============================== prooftrans (BV) ======================= Prover9 (64) version Aug-2007, Aug 2007. Process 26338 was started by veroff on titan, Sat Jan 19 14:19:12 2008 The command was "prover9 -f 4.11.in". ============================== end of head =========================== op(400,infix,[^,v,"->","=>",*]). op(200,prefix,[~,-]). ============================== end of input ========================== ============================== PROOF ================================= % -------- Comments from original proof -------- % Proof 1 at 0.26 (+ 0.02) seconds: "(N1)". % Length of proof is 105. % Level of proof is 26. % Maximum clause weight is 19. % Given clauses 122. 1 (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)") # label(non_clause) # label(goal). [goal]. 2 (x v y) v z = x v (y v z) # label("(D1)"). [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]. 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]. 18 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. 19 x v 0 = x # label("(D9)"). [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)]. 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]. 35 x => (y => z) = y => (x => z) # label("(3.18)"). [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]. 41 (x => y) ^ y = y # label("(3.13)"). [assumption]. 42 (x v y) => y = x => y # label("(4.6)"). [assumption]. 47 x -> x = 1 # label("(N2)"). [assumption]. 50 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. 51 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. 52 (c1 ^ ~ c1) ^ (c2 v ~ c2) != c1 ^ ~ c1 # label("(N1)") # answer("(N1)"). [deny(1)]. 79 x v x = x. [para(7(a,1),6(a,1,2))]. 126 (x ^ y) => x = 1. [para(5(a,1),18(a,1,1))]. 145 (x * x) => y = x -> y. [para(21(a,1),15(a,2))]. 146 (x * y) => (x => (y => z)) = (x * y) -> z. [para(15(a,1),21(a,1,2))]. 251 ~ x => 0 = x. [para(29(a,1),23(a,2))]. 369 x => ((y ^ (x => z)) => z) = 1. [para(35(a,1),18(a,1))]. 372 x => (y => (x => z)) = y => (x -> z). [para(21(a,1),35(a,1,2)),flip(a)]. 391 x => (y => (z => u)) = z => (x => (y => u)). [para(35(a,1),35(a,1,2))]. 510 ~ x ^ (y => 0) = (x v y) => 0. [para(23(a,1),39(a,1,1))]. 595 (x v y) => x = y => x. [para(4(a,1),42(a,1,1))]. 764 (x ^ y) -> z = y -> (x -> z). [para(5(a,1),50(a,1,1))]. 800 x ^ (y v ~ x) = x ^ (x -> y). [para(4(a,1),51(a,1,2))]. 805 x ^ (x -> 0) = x ^ ~ x. [para(19(a,1),51(a,1,2)),flip(a)]. 819 x v (x v y) = x v y. [para(79(a,1),2(a,1,1)),flip(a)]. 849 (~ x v y) => 0 = x ^ (y => 0). [para(251(a,1),39(a,1,1)),flip(a)]. 876 (x => (y ^ z)) => (x => y) = 1. [para(37(a,1),126(a,1,1))]. 925 (x -> y) ^ y = y. [para(145(a,1),41(a,1,1))]. 938 x v ((y -> x) ^ z) = (y -> x) ^ (x v z). [para(925(a,1),28(a,1,1))]. 960 (x v y) => (y => x) = (x v y) -> x. [para(595(a,1),21(a,1,2))]. 1014 x v (y ^ (x v z)) = (x v y) ^ (x v z). [para(819(a,1),27(a,1,2)),flip(a)]. 1056 x ^ (1 => ((y ^ (x => z)) => z)) = x. [para(369(a,1),17(a,1,2,1))]. 1097 (x v y) -> (x -> z) = x -> z. [para(7(a,1),764(a,1,1)),flip(a)]. 1106 x -> ((y => x) -> z) = x -> z. [para(41(a,1),764(a,1,1)),flip(a)]. 1127 x -> (y -> z) = y -> (x -> z). [para(764(a,1),50(a,1))]. 1164 x => ((x => (y ^ z)) => y) = 1. [para(876(a,1),35(a,1)),flip(a)]. 1195 x -> ((y * (y => x)) -> z) = (y * (y => x)) -> z. [para(38(a,1),1097(a,1,1))]. 1342 (x v y) => 0 = ~ x ^ ~ y. [para(23(a,1),510(a,1,2)),flip(a)]. 1377 ~ x ^ ~ y = ~ (x v y). [para(1342(a,1),23(a,1))]. 1420 ~ (~ x v y) = x ^ ~ y. [para(29(a,1),1377(a,1,1)),flip(a)]. 1449 (x ^ ~ y) => 0 = ~ x v y. [para(1420(a,1),251(a,1,1))]. 