)]. 929 (x * (y @ z)) @ 1 = (x * y) @ 1. [para(697(a,1),928(a,1,1,1)),flip(a)]. 930 ((x * y) @ 1) @ z = (x * (y @ u)) @ z. [para(929(a,1),750(a,1,1))]. 931 (x * (y @ z)) @ u = (x * y) @ u. [para(930(a,1),750(a,1))]. 933 (x @ (x / y)) / ((x / y) \ ((x @ 1) * ((x @ 1) \ (x / y)))) = x / y. [para(351(a,1),857(a,1,1))]. 934 (x @ (x / y)) / ((x / y) \ ((x @ 1) * (1 @ (x / y)))) = x / y. [para(856(a,1),933(a,1,2,2,2))]. 935 (x @ (x / y)) / ((x / y) \ (x @ (x / y))) = x / y. [para(351(a,1),934(a,1,2,2))]. 936 (x * ((x \ y) / z)) \ ((x * ((x \ y) / z)) V (y / z)) = 1. [para(19(a,1),897(a,1,1))]. 937 (x * ((x \ y) / z)) \ ((y / z) V (x * ((x \ y) / z))) = 1. [para(21(a,1),936(a,1,2))]. 938 ((x \ y) / z) \ (x \ ((y / z) V (x * ((x \ y) / z)))) = 1. [para(937(a,1),606(a,1)),flip(a)]. 939 (x / (x @ ((y V 1) \ x))) * (x @ ((y V 1) \ x)) = x ^ (((y V 1) * x) @ x). [para(892(a,1),921(a,2,2))]. 940 x ^ (((y V 1) * x) @ x) = (y V 1) * (x @ ((y V 1) \ x)). [para(919(a,1),939(a,1,1)),flip(a)]. 941 x ^ (((y V 1) * x) @ x) = ((y V 1) * x) @ x. [para(940(a,2),892(a,1))]. 943 ((x V 1) * y) @ y = y. [para(941(a,1),855(a,1))]. 944 ((x V 1) * y) @ z = y @ z. [para(943(a,1),750(a,1,1)),flip(a)]. 945 ((x V 1) \ y) @ z = y @ z. [para(663(a,1),944(a,1,1)),flip(a)]. 946 (((x V 1) \ y) * z) @ u = (y * z) @ u. [para(859(a,1),945(a,1,1))]. 947 x / ((y * x) @ x) = 1. [para(946(a,1),910(a,1,2))]. 948 x V (1 * ((y * x) @ x)) = x. [para(947(a,1),105(a,1,2,1))]. 949 ((x * y) @ y) \ (1 \ y) = 1. [para(947(a,1),534(a,1,2,1))]. 950 x V ((y * x) @ x) = x. [para(29(a,1),948(a,1,2))]. 952 x V ((y @ x) * (x @ 1)) = x. [para(838(a,1),950(a,1,2))]. 953 ((x * y) @ y) \ y = 1. [para(119(a,1),949(a,1,2))]. 954 (((x * y) @ y) @ 1) \ y = (1 @ ((x * y) @ y)) * 1. [para(953(a,1),811(a,1,2)),flip(a)]. 955 (((x * y) @ y) @ 1) \ y = (1 @ y) * 1. [para(539(a,1),954(a,2,1))]. 956 (((x * y) @ y) @ 1) \ y = 1 @ y. [para(955(a,2),30(a,1))]. 958 ((x * y) @ 1) \ y = 1 @ y. [para(750(a,1),956(a,1,1))]. 959 ((x \ 1) * x) \ x = 1 @ x. [para(627(a,1),958(a,1,1))]. 961 x \ ((1 @ x) * x) = 1 @ x. [para(959(a,1),810(a,1)),flip(a)]. 962 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) / (1 @ x)). [para(961(a,1),150(a,1,2,2)),flip(a)]. 963 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V (((1 @ x) * x) * (x \ 1)). [para(831(a,1),962(a,2,2))]. 964 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x * (x \ 1))). [para(28(a,1),963(a,2,2))]. 