============================== prooftrans (BV) ======================= Prover9 (64) version Aug-2007, Aug 2007. Process 32355 was started by veroff on io, Mon Jan 21 11:14:39 2008 The command was "prover9 -f 3.1_alt.in". ============================== end of head =========================== op(400,infix,[^,v,"->","=>",*]). op(200,prefix,[~,-]). ============================== end of input ========================== ============================== PROOF ================================= % -------- Comments from original proof -------- % Proof 1 at 0.04 (+ 0.00) seconds: "(3.10)". % Length of proof is 18. % Level of proof is 7. % Maximum clause weight is 13. % Given clauses 48. 1 (x ^ y) -> y = 1 # label("(3.10)") # label(non_clause) # label(goal). [goal]. 8 x v y = y v x # label("(D3)"). [assumption]. 9 x ^ y = y ^ x # label("(D4)"). [assumption]. 10 x v (x ^ y) = x # label("(D5)"). [assumption]. 11 x ^ (x v y) = x # label("(D6)"). [assumption]. 15 x ^ 1 = x # label("(D10)"). [assumption]. 22 x -> x = 1 # label("(N2)"). [assumption]. 25 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 45 (c1 ^ c2) -> c2 != 1 # label("(3.10)") # answer("(3.10)"). [deny(1)]. 69 x v (y ^ x) = x. [para(9(a,1),10(a,1,2))]. 122 1 ^ x = x. [para(15(a,1),9(a,1)),flip(a)]. 243 1 ^ (x -> y) = x -> (x ^ y). [para(22(a,1),25(a,1,1))]. 734 x -> (x ^ y) = x -> y. [para(243(a,1),122(a,1))]. 745 x -> (x v y) = x -> x. [para(11(a,1),734(a,1,2)),flip(a)]. 798 x -> (x v y) = 1. [para(22(a,1),745(a,2))]. 831 x -> (y v x) = 1. [para(8(a,1),798(a,1,2))]. 883 (x ^ y) -> y = 1 # label("(3.10)"). [para(69(a,1),831(a,1,2))]. 884 $F # answer("(3.10)"). [resolve(883,a,45,a)]. ============================== end of proof ========================== ============================== PROOF ================================= % -------- Comments from original proof -------- % Proof 2 at 0.08 (+ 0.00) seconds: "(3.11)". % Length of proof is 38. % Level of proof is 12. % Maximum clause weight is 13. % Given clauses 68. 2 (x ^ y) => y = 1 # label("(3.11)") # label(non_clause) # label(goal). [goal]. 8 x v y = y v x # label("(D3)"). [assumption]. 9 x ^ y = y ^ x # label("(D4)"). [assumption]. 10 x v (x ^ y) = x # label("(D5)"). [assumption]. 11 x ^ (x v y) = x # label("(D6)"). [assumption]. 15 x ^ 1 = x # label("(D10)"). [assumption]. 17 ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. 22 x -> x = 1 # label("(N2)"). [assumption]. 25 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 26 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. 31 x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). [assumption]. 32 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(31),flip(a)]. 33 x -> 1 = 1 # label("(3.1)"). [assumption]. 34 1 -> x = x # label("(3.2)"). [assumption]. 38 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. 46 (c3 ^ c4) => c4 != 1 # label("(3.11)") # answer("(3.11)"). [deny(2)]. 69 x v (y ^ x) = x. [para(9(a,1),10(a,1,2))]. 122 1 ^ x = x. [para(15(a,1),9(a,1)),flip(a)]. 238 (x -> y) ^ (x -> 1) = x -> y. [para(15(a,1),25(a,2,2))]. 243 1 ^ (x -> y) = x -> (x ^ y). [para(22(a,1),25(a,1,1))]. 433 x ^ (y -> x) = (1 v y) -> x. [para(34(a,1),38(a,1,1))]. 543 1 v x = 1. [para(122(a,1),10(a,1,2))]. 734 x -> (x ^ y) = x -> y. [para(243(a,1),122(a,1))]. 