============================== Prover9 =============================== Prover9 (64) version Aug-2007, Aug 2007. Process 26326 was started by veroff on titan, Sat Jan 19 14:19:09 2008 The command was "prover9 -f 4.9.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file 4.9.in clear(auto). % clear(auto) -> clear(auto_inference). % clear(auto_inference) -> clear(predicate_elim). % clear(auto_inference) -> assign(eq_defs, pass). % clear(auto) -> clear(auto_limits). % clear(auto_limits) -> assign(max_weight, 2147483647). % clear(auto_limits) -> assign(sos_limit, -1). % clear(auto) -> clear(auto_denials). % clear(auto) -> clear(auto_process). op(400,infix,[^,v,"->","=>",*]). op(200,prefix,[~,-]). lex([0,1,*,^,=>,->,v,~,-]). assign(order,kbo). set(lex_order_vars). set(paramodulation). % set(paramodulation) -> set(back_demod). set(para_units_only). % set(para_units_only) -> assign(para_lit_limit, 1). clear(back_demod). set(hyper_resolution). % set(hyper_resolution) -> set(pos_hyper_resolution). set(auto_denials). set(restrict_denials). formulas(assumptions). (x v y) v z = x v (y v z) # label("(D1)"). (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). x v y = y v x # label("(D3)"). x ^ y = y ^ x # label("(D4)"). x v (x ^ y) = x # label("(D5)"). x ^ (x v y) = x # label("(D6)"). (x => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). 1 => x = x # label("(BCK2)"). x => 1 = 1 # label("(BCK3)"). x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). x * 1 = x # label("(M1)"). x * y = y * x # label("(M2)"). (x * y) * z = x * (y * z) # label("(M3)"). (x * y) => z = x => (y => z) # label("(P)"). x ^ 1 = x # label("(D10)"). x ^ ((x => y) => y) = x # label("(3.7)"). (x ^ y) => y = 1 # label("(3.11)"). x v 0 = x # label("(D9)"). x -> y = x => (x => y) # label("(-> def)"). ~ x = x => 0 # label("(~ def)"). -x = x => (x => 0) # label("(- def)"). x => (x => (x => y)) = x => (x => y) # label("(E_2)"). (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). ~ ~ x = x # label("(DN)"). (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). x => ((x => y) => y) = 1 # label("(BCK5)"). x => x = 1 # label("(3.16)"). x => (y => x) = 1 # label("(3.17)"). x => (y => z) = y => (x => z) # label("(3.18)"). (x * y) v (x * z) = x * (y v z) # label("(4.1)"). (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). (x * (x => y)) v y = y # label("(4.3)"). (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). end_of_list. formulas(assumptions). (x => y) ^ y = y # label("(3.13)"). (x v y) => y = x => y # label("(4.6)"). x -> (y => z) = y => (x -> z) # label("(4.7)"). x ^ (~ x => y) = x # label("(4.8)"). x -> ((y v x) -> z) = x -> z # label("(4.9)"). x -> (~ x v y) = x -> y # label("(4.10)"). x -> x = 1 # label("(N2)"). (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). (x ^ y) -> z = x -> (y -> z) # label("(N6)"). end_of_list. formulas(goals). x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). end_of_list. formulas(hints). (x * x) => y = x -> y. (x * y) => (x => (y => z)) = (x * y) -> z. ~ (x * y) = x => (y => 0). x => (y => 0) = ~ (x * y). ~ x => 0 = x. x => (y -> z) = y => (x => (y => z)). x => (y => (x => z)) = y => (x -> z). x => ~ y = y => (x => 0). x => (y => 0) = y => ~ x. ~ x ^ (y => 0) = (x v y) => 0. x -> (x -> y) = x -> y. x -> (y v ~ x) = x -> y. (x -> (y -> z)) ^ (~ z -> ~ (x ^ y)) = (x ^ y) => z. (x -> y) ^ y = y. x => ~ y = ~ (x * y). ~ (x * y) = x => ~ y. ~ (x => ~ y) = x * y. ~ (x => y) = x * ~ y. ~ (x -> y) = x * ~ (x => y). x * ~ (x => y) = ~ (x -> y). x => y = ~ y => ~ x. ~ x => ~ y = y => x. x -> (y => z) = ~ z => (x -> ~ y). ~ x => (y -> ~ z) = y -> (z => x). (x -> y) ^ (y v ~ x) = y v ~ x. x * (x * ~ y) = ~ (x -> y). ~ x ^ ~ y = (x v y) => 0. (x v y) => 0 = ~ x ^ ~ y. ~ x ^ ~ y = ~ (x v y). x ^ ~ y = ~ (~ x v y). ~ (~ x v y) = x ^ ~ y. ~ (x v y) -> z = ~ x -> (~ y -> z). ~ x -> (~ y -> z) = ~ (x v y) -> z. ~ (x ^ ~ y) = ~ x v y. ~ (x ^ y) = ~ x v ~ y. ~ x v ~ y = ~ (x ^ y). x -> ~ (x ^ y) = x -> ~ y. ~ x => (y -> ~ z) = z => (y -> x). ~ x => (y -> ~ z) = (y ^ z) => (y -> x). x -> (~ y -> z) = ~ (~ x v y) -> z. ~ (~ x v y) -> z = x -> (~ y -> z). ~ (x v ~ y) -> z = y -> (~ x -> z). (x * y) -> z = x => (y => (x => (y => z))). x => (y => (x => (y => z))) = (x * y) -> z. (x ^ y) => (x -> z) = y => (x -> z). (x ^ y) => (y -> z) = x => (y -> z). x => (x => (y -> z)) = (x * y) -> z. (x * y) -> z = x -> (y -> z). ~ (x -> y) -> z = x -> ((x * ~ y) -> z). x -> ((x * ~ y) -> z) = ~ (x -> y) -> z. x -> (x -> (~ y -> z)) = ~ (x -> y) -> z. ~ (x -> y) -> z = x -> (~ y -> z). (x -> (y -> z)) ^ (~ (y -> z) -> ~ (x ^ y)) = (x ^ y) => (y -> z). (x -> (y -> z)) ^ (~ (z v ~ y) -> ~ (x ^ y)) = (x ^ y) => (z v ~ y). (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = (x ^ y) => (y -> z). (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = x => (y -> z). (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = (x ^ y) => (z v ~ y). (x ^ y) => (z v ~ y) = x => (y -> z). (x ^ y) ^ ((x => (y -> z)) => (z v ~ y)) = x ^ y. ((x -> y) ^ x) ^ (1 => (y v ~ x)) = (x -> y) ^ x. (x ^ (x -> y)) ^ (1 => (y v ~ x)) = (x -> y) ^ x. (x ^ (x -> y)) ^ (y v ~ x) = (x -> y) ^ x. (x -> y) ^ x = x ^ ((x -> y) ^ (y v ~ x)). x ^ ((x -> y) ^ (y v ~ x)) = (x -> y) ^ x. x ^ (y v ~ x) = (x -> y) ^ x. x ^ (~ x v y) = (x -> y) ^ x. end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)") # 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 v y) v z = x v (y v z) # label("(D1)"). [assumption]. (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. x v y = y v x # label("(D3)"). [assumption]. x ^ y = y ^ x # label("(D4)"). [assumption]. x v (x ^ y) = x # label("(D5)"). [assumption]. x ^ (x v y) = x # label("(D6)"). [assumption]. (x => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). [assumption]. 1 => x = x # label("(BCK2)"). [assumption]. x => 1 = 1 # label("(BCK3)"). [assumption]. x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. x * 1 = x # label("(M1)"). [assumption]. x * y = y * x # label("(M2)"). [assumption]. (x * y) * z = x * (y * z) # label("(M3)"). [assumption]. (x * y) => z = x => (y => z) # label("(P)"). [assumption]. x ^ 1 = x # label("(D10)"). [assumption]. x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. (x ^ y) => y = 1 # label("(3.11)"). [assumption]. x v 0 = x # label("(D9)"). [assumption]. x -> y = x => (x => y) # label("(-> def)"). [assumption]. ~ x = x => 0 # label("(~ def)"). [assumption]. -x = x => (x => 0) # label("(- def)"). [assumption]. x => (x => (x => y)) = x => (x => y) # label("(E_2)"). [assumption]. (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. ~ ~ x = x # label("(DN)"). [assumption]. (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). [assumption]. (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. x => ((x => y) => y) = 1 # label("(BCK5)"). [assumption]. x => x = 1 # label("(3.16)"). [assumption]. x => (y => x) = 1 # label("(3.17)"). [assumption]. x => (y => z) = y => (x => z) # label("(3.18)"). [assumption]. (x * y) v (x * z) = x * (y v z) # label("(4.1)"). [assumption]. (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). [assumption]. (x * (x => y)) v y = y # label("(4.3)"). [assumption]. (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). [assumption]. ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). [assumption]. (x => y) ^ y = y # label("(3.13)"). [assumption]. (x v y) => y = x => y # label("(4.6)"). [assumption]. x -> (y => z) = y => (x -> z) # label("(4.7)"). [assumption]. x ^ (~ x => y) = x # label("(4.8)"). [assumption]. x -> ((y v x) -> z) = x -> z # label("(4.9)"). [assumption]. x -> (~ x v y) = x -> y # label("(4.10)"). [assumption]. x -> x = 1 # label("(N2)"). [assumption]. (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. c1 ^ (~ c1 v c2) != c1 ^ (c1 -> c2) # label("(N4)"). [deny(1)]. end_of_list. formulas(demodulators). end_of_list. % 66 hints input. Auto_denials: % copying label "(N4)" to answer in negative clause Term ordering decisions: Function symbol KB weights: 0=1. 1=1. c1=1. c2=1. *=1. ^=1. =>=1. ->=1. v=1. ~=1. -=1. Predicate symbol precedence: predicate_order([ = ]). Function symbol precedence: function_order([ 0, 1, c1, c2, *, ^, =>, ->, v, ~, - ]). Skipping inverse_order, because there is a function_order (lex) command. % Operation v is commutative; C redundancy checks enabled. % Operation ^ is commutative; C redundancy checks enabled. % Operation * is commutative; C redundancy checks enabled. restricted denial: (wt=12): 51 c1 ^ (~ c1 v c2) != c1 ^ (c1 -> c2) # label("(N4)") # answer("(N4)"). [deny(1)]. ============================== end of process initial clauses ======== ============================== CLAUSES FOR SEARCH ==================== % Clauses after input processing: formulas(usable). 51 c1 ^ (~ c1 v c2) != c1 ^ (c1 -> c2) # label("(N4)") # answer("(N4)"). [deny(1)]. end_of_list. formulas(sos). 2 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. 3 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. 4 x v y = y v x # label("(D3)"). [assumption]. 5 x ^ y = y ^ x # label("(D4)"). [assumption]. 6 x v (x ^ y) = x # label("(D5)"). [assumption]. 7 x ^ (x v y) = x # label("(D6)"). [assumption]. 8 (x => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). [assumption]. 9 1 => x = x # label("(BCK2)"). [assumption]. 10 x => 1 = 1 # label("(BCK3)"). [assumption]. 11 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. 12 x * 1 = x # label("(M1)"). [assumption]. 13 x * y = y * x # label("(M2)"). [assumption]. 14 (x * y) * z = x * (y * z) # label("(M3)"). [assumption]. 15 (x * y) => z = x => (y => z) # label("(P)"). [assumption]. 16 x ^ 1 = x # label("(D10)"). [assumption]. 17 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. 18 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. 19 x v 0 = x # label("(D9)"). [assumption]. 