965 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x V ((1 @ x) * (x @ 1)). [para(324(a,1),964(a,2,2,2))]. 966 ((1 @ x) * x) / (x \ ((1 @ x) * x)) = x. [para(965(a,2),952(a,1))]. 967 ((1 @ x) * x) / (1 @ x) = x. [para(961(a,1),966(a,1,2))]. 968 ((1 @ x) * x) * (x \ 1) = x. [para(967(a,1),831(a,1)),flip(a)]. 969 (1 @ x) * (x * (x \ 1)) = x. [para(968(a,1),28(a,1)),flip(a)]. 970 (1 @ x) * (x @ 1) = x. [para(324(a,1),969(a,1,2))]. 971 (1 @ x) * ((y @ x) @ 1) = y @ x. [para(539(a,1),970(a,1,1))]. 972 (1 @ x) \ x = x @ 1. [para(970(a,1),806(a,1,2))]. 973 (1 @ x) * (y @ 1) = y @ x. [para(750(a,1),971(a,1,2))]. 974 ((1 @ x) * (y @ 1)) @ z = y @ z. [para(973(a,2),750(a,1,1))]. 976 ((1 @ x) * y) @ z = y @ z. [para(974(a,1),931(a,1)),flip(a)]. 977 ((x \ 1) * y) @ z = y @ z. [para(439(a,1),976(a,1,1,1))]. 979 x @ (y * x) = (1 @ y) * x. [para(976(a,1),705(a,1))]. 980 ((x @ 1) \ y) @ z = (x \ y) @ z. [para(811(a,1),976(a,1,1))]. 981 ((x @ 1) \ y) @ z = ((x @ u) \ y) @ z. [para(813(a,1),976(a,1,1))]. 982 x @ (y \ x) = (y \ 1) * x. [para(977(a,1),703(a,1))]. 983 (1 @ (x / y)) * y = y @ x. [para(979(a,1),582(a,1))]. 984 (x \ y) V (y @ (x \ y)) = x \ y. [para(982(a,2),266(a,1,2))]. 985 x @ (y \ x) = (1 / y) * x. [para(336(a,2),982(a,2,1))]. 986 x / (x @ (y \ x)) = 1 @ y. [para(982(a,2),382(a,1,2))]. 988 (x @ y) / x = 1 @ (y / x). [para(983(a,1),743(a,1,1))]. 989 1 V (((1 / x) * y) * z) = 1 V ((x \ y) * z). [para(985(a,1),605(a,1,2,1))]. 990 x / (x @ y) = 1 @ (x / y). [para(556(a,1),986(a,1,2))]. 993 ((x @ y) / x) \ (y / x) = (y / x) @ 1. [para(988(a,2),972(a,1,1))]. 994 (x @ y) V ((1 @ (x / y)) \ x) = (x / (x @ y)) \ x. [para(990(a,1),157(a,1,2,1))]. 998 (x @ y) / x = y / (y @ x). [para(990(a,2),988(a,2))]. 999 ((x @ y) \ z) @ u = (x \ z) @ u. [para(981(a,1),980(a,1))]. 1001 (x @ y) \ y = (x \ y) @ 1. [para(999(a,1),574(a,1)),flip(a)]. 1002 (x \ y) @ (z \ y) = (x @ z) \ y. [para(999(a,1),707(a,1))]. 1003 (x \ (y @ z)) @ (u \ z) = (x @ u) \ (y @ z). [para(999(a,1),759(a,1))]. 1004 (((x \ 1) * y) \ z) @ u = (y \ z) @ u. [para(982(a,1),999(a,1,1,1))]. 1006 (x @ 1) \ y = (x \ y) @ y. [para(119(a,1),1002(a,1,2)),flip(a)]. 1007 (x @ (y / z)) \ y = (x \ y) @ z. [para(1002(a,1),556(a,1))]. 1009 x ^ ((y \ x) @ x) = x. [para(1006(a,1),330(a,1,2))]. 1010 (x @ y) ^ ((z \ (x @ y)) @ y) = x @ y. [para(539(a,1),1009(a,1,2))]. 1011 (x \ ((1 @ y) * z)) @ u = (x \ z) @ u. [para(810(a,1),1004(a,1,1))]. 