745 x -> (x v y) = x -> x. [para(11(a,1),734(a,1,2)),flip(a)]. 798 x -> (x v y) = 1. [para(22(a,1),745(a,2))]. 831 x -> (y v x) = 1. [para(8(a,1),798(a,1,2))]. 833 ~ x -> ~ (x ^ y) = 1. [para(17(a,1),798(a,1,2))]. 883 (x ^ y) -> y = 1 # label("(3.10)"). [para(69(a,1),831(a,1,2))]. 941 ((x ^ y) -> x) ^ 1 = (x ^ y) => x. [para(833(a,1),32(a,1,2))]. 1145 x ^ (y -> x) = 1 -> x. [para(543(a,1),433(a,2,1))]. 1189 x ^ (y -> x) = x. [para(34(a,1),1145(a,2))]. 1230 x -> (y -> x) = 1. [para(1189(a,1),883(a,1,1))]. 1277 (x -> (y -> x)) ^ (x -> 1) = 1. [para(1230(a,1),238(a,2))]. 1456 (x -> (y -> x)) ^ 1 = 1. [para(33(a,1),1277(a,1,2))]. 1630 (x -> (y -> x)) ^ 1 = (x ^ y) => x. [para(26(a,1),941(a,1,1))]. 1818 (x ^ y) => x = 1. [para(1630(a,1),1456(a,1))]. 1829 (x ^ y) => y = 1 # label("(3.11)"). [para(9(a,1),1818(a,1,1))]. 1830 $F # answer("(3.11)"). [resolve(1829,a,46,a)]. ============================== end of proof ========================== ============================== PROOF ================================= % -------- Comments from original proof -------- % Proof 3 at 0.10 (+ 0.00) seconds: "(3.12)". % Length of proof is 12. % Level of proof is 6. % Maximum clause weight is 14. % Given clauses 79. 3 ~ x => ~ y = y => x # label("(3.12)") # label(non_clause) # label(goal). [goal]. 9 x ^ y = y ^ x # label("(D4)"). [assumption]. 19 ~ ~ x = x # label("(DN)"). [assumption]. 31 x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). [assumption]. 32 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(31),flip(a)]. 47 ~ c5 => ~ c6 != c6 => c5 # label("(3.12)") # answer("(3.12)"). [deny(3)]. 299 (~ x -> ~ y) ^ (y -> x) = y => x. [para(32(a,1),9(a,1)),flip(a)]. 305 (x -> ~ y) ^ (y -> ~ x) = x => ~ y. [para(19(a,1),32(a,1,2,1))]. 1558 (x -> ~ y) ^ (y -> ~ x) = y => ~ x. [para(19(a,1),299(a,1,1,1))]. 2306 x => ~ y = y => ~ x. [para(1558(a,1),305(a,1))]. 2307 ~ x => ~ y = y => x # label("(3.12)"). [para(19(a,1),2306(a,1,2)),flip(a)]. 2308 $F # answer("(3.12)"). [resolve(2307,a,47,a)]. ============================== end of proof ========================== ============================== PROOF ================================= % -------- Comments from original proof -------- % Proof 4 at 0.10 (+ 0.00) seconds: "(3.13)". % Length of proof is 45. % Level of proof is 13. % Maximum clause weight is 17. % Given clauses 82. 4 (x => y) ^ y = y # label("(3.13)") # label(non_clause) # label(goal). [goal]. 6 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. 7 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. 8 x v y = y v x # label("(D3)"). [assumption]. 9 x ^ y = y ^ x # label("(D4)"). [assumption]. 10 x v (x ^ y) = x # label("(D5)"). [assumption]. 11 x ^ (x v y) = x # label("(D6)"). [assumption]. 15 x ^ 1 = x # label("(D10)"). [assumption]. 19 ~ ~ x = x # label("(DN)"). [assumption]. 22 x -> x = 1 # label("(N2)"). [assumption]. 24 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. 25 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 31 x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). [assumption]. 32 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(31),flip(a)]. 34 1 -> x = x # label("(3.2)"). [assumption]. 37 x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. 38 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. 