21 x => (x => y) = x -> y. [copy(20),flip(a)]. 23 x => 0 = ~ x. [copy(22),flip(a)]. 25 x => (x => 0) = -x. [copy(24),flip(a)]. 26 x => (x => (x => y)) = x => (x => y) # label("(E_2)"). [assumption]. 27 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. 28 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. 29 ~ ~ x = x # label("(DN)"). [assumption]. 30 (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). [assumption]. 31 (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. 32 x => ((x => y) => y) = 1 # label("(BCK5)"). [assumption]. 33 x => x = 1 # label("(3.16)"). [assumption]. 34 x => (y => x) = 1 # label("(3.17)"). [assumption]. 35 x => (y => z) = y => (x => z) # label("(3.18)"). [assumption]. 36 (x * y) v (x * z) = x * (y v z) # label("(4.1)"). [assumption]. 37 (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). [assumption]. 38 (x * (x => y)) v y = y # label("(4.3)"). [assumption]. 39 (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). [assumption]. 40 ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). [assumption]. 41 (x => y) ^ y = y # label("(3.13)"). [assumption]. 42 (x v y) => y = x => y # label("(4.6)"). [assumption]. 43 x -> (y => z) = y => (x -> z) # label("(4.7)"). [assumption]. 44 x ^ (~ x => y) = x # label("(4.8)"). [assumption]. 45 x -> ((y v x) -> z) = x -> z # label("(4.9)"). [assumption]. 46 x -> (~ x v y) = x -> y # label("(4.10)"). [assumption]. 47 x -> x = 1 # label("(N2)"). [assumption]. 48 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 49 (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. 50 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. end_of_list. formulas(demodulators). end_of_list. % 66 hints processed (15 are redundant). ============================== end of clauses for search ============= ============================== SEARCH ================================ % Starting search at 0.01 seconds. given #1 (I,wt=11): 2 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. given #2 (I,wt=11): 3 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. given #3 (I,wt=7): 4 x v y = y v x # label("(D3)"). [assumption]. given #4 (I,wt=7): 5 x ^ y = y ^ x # label("(D4)"). [assumption]. given #5 (I,wt=7): 6 x v (x ^ y) = x # label("(D5)"). [assumption]. given #6 (I,wt=7): 7 x ^ (x v y) = x # label("(D6)"). [assumption]. given #7 (I,wt=13): 8 (x => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). [assumption]. given #8 (I,wt=5): 9 1 => x = x # label("(BCK2)"). [assumption]. given #9 (I,wt=5): 10 x => 1 = 1 # label("(BCK3)"). [assumption]. given #10 (I,wt=13): 11 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. given #11 (I,wt=5): 12 x * 1 = x # label("(M1)"). [assumption]. given #12 (I,wt=7): 13 x * y = y * x # label("(M2)"). [assumption]. given #13 (I,wt=11): 14 (x * y) * z = x * (y * z) # label("(M3)"). [assumption]. given #14 (I,wt=11): 15 (x * y) => z = x => (y => z) # label("(P)"). [assumption]. given #15 (I,wt=5): 16 x ^ 1 = x # label("(D10)"). [assumption]. given #16 (I,wt=9): 17 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. given #17 (I,wt=7): 18 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. given #18 (I,wt=5): 19 x v 0 = x # label("(D9)"). [assumption]. given #19 (I,wt=9): 21 x => (x => y) = x -> y. [copy(20),flip(a)]. given #20 (I,wt=6): 23 x => 0 = ~ x. [copy(22),flip(a)]. given #21 (I,wt=8): 25 x => (x => 0) = -x. [copy(24),flip(a)]. given #22 (I,wt=13): 26 x => (x => (x => y)) = x => (x => y) # label("(E_2)"). [assumption]. given #23 (I,wt=13): 27 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. given #24 (I,wt=13): 28 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. given #25 (I,wt=5): 29 ~ ~ x = x # label("(DN)"). [assumption]. given #26 (I,wt=13): 30 (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). [assumption]. given #27 (I,wt=13): 31 (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. given #28 (I,wt=9): 32 x => ((x => y) => y) = 1 # label("(BCK5)"). [assumption]. given #29 (I,wt=5): 33 x => x = 1 # label("(3.16)"). [assumption]. given #30 (I,wt=7): 34 x => (y => x) = 1 # label("(3.17)"). [assumption]. given #31 (I,wt=11): 35 x => (y => z) = y => (x => z) # label("(3.18)"). [assumption]. given #32 (I,wt=13): 36 (x * y) v (x * z) = x * (y v z) # label("(4.1)"). [assumption]. given #33 (I,wt=13): 37 (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). [assumption]. given #34 (I,wt=9): 38 (x * (x => y)) v y = y # label("(4.3)"). [assumption]. given #35 (I,wt=13): 39 (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). [assumption]. given #36 (I,wt=13): 40 ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). [assumption]. given #37 (I,wt=7): 41 (x => y) ^ y = y # label("(3.13)"). [assumption]. given #38 (I,wt=9): 42 (x v y) => y = x => y # label("(4.6)"). [assumption]. given #39 (I,wt=11): 43 x -> (y => z) = y => (x -> z) # label("(4.7)"). [assumption]. given #40 (I,wt=8): 44 x ^ (~ x => y) = x # label("(4.8)"). [assumption]. given #41 (I,wt=11): 45 x -> ((y v x) -> z) = x -> z # label("(4.9)"). [assumption]. given #42 (I,wt=10): 46 x -> (~ x v y) = x -> y # label("(4.10)"). [assumption]. given #43 (I,wt=5): 47 x -> x = 1 # label("(N2)"). [assumption]. given #44 (I,wt=13): 48 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. given #45 (I,wt=13): 49 (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. given #46 (I,wt=11): 50 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. given #47 (H,wt=6): 250 ~ x => 0 = x. [para(29(a,1),23(a,2))]. given #48 (H,wt=9): 144 (x * x) => y = x -> y. [para(21(a,1),15(a,2))]. given #49 (H,wt=7): 857 (x -> y) ^ y = y. [para(144(a,1),41(a,1,1))]. given #50 (H,wt=9): 686 x -> (x -> y) = x -> y. [para(38(a,1),45(a,1,2,1))]. given #51 (H,wt=10): 152 x => (y => 0) = ~ (x * y). [para(23(a,1),15(a,1)),flip(a)]. given #52 (H,wt=9): 919 ~ (x * y) = x => ~ y. [para(23(a,1),152(a,1,2)),flip(a)]. given #53 (H,wt=9): 960 ~ (x => ~ y) = x * y. [para(919(a,1),29(a,1,1))]. given #54 (H,wt=9): 979 ~ (x => y) = x * ~ y. [para(29(a,1),960(a,1,1,2))]. given #55 (H,wt=10): 372 x => (y => 0) = y => ~ x. [para(23(a,1),35(a,1,2)),flip(a)]. given #56 (H,wt=9): 1068 ~ x => ~ y = y => x. [para(250(a,1),372(a,1,2)),flip(a)]. given #57 (H,wt=10): 692 x -> (y v ~ x) = x -> y. [para(4(a,1),46(a,1,2))]. given #58 (H,wt=11): 998 x * ~ (x => y) = ~ (x -> y). [para(21(a,1),979(a,1,1)),flip(a)]. given #59 (H,wt=11): 1196 x * (x * ~ y) = ~ (x -> y). [para(979(a,1),998(a,1,2))]. given #60 (H,wt=12): 509 ~ x ^ (y => 0) = (x v y) => 