1012 (x \ (y @ z)) @ u = (x \ y) @ u. [para(983(a,1),1011(a,1,1,2))]. 1013 (x @ y) ^ ((z \ x) @ y) = x @ y. [para(1012(a,1),1010(a,1,2))]. 1014 (x @ y) ^ ((x V z) @ y) = x @ y. [para(34(a,1),1013(a,1,2,1))]. 1015 (x @ y) ^ ((z V x) @ y) = x @ y. [para(81(a,1),1013(a,1,2,1))]. 1017 (x @ y) V ((x V z) @ y) = (x V z) @ y. [para(1014(a,1),65(a,1,1))]. 1018 (x @ y) V ((z V x) @ y) = (z V x) @ y. [para(1015(a,1),65(a,1,1))]. 1019 (x V y) V ((x V y) @ z) = (x V y) V (x @ z). [para(1017(a,1),876(a,1,2))]. 1020 (x @ y) V ((z V (x @ u)) @ y) = (z V (x @ u)) @ y. [para(750(a,1),1018(a,1,1))]. 1021 1 V ((1 / x) * (y * z)) = 1 V ((x \ y) * z). [para(28(a,1),989(a,1,2))]. 1023 (x \ y) @ (z \ u) = (x @ z) \ (y @ u) # label("Goal 6"). [para(1003(a,1),1012(a,1)),flip(a)]. 1025 (x \ y) @ (x \ z) = x \ (y @ z). [para(697(a,1),1023(a,2,1))]. 1027 x \ (y @ x) = (x \ y) @ 1. [para(278(a,1),1025(a,1,2)),flip(a)]. 1028 1 V ((x \ (y @ z)) * u) = 1 V ((x \ z) * u). [para(1025(a,1),605(a,1,2,1))]. 1029 (x @ (x / y)) / (((x / y) \ x) @ 1) = x / y. [para(1027(a,1),935(a,1,2))]. 1030 x V (y V ((x V y) @ z)) = (x V y) V (x @ z). [para(1019(a,1),17(a,1)),flip(a)]. 1031 ((x V y) V (x @ z)) @ u = (x V y) @ u. [para(1019(a,1),870(a,1,1))]. 1032 (x V (y V (x @ z))) @ u = (x V y) @ u. [para(17(a,1),1031(a,1,1))]. 1033 (x V ((y @ (x ^ z)) V (x @ u))) @ w = (x V (y @ x)) @ w. [para(866(a,1),1032(a,2,1))]. 1034 1 V (x \ (y * z)) = 1 V ((x \ y) * z). [para(1021(a,1),757(a,1)),flip(a)]. 1035 1 V (((x ^ (y V ((z @ 1) * x))) \ (z @ 1)) * x) = 1. [para(1034(a,1),625(a,1))]. 1036 x V (y V ((x V y) @ z)) = x V (y V (x @ z)). [para(1030(a,2),17(a,1))]. 1037 (x @ y) V ((1 \ x) @ y) = (x / (x @ y)) \ x. [para(1007(a,1),994(a,1,2))]. 1038 (x @ y) V (x @ y) = (x / (x @ y)) \ x. [para(119(a,1),1037(a,1,2,1))]. 1039 (x / (x @ y)) \ x = x @ y. [para(1038(a,1),19(a,1))]. 1040 ((x @ y) / x) \ y = y @ x. [para(998(a,2),1039(a,1,1))]. 1041 (x V (y @ z)) V ((x V (y @ z)) @ u) = (x V (y @ z)) V (y @ u). [para(1020(a,1),876(a,1,2))]. 1042 1 V (((x ^ (y V ((z @ 1) * x))) \ 1) * x) = 1. [para(1035(a,1),1028(a,1)),flip(a)]. 1043 1 V ((x ^ (y V ((z @ 1) * x))) \ x) = 1. [para(1042(a,1),756(a,1)),flip(a)]. 1045 (x ^ (y V ((z @ 1) * x))) \ x = 1. [para(1043(a,1),130(a,1)),flip(a)]. 1046 ((x ^ (1 @ (1 \ y))) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(722(a,1),1045(a,1,1,2))]. 1047 ((x ^ (1 @ y)) ^ y) \ (x ^ (1 @ (1 \ y))) = 1. [para(119(a,1),1046(a,1,1,1,2,2))]. 