48 (c7 => c8) ^ c8 != c8 # label("(3.13)") # answer("(3.13)"). [deny(4)]. 69 x v (y ^ x) = x. [para(9(a,1),10(a,1,2))]. 74 x ^ (y v x) = x. [para(8(a,1),11(a,1,2))]. 122 1 ^ x = x. [para(15(a,1),9(a,1)),flip(a)]. 213 x ^ (y v ~ x) = x ^ (x -> y). [para(8(a,1),24(a,1,2))]. 243 1 ^ (x -> y) = x -> (x ^ y). [para(22(a,1),25(a,1,1))]. 297 (x -> y) ^ ((~ y -> ~ x) ^ z) = (x => y) ^ z. [para(32(a,1),7(a,1,1)),flip(a)]. 433 x ^ (y -> x) = (1 v y) -> x. [para(34(a,1),38(a,1,1))]. 543 1 v x = 1. [para(122(a,1),10(a,1,2))]. 589 x ^ (y v (z v x)) = x. [para(6(a,1),74(a,1,2))]. 611 x ^ (y v (x v z)) = x. [para(8(a,1),589(a,1,2,2))]. 734 x -> (x ^ y) = x -> y. [para(243(a,1),122(a,1))]. 774 x -> (x ^ (x -> y)) = x -> y. [para(37(a,1),734(a,2))]. 1145 x ^ (y -> x) = 1 -> x. [para(543(a,1),433(a,2,1))]. 1189 x ^ (y -> x) = x. [para(34(a,1),1145(a,2))]. 1215 (x -> y) ^ y = y. [para(1189(a,1),9(a,1)),flip(a)]. 1225 (x -> y) v y = x -> y. [para(1189(a,1),69(a,1,2))]. 1319 x ^ (y -> (x v z)) = x. [para(1225(a,1),611(a,1,2))]. 1535 x -> (x ^ (x -> y)) = x -> (y v ~ x). [para(213(a,1),734(a,1,2))]. 1996 x -> (y v ~ x) = x -> y. [para(1535(a,1),774(a,1))]. 2006 x -> (~ x v y) = x -> y. [para(8(a,1),1996(a,1,2))]. 2133 ~ x ^ (x -> y) = ~ x. [para(2006(a,1),1319(a,1,2))]. 2154 x ^ (~ x -> y) = ~ ~ x. [para(19(a,1),2133(a,1,1))]. 2191 x ^ (~ x -> y) = x. [para(19(a,1),2154(a,2))]. 2218 (~ x -> y) ^ x = x. [para(2191(a,1),9(a,1)),flip(a)]. 2429 (x -> y) ^ y = (x => y) ^ y. [para(2218(a,1),297(a,1,2))]. 2474 (x => y) ^ y = y # label("(3.13)"). [para(2429(a,1),1215(a,1))]. 2475 $F # answer("(3.13)"). [resolve(2474,a,48,a)]. ============================== end of proof ========================== ============================== PROOF ================================= % -------- Comments from original proof -------- % Proof 5 at 0.11 (+ 0.01) seconds: "(3.14)". % Length of proof is 20. % Level of proof is 6. % Maximum clause weight is 17. % Given clauses 85. 5 x => (x ^ y) = x => y # label("(3.14)") # label(non_clause) # label(goal). [goal]. 9 x ^ y = y ^ x # label("(D4)"). [assumption]. 15 x ^ 1 = x # label("(D10)"). [assumption]. 17 ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. 22 x -> x = 1 # label("(N2)"). [assumption]. 25 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 31 x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). [assumption]. 32 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(31),flip(a)]. 38 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. 49 c9 => (c9 ^ c10) != c9 => c10 # label("(3.14)") # answer("(3.14)"). [deny(5)]. 122 1 ^ x = x. [para(15(a,1),9(a,1)),flip(a)]. 243 1 ^ (x -> y) = x -> (x ^ y). [para(22(a,1),25(a,1,1))]. 416 1 ^ (x -> y) = (y v x) -> y. [para(22(a,1),38(a,1,1))]. 734 x -> (x ^ y) = x -> y. [para(243(a,1),122(a,1))]. 761 (x -> y) ^ (~ (x ^ y) -> ~ x) = x => (x ^ y). [para(734(a,1),32(a,1,1))]. 1016 (x v y) -> x = y -> x. [para(416(a,1),122(a,1))]. 1040 ~ (x ^ y) -> ~ x = ~ y -> ~ x. [para(17(a,1),1016(a,1,1))]. 2614 (x -> y) ^ (~ y -> ~ x) = x => (x ^ y). [para(1040(a,1),761(a,1,2))]. 2668 x => (x ^ y) = x => y # label("(3.14)"). [para(2614(a,1),32(a,1))]. 2669 $F # answer("(3.14)"). [resolve(2668,a,49,a)]. ============================== end of proof ==========================