0. [para(23(a,1),39(a,1,1))]. given #61 (H,wt=11): 1236 (x v y) => 0 = ~ x ^ ~ y. [para(23(a,1),509(a,1,2)),flip(a)]. given #62 (H,wt=10): 1265 ~ x ^ ~ y = ~ (x v y). [para(1236(a,1),23(a,1))]. given #63 (H,wt=10): 1313 ~ (~ x v y) = x ^ ~ y. [para(29(a,1),1265(a,1,1)),flip(a)]. given #64 (H,wt=10): 1331 ~ (x ^ ~ y) = ~ x v y. [para(1313(a,1),29(a,1,1))]. given #65 (H,wt=10): 1368 ~ x v ~ y = ~ (x ^ y). [para(29(a,1),1331(a,1,1,2)),flip(a)]. given #66 (H,wt=11): 1420 x -> ~ (x ^ y) = x -> ~ y. [para(1368(a,1),46(a,1,2))]. given #67 (H,wt=13): 371 x => (y => (x => z)) = y => (x -> z). [para(21(a,1),35(a,1,2)),flip(a)]. given #68 (H,wt=13): 1115 ~ x => (y -> ~ z) = y -> (z => x). [para(1068(a,1),43(a,1,2)),flip(a)]. given #69 (H,wt=13): 1150 (x -> y) ^ (y v ~ x) = y v ~ x. [para(692(a,1),857(a,1,1))]. given #70 (H,wt=13): 1610 ~ x => (y -> ~ z) = z => (y -> x). [para(43(a,1),1115(a,2))]. given #71 (H,wt=14): 1319 ~ x -> (~ y -> z) = ~ (x v y) -> z. [para(1265(a,1),50(a,1,1)),flip(a)]. given #72 (H,wt=14): 1744 ~ (~ x v y) -> z = x -> (~ y -> z). [para(29(a,1),1319(a,1,1)),flip(a)]. given #73 (H,wt=14): 1796 ~ (x v ~ y) -> z = y -> (~ x -> z). [para(4(a,1),1744(a,1,1,1))]. given #74 (H,wt=15): 145 (x * y) => (x => (y => z)) = (x * y) -> z. [para(15(a,1),21(a,1,2))]. given #75 (H,wt=15): 1732 ~ x => (y -> ~ z) = (y ^ z) => (y -> x). [para(1420(a,1),1610(a,1,2))]. given #76 (H,wt=13): 2113 (x ^ y) => (x -> z) = y => (x -> z). [para(1732(a,1),1610(a,1))]. given #77 (H,wt=13): 2124 (x ^ y) => (y -> z) = x => (y -> z). [para(5(a,1),2113(a,1,1))]. given #78 (H,wt=15): 1939 x => (y => (x => (y => z))) = (x * y) -> z. [para(145(a,1),15(a,1)),flip(a)]. given #79 (H,wt=13): 2371 x => (x => (y -> z)) = (x * y) -> z. [para(371(a,1),1939(a,1,2))]. given #80 (H,wt=11): 2399 (x * y) -> z = x -> (y -> z). [para(2371(a,1),21(a,1))]. given #81 (H,wt=15): 2498 x -> ((x * ~ y) -> z) = ~ (x -> y) -> z. [para(1196(a,1),2399(a,1,1)),flip(a)]. given #82 (H,wt=15): 2601 x -> (x -> (~ y -> z)) = ~ (x -> y) -> z. [para(2399(a,1),2498(a,1,2))]. given #83 (H,wt=13): 2637 ~ (x -> y) -> z = x -> (~ y -> z). [para(2601(a,1),686(a,1))]. given #84 (H,wt=19): 768 (x -> (y -> z)) ^ (~ z -> ~ (x ^ y)) = (x ^ y) => z. [para(50(a,1),30(a,1,1))]. given #85 (H,wt=23): 2858 (x -> (y -> z)) ^ (~ (y -> z) -> ~ (x ^ y)) = (x ^ y) => (y -> z). [para(686(a,1),768(a,1,1,2))]. given #86 (H,wt=23): 3017 (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = (x ^ y) => (y -> z). [para(2637(a,1),2858(a,1,2))]. given #87 (H,wt=21): 3122 (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = x => (y -> z). [para(2124(a,1),3017(a,2))]. given #88 (H,wt=25): 2865 (x -> (y -> z)) ^ (~ (z v ~ y) -> ~ (x ^ y)) = (x ^ y) => (z v ~ y). [para(692(a,1),768(a,1,1,2))]. given #89 (H,wt=24): 3348 (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = (x ^ y) => (z v ~ y). [para(1796(a,1),2865(a,1,2))]. given #90 (H,wt=14): 3496 (x ^ y) => (z v ~ y) = x => (y -> z). [para(3348(a,1),3122(a,1))]. given #91 (H,wt=18): 3517 (x ^ y) ^ ((x => (y -> z)) => (z v ~ y)) = x ^ y. [para(3496(a,1),17(a,1,2,1))]. given #92 (H,wt=18): 3625 ((x -> y) ^ x) ^ (1 => (y v ~ x)) = (x -> y) ^ x. [para(33(a,1),3517(a,1,2,1))]. given #93 (H,wt=18): 3712 (x ^ (x -> y)) ^ (1 => (y v ~ x)) = (x -> y) ^ x. [para(5(a,1),3625(a,1,1))]. given #94 (H,wt=16): 3788 (x ^ (x -> y)) ^ (y v ~ x) = (x -> y) ^ x. [para(9(a,1),3712(a,1,2))]. given #95 (H,wt=16): 3855 x ^ ((x -> y) ^ (y v ~ x)) = (x -> y) ^ x. [para(3788(a,1),3(a,1)),flip(a)]. given #96 (H,wt=12): 3970 x ^ (y v ~ x) = (x -> y) ^ x. [para(1150(a,1),3855(a,1,2))]. given #97 (H,wt=12): 4006 x ^ (~ x v y) = (x -> y) ^ x. [para(4(a,1),3970(a,1,2))]. -------- Proof 1 -------- "(N4)". ============================== PROOF ================================= % Proof 1 at 0.17 (+ 0.00) seconds: "(N4)". % Length of proof is 76. % Level of proof is 22. % Maximum clause weight is 25. % Given clauses 97. 1 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)") # label(non_clause) # label(goal). [goal]. 3 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. 4 x v y = y v x # label("(D3)"). [assumption]. 5 x ^ y = y ^ x # label("(D4)"). [assumption]. 9 1 => x = x # label("(BCK2)"). [assumption]. 15 (x * y) => z = x => (y => z) # label("(P)"). [assumption]. 17 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. 20 x -> y = x => (x => y) # label("(-> def)"). [assumption]. 21 x => (x => y) = x -> y. [copy(20),flip(a)]. 22 ~ x = x => 0 # label("(~ def)"). [assumption]. 23 x => 0 = ~ x. [copy(22),flip(a)]. 29 ~ ~ x = x # label("(DN)"). [assumption]. 30 (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). [assumption]. 33 x => x = 1 # label("(3.16)"). [assumption]. 35 x => (y => z) = y => (x => z) # label("(3.18)"). [assumption]. 38 (x * (x => y)) v y = y # label("(4.3)"). [assumption]. 39 (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). [assumption]. 41 (x => y) ^ y = y # label("(3.13)"). [assumption]. 43 x -> (y => z) = y => (x -> z) # label("(4.7)"). [assumption]. 45 x -> ((y v x) -> z) = x -> z # label("(4.9)"). [assumption]. 46 x -> (~ x v y) = x -> y # label("(4.10)"). [assumption]. 50 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. 51 c1 ^ (~ c1 v c2) != c1 ^ (c1 -> c2) # label("(N4)") # answer("(N4)"). [deny(1)]. 144 (x * x) => y = x -> y. [para(21(a,1),15(a,2))]. 145 (x * y) => (x => (y => z)) = (x * y) -> z. [para(15(a,1),21(a,1,2))]. 152 x => (y => 0) = ~ (x * y). [para(23(a,1),15(a,1)),flip(a)]. 250 ~ x => 0 = x. [para(29(a,1),23(a,2))]. 371 x => (y => (x => z)) = y => (x -> z). [para(21(a,1),35(a,1,2)),flip(a)]. 372 x => (y => 0) = y => ~ x. [para(23(a,1),35(a,1,2)),flip(a)]. 509 ~ x ^ (y => 0) = (x v y) => 0. [para(23(a,1),39(a,1,1))]. 686 x -> (x -> y) = x -> y. [para(38(a,1),45(a,1,2,1))]. 692 x -> (y v ~ x) = x -> y. [para(4(a,1),46(a,1,2))]. 768 (x -> (y -> z)) ^ (~ z -> ~ (x ^ y)) = (x ^ y) => z. [para(50(a,1),30(a,1,1))]. 857 (x -> y) ^ y = y. [para(144(a,1),41(a,1,1))]. 919 ~ (x * y) = x => ~ y. [para(23(a,1),152(a,1,2)),flip(a)]. 960 ~ (x => ~ y) = x * y. [para(919(a,1),29(a,1,1))]. 979 ~ (x => y) = x * ~ y. [para(29(a,1),960(a,1,1,2))]. 998 x * ~ (x => y) = ~ (x -> y). [para(21(a,1),979(a,1,1)),flip(a)]. 1068 ~ x => ~ y = y => x. [para(250(a,1),372(a,1,2)),flip(a)]. 1115 ~ x => (y -> ~ z) = y -> (z => x). [para(1068(a,1),43(a,1,2)),flip(a)]. 1150 (x -> y) ^ (y v ~ x) = y v ~ x. [para(692(a,1),857(a,1,1))]. 1196 x * (x * ~ y) = ~ (x -> y). [para(979(a,1),998(a,1,2))]. 1236 (x v y) => 0 = ~ x ^ ~ y. [para(23(a,1),509(a,1,2)),flip(a)]. 