1548 (~ x v y) => (x ^ (y => 0)) = (~ x v y) -> 0. [para(849(a,1),21(a,1,2))]. 1723 x => ((y v x) => y) = (y v x) -> y. [para(960(a,1),35(a,1)),flip(a)]. 1773 x ^ ((y ^ (x => z)) => z) = x. [para(9(a,1),1056(a,1,2))]. 1843 x ^ (((y v x) => z) => z) = x. [para(39(a,1),1773(a,1,2,1))]. 1893 x ^ (1 => (((y v x) => (z ^ u)) => z)) = x. [para(1164(a,1),1843(a,1,2,1))]. 1947 x => (x => y) = (y v x) -> y. [para(595(a,1),1723(a,1,2))]. 1975 (x v y) -> x = y -> x. [para(1947(a,1),21(a,1))]. 2115 x => (y => (x => (y => z))) = (x * y) -> z. [para(146(a,1),15(a,1)),flip(a)]. 2347 x => (y => ((z ^ (y => (x => u))) => u)) = 1. [para(391(a,1),369(a,1))]. 2370 (x ^ ~ y) => (z => (u => 0)) = z => (u => (~ x v y)). [para(1449(a,1),391(a,1,2,2)),flip(a)]. 2437 x v (y ^ (z -> x)) = (z -> x) ^ (x v y). [para(5(a,1),938(a,1,2))]. 2538 x v (y ^ (x v z)) = x v (y ^ z). [para(27(a,1),1014(a,2))]. 2653 x v (y ^ (y -> x)) = x v (y ^ ~ y). [para(800(a,1),2538(a,1,2))]. 2748 x ^ (((y v x) => (z ^ u)) => z) = x. [para(9(a,1),1893(a,1,2))]. 2940 x => (x => (y -> z)) = (x * y) -> z. [para(372(a,1),2115(a,1,2))]. 2983 (x * y) -> z = x -> (y -> z). [para(2940(a,1),21(a,1))]. 3309 (x -> y) ^ (y v x) = y v (x ^ ~ x). [para(2437(a,1),2653(a,1))]. 3494 x -> (y -> ((y => x) -> z)) = (y * (y => x)) -> z. [para(2983(a,1),1195(a,1,2))]. 3589 x ^ (((~ y v x) -> 0) => y) = x. [para(1548(a,1),2748(a,1,2,1))]. 3632 x => (((~ (x => y) v z) -> 0) => (z => y)) = 1. [para(3589(a,1),2347(a,1,2,2,1))]. 3636 x => (((y v ~ (x => z)) -> 0) => (y => z)) = 1. [para(4(a,1),3632(a,1,2,1,1))]. 3796 x => (y => (((y v ~ (x => z)) -> 0) => z)) = 1. [para(35(a,1),3636(a,1,2))]. 3896 x => (y => (((y v ~ ~ x) -> 0) => 0)) = 1. [para(23(a,1),3796(a,1,2,2,1,1,2,1))]. 4000 x => (y => (((y v x) -> 0) => 0)) = 1. [para(29(a,1),3896(a,1,2,2,1,1,2))]. 4064 x => (y => ~ ((y v x) -> 0)) = 1. [para(23(a,1),4000(a,1,2,2))]. 4280 x -> (y -> ((x => y) -> z)) = (x * (x => y)) -> z. [para(3494(a,1),1127(a,1)),flip(a)]. 4427 (x * (x => y)) -> z = x -> (y -> z). [para(1106(a,1),4280(a,1,2)),flip(a)]. 4512 (x * (x -> y)) -> z = x -> ((x => y) -> z). [para(21(a,1),4427(a,1,1,2))]. 4596 x -> ((x => y) -> z) = x -> (y -> z). [para(4427(a,1),2983(a,1)),flip(a)]. 4885 (x * (x -> y)) -> z = x -> (y -> z). [para(4596(a,1),4512(a,2))]. 4955 x -> ((x -> y) -> z) = x -> (y -> z). [para(4885(a,1),2983(a,1)),flip(a)]. 5035 (x v y) -> (((x v y) -> x) -> z) = x -> z. [para(1097(a,1),4955(a,2))]. 5177 (x v y) -> ((y -> x) -> z) = x -> z. [para(1975(a,1),5035(a,1,2,1))]. 5257 ((x -> y) ^ (y v x)) -> z = y -> z. [para(5177(a,1),764(a,2))]. 5405 (x v (y ^ ~ y)) -> z = x -> z. [para(3309(a,1),5257(a,1,1))]. 5517 (x ^ ~ x) => (y => ~ (y -> 0)) = 1. [para(5405(a,1),4064(a,1,2,2,1))]. 5653 (x ^ ~ y) => (z => ~ u) = z => (u => (~ x v y)). [para(23(a,1),2370(a,1,2,2))]. 5906 x => ((x -> 0) => (~ y v y)) = 1. [para(5653(a,1),5517(a,1))]. 5908 x => ((x -> 0) => (y v ~ y)) = 1. [para(4(a,1),5906(a,1,2,2))]. 6003 (x ^ y) => ((x -> (y -> 0)) => (z v ~ z)) = 1. [para(50(a,1),5908(a,1,2,1))]. 6083 ((x -> 0) ^ x) => (1 => (y v ~ y)) = 1. [para(47(a,1),6003(a,1,2,1))]. 6161 (x ^ (x -> 0)) => (1 => (y v ~ y)) = 1. [para(5(a,1),6083(a,1,1))]. 6258 (x ^ ~ x) => (1 => (y v ~ y)) = 1. [para(805(a,1),6161(a,1,1))]. 6306 (x ^ ~ x) => (y v ~ y) = 1. [para(9(a,1),6258(a,1,2))]. 6369 (x ^ ~ x) ^ (1 => (y v ~ y)) = x ^ ~ x. [para(6306(a,1),17(a,1,2,1))]. 6424 (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x. [para(9(a,1),6369(a,1,2))]. 6425 $F # answer("(N1)"). [resolve(6424,a,52,a)]. ============================== end of proof ==========================