1048 (x ^ (y ^ (1 @ x))) \ (y ^ (1 @ (1 \ x))) = 1. [para(20(a,1),1047(a,1,1))]. 1049 (x ^ y) \ (y ^ (1 @ (1 \ x))) = 1. [para(305(a,1),1048(a,1,1))]. 1050 (x ^ y) \ (y ^ (1 @ x)) = 1. [para(119(a,1),1049(a,1,2,2,2))]. 1051 x * ((y ^ z) \ (z ^ (1 @ y))) = x. [para(1050(a,2),30(a,1,2))]. 1052 x @ (y ^ (1 @ z)) = x @ (z ^ y). [para(1051(a,1),560(a,1,2)),flip(a)]. 1053 x @ (y ^ (1 @ z)) = x @ (y ^ z). [para(20(a,1),1052(a,2,2))]. 1054 x @ ((y @ z) ^ u) = x @ (u ^ (1 @ z)). [para(539(a,1),1052(a,1,2,2)),flip(a)]. 1055 x @ ((y @ z) ^ u) = x @ (z ^ u). [para(1054(a,2),1052(a,1))]. 1056 x @ ((y @ z) ^ u) = x @ (u ^ z). [para(1054(a,2),1053(a,1))]. 1057 x @ (y ^ (z @ u)) = x @ (u ^ y). [para(20(a,1),1055(a,1,2))]. 1058 x @ (y ^ (z @ u)) = x @ (y ^ u). [para(20(a,1),1056(a,1,2))]. 1060 (x ^ (y @ z)) @ (z ^ x) = x ^ (y @ z). [para(1057(a,1),697(a,1))]. 1061 (x ^ (y @ z)) @ (x ^ z) = x ^ (y @ z). [para(1058(a,1),697(a,1))]. 1062 (x @ (y ^ z)) \ (y ^ (u @ z)) = (x \ (y ^ (u @ z))) @ 1. [para(1058(a,1),1001(a,1,1))]. 1063 (1 @ (x ^ y)) ^ (x ^ (z @ y)) = (x ^ (z @ y)) @ (x ^ y). [para(1061(a,1),256(a,1,2))]. 1064 x ^ ((y @ z) ^ (1 @ (x ^ z))) = (x ^ (y @ z)) @ (x ^ z). [para(1063(a,1),60(a,2))]. 1065 (x ^ (1 @ y)) @ (x ^ y) = x ^ (1 @ (x ^ y)). [para(889(a,1),1064(a,1,2)),flip(a)]. 1066 x ^ (1 @ (x ^ y)) = x ^ (1 @ y). [para(1065(a,1),1061(a,1))]. 1067 (x @ (x ^ y)) \ (x ^ (1 @ y)) = 1. [para(1066(a,1),895(a,1,2))]. 1068 (x \ (x ^ (1 @ y))) @ 1 = 1. [para(1067(a,1),1062(a,1)),flip(a)]. 1069 x \ ((x ^ (1 @ y)) @ x) = 1. [para(1068(a,1),1027(a,2))]. 1070 ((x ^ (1 @ y)) @ x) V (x * 1) = (x ^ (1 @ y)) @ x. [para(1069(a,1),99(a,1,2,2))]. 1071 ((x ^ (1 @ y)) @ x) V x = (x ^ (1 @ y)) @ x. [para(30(a,1),1070(a,1,2))]. 1072 x V ((x ^ (1 @ y)) @ x) = (x ^ (1 @ y)) @ x. [para(1071(a,1),21(a,1)),flip(a)]. 1073 (x ^ (1 @ y)) @ x = x. [para(1072(a,1),844(a,1))]. 1074 (x ^ (1 @ y)) @ z = x @ z. [para(1073(a,1),750(a,1,1)),flip(a)]. 1075 x ^ (1 @ y) = x @ (y ^ x). [para(1074(a,1),1060(a,1)),flip(a)]. 1076 x ^ (1 @ y) = x @ (x ^ y). [para(1074(a,1),1061(a,1)),flip(a)]. 1077 (x @ y) ^ (1 @ z) = x @ (z ^ (x @ y)). [para(1075(a,2),750(a,1))]. 1079 (x @ y) ^ (1 @ z) = x @ (z ^ y). [para(1077(a,2),1058(a,1))]. 1082 x ^ (y @ (x ^ z)) = x ^ (y @ z). [para(1079(a,1),305(a,1,2))]. 1084 x @ (x ^ y) = x ^ (x @ y). [para(1082(a,1),738(a,1)),flip(a)]. 1086 x ^ (1 @ y) = x ^ (x @ y). [para(1084(a,1),1076(a,2))]. 