============================== Prover9 =============================== Prover9 (64) version Aug-2007, Aug 2007. Process 32327 was started by veroff on io, Mon Jan 21 11:14:29 2008 The command was "prover9 -f 3.1.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file 3.1.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). 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 v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). x v 0 = x # label("(D9)"). x ^ 1 = x # label("(D10)"). x ^ 0 = 0 # label("(2.1)"). ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). ~ ~ x = x # label("(DN)"). ~ 1 = 0 # label("(2.2)"). (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). x -> x = 1 # label("(N2)"). (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). (x ^ y) -> z = x -> (y -> z) # label("(N6)"). -x = x -> 0 # label("(N7)"). x * y = ~ (x -> ~ y) v ~ (y -> ~ x) # label("(* def)"). x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). x -> 1 = 1 # label("(3.1)"). 1 -> x = x # label("(3.2)"). (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). x -> (y -> z) = y -> (x -> z) # label("(3.4)"). x -> (x -> y) = x -> y # label("(3.5)"). (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). x ^ ((x => y) => y) = x # label("(3.7)"). x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). 1 => x = x # label("(BCK2)"). x => 1 = 1 # label("(BCK3)"). x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). end_of_list. formulas(assumptions). (A ^ B) -> B != 1 | (A ^ B) => B != 1 | ~ A => ~ B != B => A | (A => B) ^ B != B | A => (A ^ B) != A => B # answer("(Lemma 3.1, 5 parts)"). end_of_list. formulas(hints). (x ^ y) -> y = 1 # label("(3.10)"). (x ^ y) => y = 1 # label("(3.11)"). ~ x => ~ y = y => x # label("(3.12)"). (x => y) ^ y = y # label("(3.13)"). x => (x ^ y) = x => y # label("(3.14)"). (x -> ~ y) ^ (y -> ~ x) = x => ~ y. 1 ^ (x -> y) = x -> (x ^ y). (x => y) ^ z = (x -> y) ^ ((~ y -> ~ x) ^ z). (x -> y) ^ ((~ y -> ~ x) ^ z) = (x => y) ^ z. x ^ (y v ~ x) = x ^ (x -> y). (~ x -> ~ y) ^ (y -> x) = y => x. x v (y ^ x) = x. x ^ (y v x) = x. (x -> y) ^ (x -> 1) = x -> y. x = 1 ^ x. 1 ^ x = x. 1 ^ (x -> y) = (y v x) -> y. x ^ (y -> x) = (1 v y) -> x. 1 v x = 1. x ^ (y v (z v x)) = x. x ^ (y v (x v z)) = x. x -> (x ^ y) = x -> y. (x -> y) ^ (~ (x ^ y) -> ~ x) = x => (x ^ y). x -> x = x -> (x v y). x -> (x v y) = x -> x. x -> (x ^ (x -> y)) = x -> y. x -> (x v y) = 1. ~ x -> ~ (x ^ y) = 1. x -> (y v x) = 1. (x ^ y) -> y = 1. ((x ^ y) -> x) ^ 1 = (x ^ y) => x. (x v y) -> x = y -> x. ~ (x ^ y) -> ~ x = ~ y -> ~ x. x ^ (y -> x) = 1 -> x. x ^ (y -> x) = x. x = (y -> x) ^ x. (x -> y) ^ y = y. (x -> y) v y = x -> y. x -> (y -> x) = 1. (x -> (y -> x)) ^ (x -> 1) = 1. x ^ (y -> (x v z)) = x. (x -> (y -> x)) ^ 1 = 1. x -> (x ^ (x -> y)) = x -> (y v ~ x). (x -> ~ y) ^ (y -> ~ x) = y => ~ x. (x -> (y -> x)) ^ 1 = (x ^ y) => x. (x ^ y) => x = 1. (x ^ y) => y = 1. x -> (y v ~ x) = x -> y. x -> (~ x v y) = x -> y. ~ x ^ (x -> y) = ~ x. x ^ (~ x -> y) = ~ ~ x. x ^ (~ x -> y) = x. x = (~ x -> y) ^ x. (~ x -> y) ^ x = x. x => ~ y = y => ~ x. x => y = ~ y => ~ x. ~ x => ~ y = y => x. (x -> y) ^ y = (x => y) ^ y. (x => y) ^ y = y. (x -> y) ^ (~ y -> ~ x) = x => (x ^ y). x => (x ^ y) = x => y. end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: ============================== 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 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 v 0 = x # label("(D9)"). [assumption]. x ^ 1 = x # label("(D10)"). [assumption]. x ^ 0 = 0 # label("(2.1)"). [assumption]. ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). [assumption]. ~ ~ x = x # label("(DN)"). [assumption]. ~ 1 = 0 # label("(2.2)"). [assumption]. (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). [assumption]. x -> x = 1 # label("(N2)"). [assumption]. (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. -x = x -> 0 # label("(N7)"). [assumption]. x * y = ~ (x -> ~ y) v ~ (y -> ~ x) # label("(* def)"). [assumption]. x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). [assumption]. x -> 1 = 1 # label("(3.1)"). [assumption]. 