============================== prooftrans (BV) ======================= Prover9 (64) version Aug-2007, Aug 2007. Process 26254 was started by veroff on titan, Sat Jan 19 14:18:50 2008 The command was "prover9 -f 4.4.in". ============================== end of head =========================== op(400,infix,[^,v,"->","=>",*]). op(200,prefix,[~,-]). ============================== end of input ========================== ============================== PROOF ================================= % -------- Comments from original proof -------- % Proof 1 at 0.13 (+ 0.01) seconds: "(Lemma 4.4, 2 parts)". % Length of proof is 79. % Level of proof is 18. % Maximum clause weight is 21. % Given clauses 97. 3 x v y = y v x # label("(D3)"). [assumption]. 4 x ^ y = y ^ x # label("(D4)"). [assumption]. 5 x v (x ^ y) = x # label("(D5)"). [assumption]. 8 1 => x = x # label("(BCK2)"). [assumption]. 12 x * y = y * x # label("(M2)"). [assumption]. 14 (x * y) => z = x => (y => z) # label("(P)"). [assumption]. 16 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. 19 x -> y = x => (x => y) # label("(-> def)"). [assumption]. 20 x => (x => y) = x -> y. [copy(19),flip(a)]. 21 ~ x = x => 0 # label("(~ def)"). [assumption]. 22 x => 0 = ~ x. [copy(21),flip(a)]. 25 x => (x => (x => y)) = x => (x => y) # label("(E_2)"). [assumption]. 28 ~ ~ x = x # label("(DN)"). [assumption]. 30 (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. 34 x => (y => z) = y => (x => z) # label("(3.18)"). [assumption]. 36 (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). [assumption]. 38 (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). [assumption]. 39 ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). [assumption]. 40 (x => y) ^ y = y # label("(3.13)"). [assumption]. 41 (x v y) => y = x => y # label("(4.6)"). [assumption]. 42 x -> (y => z) = y => (x -> z) # label("(4.7)"). [assumption]. 44 A -> ((B v A) -> C) != A -> C | A -> (~ A v B) != A -> B # answer("(Lemma 4.4, 2 parts)"). [assumption]. 63 (x ^ y) v x = x. [para(5(a,1),3(a,1)),flip(a)]. 146 x => (y => 0) = ~ (x * y). [para(22(a,1),14(a,1)),flip(a)]. 187 x => (x -> y) = x => (x => y). [para(20(a,1),25(a,1,2))]. 376 (x => y) => (z => ((z => x) => y)) = 1. [para(34(a,1),30(a,1,2))]. 428 x => ((x => y) ^ z) = (x -> y) ^ (x => z). [para(20(a,1),36(a,1,1)),flip(a)]. 504 ~ x ^ (y => 0) = (x v y) => 0. [para(22(a,1),38(a,1,1))]. 540 (x * (y => z)) => (y => (z * x)) = 1. [para(12(a,1),39(a,1,1))]. 576 x ^ (y => x) = x. [para(40(a,1),4(a,1)),flip(a)]. 643 x => (y -> (x => z)) = y -> (x -> z). [para(20(a,1),42(a,1,2)),flip(a)]. 689 x v (y => x) = y => x. [para(40(a,1),63(a,1,1))]. 760 ~ (x * y) = x => ~ y. [para(22(a,1),146(a,1,2)),flip(a)]. 801 ~ (x => ~ y) = x * y. [para(760(a,1),28(a,1,1))]. 821 (x v ~ y) * y = ~ (x => ~ y). [para(41(a,1),801(a,1,1)),flip(a)]. 841 x => (x -> y) = x -> y. [para(20(a,1),187(a,2))]. 891 x => (y => (x -> z)) = y => (x -> z). [para(841(a,1),34(a,1,2)),flip(a)]. 