1265 ~ x ^ ~ y = ~ (x v y). [para(1236(a,1),23(a,1))]. 1313 ~ (~ x v y) = x ^ ~ y. [para(29(a,1),1265(a,1,1)),flip(a)]. 1319 ~ x -> (~ y -> z) = ~ (x v y) -> z. [para(1265(a,1),50(a,1,1)),flip(a)]. 1331 ~ (x ^ ~ y) = ~ x v y. [para(1313(a,1),29(a,1,1))]. 1368 ~ x v ~ y = ~ (x ^ y). [para(29(a,1),1331(a,1,1,2)),flip(a)]. 1420 x -> ~ (x ^ y) = x -> ~ y. [para(1368(a,1),46(a,1,2))]. 1610 ~ x => (y -> ~ z) = z => (y -> x). [para(43(a,1),1115(a,2))]. 1732 ~ x => (y -> ~ z) = (y ^ z) => (y -> x). [para(1420(a,1),1610(a,1,2))]. 1744 ~ (~ x v y) -> z = x -> (~ y -> z). [para(29(a,1),1319(a,1,1)),flip(a)]. 1796 ~ (x v ~ y) -> z = y -> (~ x -> z). [para(4(a,1),1744(a,1,1,1))]. 1939 x => (y => (x => (y => z))) = (x * y) -> z. [para(145(a,1),15(a,1)),flip(a)]. 2113 (x ^ y) => (x -> z) = y => (x -> z). [para(1732(a,1),1610(a,1))]. 2124 (x ^ y) => (y -> z) = x => (y -> z). [para(5(a,1),2113(a,1,1))]. 2371 x => (x => (y -> z)) = (x * y) -> z. [para(371(a,1),1939(a,1,2))]. 2399 (x * y) -> z = x -> (y -> z). [para(2371(a,1),21(a,1))]. 2498 x -> ((x * ~ y) -> z) = ~ (x -> y) -> z. [para(1196(a,1),2399(a,1,1)),flip(a)]. 2601 x -> (x -> (~ y -> z)) = ~ (x -> y) -> z. [para(2399(a,1),2498(a,1,2))]. 2637 ~ (x -> y) -> z = x -> (~ y -> z). [para(2601(a,1),686(a,1))]. 2858 (x -> (y -> z)) ^ (~ (y -> z) -> ~ (x ^ y)) = (x ^ y) => (y -> z). [para(686(a,1),768(a,1,1,2))]. 2865 (x -> (y -> z)) ^ (~ (z v ~ y) -> ~ (x ^ y)) = (x ^ y) => (z v ~ y). [para(692(a,1),768(a,1,1,2))]. 3017 (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = (x ^ y) => (y -> z). [para(2637(a,1),2858(a,1,2))]. 3122 (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = x => (y -> z). [para(2124(a,1),3017(a,2))]. 3348 (x -> (y -> z)) ^ (y -> (~ z -> ~ (x ^ y))) = (x ^ y) => (z v ~ y). [para(1796(a,1),2865(a,1,2))]. 3496 (x ^ y) => (z v ~ y) = x => (y -> z). [para(3348(a,1),3122(a,1))]. 3517 (x ^ y) ^ ((x => (y -> z)) => (z v ~ y)) = x ^ y. [para(3496(a,1),17(a,1,2,1))]. 3625 ((x -> y) ^ x) ^ (1 => (y v ~ x)) = (x -> y) ^ x. [para(33(a,1),3517(a,1,2,1))]. 3712 (x ^ (x -> y)) ^ (1 => (y v ~ x)) = (x -> y) ^ x. [para(5(a,1),3625(a,1,1))]. 3788 (x ^ (x -> y)) ^ (y v ~ x) = (x -> y) ^ x. [para(9(a,1),3712(a,1,2))]. 3855 x ^ ((x -> y) ^ (y v ~ x)) = (x -> y) ^ x. [para(3788(a,1),3(a,1)),flip(a)]. 3970 x ^ (y v ~ x) = (x -> y) ^ x. [para(1150(a,1),3855(a,1,2))]. 4006 x ^ (~ x v y) = (x -> y) ^ x. [para(4(a,1),3970(a,1,2))]. 4064 x ^ (~ x v y) = x ^ (x -> y). [para(5(a,1),4006(a,2))]. 4065 $F # answer("(N4)"). [resolve(4064,a,51,a)]. ============================== end of proof ========================== ============================== STATISTICS ============================ Given=97. Generated=4702. Kept=4060. proofs=1. Usable=98. Sos=3922. Demods=0. Limbo=4, Disabled=82. Hints=66. Weight_deleted=0. Literals_deleted=0. Forward_subsumed=642. Back_subsumed=35. 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=1. Megabytes=6.20. User_CPU=0.17, System_CPU=0.00, Wall_clock=0. ============================== end of statistics ===================== ============================== end of search ========================= THEOREM PROVED THEOREM PROVED Exiting with 1 proof. ------ process 26326 exit (max_proofs) ------  Process 26326 exit (max_proofs) Sat Jan 19 14:19:09 2008