1 -> x = x # label("(3.2)"). [assumption]. (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). [assumption]. x -> (y -> z) = y -> (x -> z) # label("(3.4)"). [assumption]. x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). [assumption]. ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). [assumption]. 1 => x = x # label("(BCK2)"). [assumption]. x => 1 = 1 # label("(BCK3)"). [assumption]. x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. (A ^ B) -> B != 1 | (A ^ B) => B != 1 | ~ A => ~ B != B => A | (A => B) ^ B != B | A => (A ^ B) != A => B # answer("(Lemma 3.1, 5 parts)"). [assumption]. end_of_list. formulas(demodulators). end_of_list. WARNING, function symbols not in function_order (lex) command: A, B. % 61 hints input. Auto_denials: (no changes). Term ordering decisions: WARNING, function symbols not in function_order (lex) command: A, B. Function symbol KB weights: A=1. B=1. 0=1. 1=1. *=1. ^=1. =>=1. ->=1. v=1. ~=1. -=1. Predicate symbol precedence: predicate_order([ = ]). Function symbol precedence: function_order([ A, B, 0, 1, *, ^, =>, ->, 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. ============================== end of process initial clauses ======== ============================== CLAUSES FOR SEARCH ==================== % Clauses after input processing: formulas(usable). end_of_list. formulas(sos). 1 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. 2 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. 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]. 6 x ^ (x v y) = x # label("(D6)"). [assumption]. 7 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. 8 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. 9 x v 0 = x # label("(D9)"). [assumption]. 10 x ^ 1 = x # label("(D10)"). [assumption]. 11 x ^ 0 = 0 # label("(2.1)"). [assumption]. 12 ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. 13 ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). [assumption]. 14 ~ ~ x = x # label("(DN)"). [assumption]. 15 ~ 1 = 0 # label("(2.2)"). [assumption]. 16 (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). [assumption]. 17 x -> x = 1 # label("(N2)"). [assumption]. 18 (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. 19 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. 20 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 21 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. 23 x -> 0 = -x. [copy(22),flip(a)]. 25 ~ (x -> ~ y) v ~ (y -> ~ x) = x * y. [copy(24),flip(a)]. 27 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(26),flip(a)]. 28 x -> 1 = 1 # label("(3.1)"). [assumption]. 29 1 -> x = x # label("(3.2)"). [assumption]. 30 (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). [assumption]. 31 x -> (y -> z) = y -> (x -> z) # label("(3.4)"). [assumption]. 32 x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. 33 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. 34 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. 35 x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). [assumption]. 36 ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). [assumption]. 37 1 => x = x # label("(BCK2)"). [assumption]. 38 x => 1 = 1 # label("(BCK3)"). [assumption]. 39 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. 