1087 x ^ (y ^ (y @ x)) = x ^ y. [para(1086(a,1),305(a,1,2))]. 1088 x ^ ((x @ y) ^ y) = y ^ x. [para(1087(a,1),60(a,2))]. 1089 x ^ (y ^ (x @ y)) = y ^ x. [para(20(a,1),1088(a,1,2))]. 1090 x ^ (y ^ (x @ y)) = x ^ y. [para(1089(a,2),20(a,1))]. 1091 (x ^ y) ^ (x @ y) = x ^ y. [para(1090(a,1),16(a,2))]. 1092 (x ^ y) V (x @ y) = x @ y. [para(1091(a,1),65(a,1,1))]. 1093 (x ^ y) V (y @ x) = y @ x. [para(20(a,1),1092(a,1,1))]. 1094 (x * y) V (((z V (x * (y * u))) / u) @ (x * y)) = ((z V (x * (y * u))) / u) @ (x * y). [para(163(a,1),1093(a,1,1))]. 1109 x V ((y @ z) V ((x V (y @ z)) @ u)) = (x V (y @ z)) V (y @ u). [para(1041(a,1),17(a,1)),flip(a)]. 1110 (x V (y @ z)) V (y @ u) = x V ((y @ z) V (x @ u)). [para(1109(a,1),1036(a,1))]. 1111 x V ((y @ z) V (y @ u)) = x V ((y @ z) V (x @ u)). [para(1110(a,1),17(a,1)),flip(a)]. 1112 x V ((y @ z) V (x @ u)) = x V (y @ (z V u)). [para(885(a,1),1111(a,1,2)),flip(a)]. 1113 (x V (y @ ((x ^ z) V u))) @ w = (x V (y @ x)) @ w. [para(1112(a,1),1033(a,1,1))]. 1114 (x V (y @ z)) @ u = (x V (y @ x)) @ u. [para(65(a,1),1113(a,1,1,2,2))]. 1115 (x V (y @ x)) @ z = (x V y) @ z. [para(697(a,1),1114(a,1,1,2)),flip(a)]. 1118 (x V (y @ z)) @ u = (x V (y @ w)) @ u. [para(1114(a,2),1114(a,2))]. 1120 (x V y) @ (x V x) = x V (y @ x). [para(1115(a,1),785(a,1))]. 1122 (x V y) @ x = x V (y @ x). [para(19(a,1),1120(a,1,2))]. 1123 (x V y) @ y = y V (x @ y). [para(21(a,1),1122(a,1,1))]. 1124 (x V (y \ x)) @ (y \ x) = y \ x. [para(1123(a,2),984(a,1))]. 1125 (x V (y \ x)) @ z = (y \ x) @ z. [para(1124(a,1),750(a,1,1)),flip(a)]. 1127 (x V (y @ z)) @ (x V u) = x V (y @ u). [para(1118(a,1),785(a,1))]. 1128 (x V (y @ z)) @ u = (y V x) @ u. [para(860(a,1),1118(a,1,1)),flip(a)]. 1133 ((x @ y) V z) @ u = (x V z) @ u. [para(21(a,1),1128(a,1,1))]. 1135 (x V y) @ (x V z) = x V (y @ z). [para(697(a,1),1127(a,1,1,2))]. 1136 (x V y) @ (z V x) = x V (y @ z). [para(21(a,1),1135(a,1,2))]. 1137 ((x @ y) V z) @ (y V u) = (x @ y) V (z @ u). [para(1135(a,1),784(a,1)),flip(a)]. 1138 (x \ y) @ (y V z) = y V ((x \ y) @ z). [para(1135(a,1),1125(a,1)),flip(a)]. 1139 (x V y) \ (x V ((z \ x) @ y)) = ((x V y) \ (z \ x)) @ 1. [para(1138(a,1),1027(a,1,2))]. 1140 (x V y) @ (z V u) = (x @ z) V (y @ u). [para(1137(a,1),1133(a,1)),flip(a)]. 1142 (x @ y) V (z @ y) = (x V z) @ y. [para(19(a,1),1140(a,1,2)),flip(a)]. 1144 (x @ y) V (z @ x) = x V (z @ y). [para(1140(a,1),1136(a,1))]. 