895 (x -> y) ^ (z => (x -> y)) = (x v z) => (x -> y). [para(841(a,1),38(a,1,1))]. 918 (x v y) => 0 = ~ x ^ ~ y. [para(22(a,1),504(a,1,2)),flip(a)]. 946 ~ x ^ ~ y = ~ (x v y). [para(918(a,1),22(a,1))]. 996 ~ (~ x v y) = x ^ ~ y. [para(28(a,1),946(a,1,1)),flip(a)]. 1009 ~ (x v ~ y) = y ^ ~ x. [para(3(a,1),996(a,1,1))]. 1044 ~ (x v (y => ~ z)) = (y * z) ^ ~ x. [para(760(a,1),1009(a,1,1,2))]. 1091 (x v ~ y) * y = x * y. [para(801(a,1),821(a,2))]. 1100 x * (y v ~ x) = y * x. [para(1091(a,1),12(a,1)),flip(a)]. 1190 x => ((y => z) => ((x => y) => z)) = 1. [para(376(a,1),34(a,1)),flip(a)]. 1255 x => ((y * (x => z)) => (z * y)) = 1. [para(540(a,1),34(a,1)),flip(a)]. 1327 x => (x => (y -> z)) = y -> (x -> z). [para(42(a,1),643(a,1,2))]. 1406 x => (((x => y) => z) => ((x -> y) => z)) = 1. [para(20(a,1),1190(a,1,2,2,1))]. 1490 x => (y => ((x => z) => (z * y))) = 1. [para(14(a,1),1255(a,1,2))]. 1556 x -> (y -> z) = y -> (x -> z). [para(1327(a,1),20(a,1))]. 1625 x -> ((x => y) => (y * x)) = 1. [para(1490(a,1),20(a,1)),flip(a)]. 1687 x -> ((x => y) => (x * y)) = 1. [para(12(a,1),1625(a,1,2,2))]. 1745 (x => y) => (x -> (x * y)) = 1. [para(1687(a,1),42(a,1)),flip(a)]. 1857 (x * y) ^ ~ ~ y = ~ (x => ~ y). [para(689(a,1),1044(a,1,1)),flip(a)]. 1913 (x * y) ^ ~ ~ y = x * y. [para(801(a,1),1857(a,2))]. 1940 (x * y) ^ y = x * y. [para(28(a,1),1913(a,1,2))]. 1980 x => (y ^ (x => z)) = (x -> z) ^ (x => y). [para(4(a,1),428(a,1,2))]. 2143 x => ((x -> y) => (((x => y) => z) => z)) = 1. [para(34(a,1),1406(a,1,2))]. 2230 x => ((x => y) ^ (x => z)) = (x -> z) ^ (x -> y). [para(20(a,1),1980(a,2,2))]. 2432 x => ((x -> y) => (1 => (x -> (x * y)))) = 1. [para(1745(a,1),2143(a,1,2,2,1))]. 2450 x => ((x -> y) => (x -> (x * y))) = 1. [para(8(a,1),2432(a,1,2,2))]. 2544 (x -> y) => (x -> (x * y)) = 1. [para(2450(a,1),891(a,1)),flip(a)]. 2569 (x -> y) ^ (1 => (x -> (x * y))) = x -> y. [para(2544(a,1),16(a,1,2,1))]. 2619 (x -> y) ^ (x -> (x * y)) = x -> y. [para(8(a,1),2569(a,1,2))]. 2691 (x v y) => (x -> z) = x -> z. [para(895(a,1),576(a,1))]. 2718 (x v y) -> (x -> z) = (x v y) => (x -> z). [para(2691(a,1),20(a,1,2)),flip(a)]. 2816 (x v y) -> (x -> z) = x -> z. [para(2691(a,1),2718(a,2))]. 2820 (x v y) -> (y -> z) = y -> z. [para(3(a,1),2816(a,1,1))]. 2889 x -> ((y v x) -> z) = x -> z # label("(4.9)"). [para(2820(a,1),1556(a,1)),flip(a)]. 3024 x => (x => (y ^ z)) = (x -> z) ^ (x -> y). [para(36(a,1),2230(a,1,2))]. 3145 (x -> y) ^ (x -> z) = x -> (z ^ y). [para(3024(a,1),20(a,1))]. 3271 x -> ((x * y) ^ y) = x -> y. [para(3145(a,1),2619(a,1))]. 3308 x -> (x * y) = x -> y. [para(1940(a,1),3271(a,1,2))]. 3356 x -> (y v ~ x) = x -> (y * x). [para(1100(a,1),3308(a,1,2)),flip(a)]. 3407 x -> (y v ~ x) = x -> (x * y). [para(12(a,1),3356(a,2,2))]. 3522 x -> (y v ~ x) = x -> y. [para(3308(a,1),3407(a,2))]. 3524 x -> (~ x v y) = x -> y # label("(4.10)"). [para(3(a,1),3522(a,1,2))]. 3589 $F # answer("(Lemma 4.4, 2 parts)"). [hyper(44,a,2889,a,b,3524,a)]. ============================== end of proof ==========================