1146 (x @ y) V (y @ z) = y V (x @ z). [para(1144(a,1),21(a,1)),flip(a)]. 1159 (x V y) \ (x V ((x / y) \ x)) = ((x V y) \ ((x / y) \ x)) @ 1. [para(700(a,1),1139(a,1,2,2))]. 1166 ((x V y) \ ((x / y) \ x)) @ 1 = 1. [para(1159(a,1),912(a,1))]. 1168 ((x V y) @ ((x / y) \ x)) \ ((x / y) \ x) = 1. [para(1166(a,1),1001(a,2))]. 1169 ((x V y) @ y) \ ((x / y) \ x) = 1. [para(556(a,1),1168(a,1,1))]. 1171 (x V (y @ x)) \ ((y / x) \ y) = 1. [para(1123(a,1),1169(a,1,1))]. 1172 (x / y) ^ (x / ((y V (x @ y)) * 1)) = x / y. [para(1171(a,1),179(a,1,2,2,2))]. 1173 (x / y) ^ (x / (y V (x @ y))) = x / y. [para(30(a,1),1172(a,1,2,2))]. 1174 x / (y V (x @ y)) = x / y. [para(1173(a,1),235(a,1)),flip(a)]. 1175 x / ((y @ z) V (x @ z)) = x / (y @ z). [para(539(a,1),1174(a,1,2,2))]. 1176 x / ((y V x) @ z) = x / (y @ z). [para(1142(a,1),1175(a,1,2))]. 1177 x / (x V (y @ x)) = x / (y @ x). [para(1146(a,1),1175(a,1,2))]. 1178 (x @ y) / ((x V z) @ u) = (x @ y) / (z @ u). [para(1128(a,1),1176(a,1,2))]. 1179 (x / (y @ x)) \ x = x V (y @ x). [para(1177(a,1),34(a,1,1))]. 1180 ((x @ y) / (y @ (x @ y))) \ (x @ y) = y V (x @ (x @ y)). [para(1179(a,2),1146(a,1))]. 1181 ((x @ y) / y) \ (x @ y) = y V (x @ (x @ y)). [para(699(a,1),1180(a,1,1,2))]. 1182 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (y * ((y \ x) / z))). [para(196(a,1),1094(a,1,2,1,1)),flip(a)]. 1183 (x \ (y / z)) / (((x \ y) / z) \ (x \ (y / z))) = (x \ y) / z. [para(607(a,1),669(a,2,2))]. 1184 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V ((x / z) @ (x / z)). [para(793(a,1),1182(a,2,2))]. 1185 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (y * ((y \ x) / z)) V (x / z). [para(697(a,1),1184(a,2,2))]. 1187 ((x V (y * (((y \ x) / z) * z))) / z) @ (y * ((y \ x) / z)) = (x / z) V (y * ((y \ x) / z)). [para(1185(a,2),21(a,1))]. 1188 (x / y) @ (z * ((z \ x) / y)) = (x / y) V (z * ((z \ x) / y)). [para(196(a,1),1187(a,1,1,1))]. 1189 (x / y) V (z * ((z \ x) / y)) = (x / y) @ (x / y). [para(1188(a,1),793(a,1))]. 1190 (x / y) V (z * ((z \ x) / y)) = x / y. [para(1189(a,2),697(a,1))]. 1192 ((x \ y) / z) \ (x \ (y / z)) = 1. [para(1190(a,1),938(a,1,2,2))]. 1193 (x \ (y / z)) / 1 = (x \ y) / z. [para(1192(a,1),1183(a,1,2))]. 1195 (x \ y) / z = x \ (y / z). [para(1193(a,1),146(a,1))]. 1197 (((x @ y) / x) \ y) / x = (y / x) @ 1. [para(1195(a,2),993(a,1))]. 1198 (((x / y) / ((x / y) @ z)) \ x) / y = (x / y) @ z. [para(1195(a,2),1039(a,1))]. 1199 (x @ y) / y = (x / y) @ 1. [para(1040(a,1),1197(a,1,1))]. 1200 x V (((y / x) @ 1) \ (y @ x)) = ((y @ x) / x) \ (y @ x). [para(1199(a,1),157(a,1,2,1))]. 1201 ((1 @ ((x / y) / z)) \ x) / y = (x / y) @ z. [para(990(a,1),1198(a,1,1,1))]. 1202 ((1 @ (x / (y * z))) \ x) / z = (x / z) @ y. [para(797(a,1),1201(a,1,1,1))]. 1203 ((1 \ x) @ (y * z)) / z = (x / z) @ y. [para(1007(a,1),1202(a,1,1))]. 1204 (x @ (y * z)) / z = (x / z) @ y. [para(119(a,1),1203(a,1,1,1))]. 1205 (x / y) @ (z / y) = (x @ z) / y. [para(582(a,1),1204(a,1,1)),flip(a)]. 1206 ((x @ y) / z) @ u = (x / z) @ u. [para(1205(a,1),750(a,1,1))]. 1208 x V (((y / x) \ y) @ (1 \ x)) = ((y @ x) / x) \ (y @ x). [para(1023(a,2),1200(a,1,2))]. 1209 ((x @ y) / y) \ (x @ y) = y V (((x / y) \ x) @ y). [para(119(a,1),1208(a,1,2,2)),flip(a)]. 1210 ((x @ y) / y) \ (x @ y) = (x / y) \ x. [para(1209(a,2),741(a,1))]. 1211 x V (y @ (y @ x)) = (y / x) \ y. [para(1210(a,1),1181(a,1)),flip(a)]. 1213 ((x / y) \ x) @ z = (x V y) @ z. [para(1211(a,1),1128(a,1,1))]. 1221 (x @ (x / y)) / ((x V y) @ 1) = x / y. [para(1213(a,1),1029(a,1,2))]. 1223 (x @ (x / y)) / (y @ 1) = x / y. [para(1221(a,1),1178(a,1)),flip(a)]. 1226 (x / (y @ 1)) @ z = (x / y) @ z. [para(1223(a,1),1206(a,1,1)),flip(a)]. 1227 (x / (y @ 1)) @ z = (x / (y @ u)) @ z. [para(750(a,1),1226(a,1,1,2))]. 1232 (x / (y @ z)) @ u = (x / y) @ u. [para(1227(a,1),1226(a,1))]. 1234 ((x @ y) / z) @ (y / u) = (x @ y) / (z @ u). [para(1232(a,1),758(a,1))]. 1236 (x / y) @ (z / u) = (x @ z) / (y @ u) # label("Goal 7"). [para(1234(a,1),1206(a,1)),flip(a)]. 1237 $F # answer("Goal 7"). [resolve(1236,a,46,a)]. ============================== end of proof ========================== ============================== STATISTICS ============================ Given=1051. Generated=1836287. Kept=1196. proofs=15. Usable=901. Sos=122. Demods=0. Limbo=1, Disabled=204. Hints=1291. Kept_by_rule=0, Deleted_by_rule=1823442. Forward_subsumed=11649. Back_subsumed=157. Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. Demod_attempts=0. Demod_rewrites=0. Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. Megabytes=3.47. User_CPU=10.64, System_CPU=0.24, Wall_clock=11. ============================== end of statistics ===================== ============================== end of search ========================= THEOREM PROVED THEOREM PROVED Exiting with 15 proofs. ------ process 4210 exit (max_proofs) ------  Process 4210 exit (max_proofs) Sun Jan 31 18:00:17 2010