40 (A ^ B) -> B != 1 | (A ^ B) => B != 1 | ~ A => ~ B != B => A | (A => B) ^ B != B | A => (A ^ B) != A => B # answer("(Lemma 3.1, 5 parts)"). [assumption]. end_of_list. formulas(demodulators). end_of_list. % 61 hints processed (11 are redundant). ============================== end of clauses for search ============= ============================== SEARCH ================================ % Starting search at 0.00 seconds. given #1 (I,wt=11): 1 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. given #2 (I,wt=11): 2 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. given #3 (I,wt=7): 3 x v y = y v x # label("(D3)"). [assumption]. given #4 (I,wt=7): 4 x ^ y = y ^ x # label("(D4)"). [assumption]. given #5 (I,wt=7): 5 x v (x ^ y) = x # label("(D5)"). [assumption]. given #6 (I,wt=7): 6 x ^ (x v y) = x # label("(D6)"). [assumption]. given #7 (I,wt=13): 7 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. given #8 (I,wt=13): 8 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. given #9 (I,wt=5): 9 x v 0 = x # label("(D9)"). [assumption]. given #10 (I,wt=5): 10 x ^ 1 = x # label("(D10)"). [assumption]. given #11 (I,wt=5): 11 x ^ 0 = 0 # label("(2.1)"). [assumption]. given #12 (I,wt=10): 12 ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. given #13 (I,wt=10): 13 ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). [assumption]. given #14 (I,wt=5): 14 ~ ~ x = x # label("(DN)"). [assumption]. given #15 (I,wt=4): 15 ~ 1 = 0 # label("(2.2)"). [assumption]. given #16 (I,wt=14): 16 (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). [assumption]. given #17 (I,wt=5): 17 x -> x = 1 # label("(N2)"). [assumption]. given #18 (I,wt=13): 18 (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. given #19 (I,wt=12): 19 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. given #20 (I,wt=13): 20 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. given #21 (I,wt=11): 21 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. given #22 (I,wt=6): 23 x -> 0 = -x. [copy(22),flip(a)]. given #23 (I,wt=15): 25 ~ (x -> ~ y) v ~ (y -> ~ x) = x * y. [copy(24),flip(a)]. given #24 (I,wt=13): 27 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(26),flip(a)]. given #25 (I,wt=5): 28 x -> 1 = 1 # label("(3.1)"). [assumption]. given #26 (I,wt=5): 29 1 -> x = x # label("(3.2)"). [assumption]. given #27 (I,wt=13): 30 (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). [assumption]. given #28 (I,wt=11): 31 x -> (y -> z) = y -> (x -> z) # label("(3.4)"). [assumption]. given #29 (I,wt=9): 32 x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. given #30 (I,wt=13): 33 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. given #31 (I,wt=9): 34 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. given #32 (I,wt=11): 35 x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). [assumption]. given #33 (I,wt=13): 36 ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). [assumption]. given #34 (I,wt=5): 37 1 => x = x # label("(BCK2)"). [assumption]. given #35 (I,wt=5): 38 x => 1 = 1 # label("(BCK3)"). [assumption]. given #36 (I,wt=13): 39 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. given #37 (I,wt=39): 40 (A ^ B) -> B != 1 | (A ^ B) => B != 1 | ~ A => ~ B != B => A | (A => B) ^ B != B | A => (A ^ B) != A => B # answer("(Lemma 3.1, 5 parts)"). [assumption]. given #38 (H,wt=5): 113 1 ^ x = x. [para(10(a,1),4(a,1)),flip(a)]. given #39 (H,wt=5): 534 1 v x = 1. [para(113(a,1),5(a,1,2))]. given #40 (H,wt=7): 60 x v (y ^ x) = x. [para(4(a,1),5(a,1,2))]. given #41 (H,wt=7): 65 x ^ (y v x) = x. [para(3(a,1),6(a,1,2))]. given #42 (H,wt=9): 580 x ^ (y v (z v x)) = x. [para(1(a,1),65(a,1,2))]. given #43 (H,wt=9): 602 x ^ (y v (x v z)) = x. [para(3(a,1),580(a,1,2,2))]. given #44 (H,wt=11): 229 (x -> y) ^ (x -> 1) = x -> y. [para(10(a,1),20(a,2,2))]. given #45 (H,wt=11): 234 1 ^ (x -> y) = x -> (x ^ y). [para(17(a,1),20(a,1,1))]. given #46 (H,wt=9): 725 x -> (x ^ y) = x -> y. [para(234(a,1),113(a,1))]. given #47 (H,wt=9): 736 x -> (x v y) = x -> x. [para(6(a,1),725(a,1,2)),flip(a)]. given #48 (H,wt=7): 789 x -> (x v y) = 1. [para(17(a,1),736(a,2))]. given #49 (H,wt=7): 822 x -> (y v x) = 1. [para(3(a,1),789(a,1,2))]. given #50 (H,wt=7): 874 (x ^ y) -> y = 1 # label("(3.10)"). [para(60(a,1),822(a,1,2))]. given #51 (H,wt=9): 824 ~ x -> ~ (x ^ y) = 1. [para(12(a,1),789(a,1,2))]. given #52 (H,wt=11): 407 1 ^ (x -> y) = (y v x) -> y. [para(17(a,1),33(a,1,1))]. given #53 (H,wt=9): 1006 (x v y) -> x = y -> x. [para(407(a,1),113(a,1))]. given #54 (H,wt=11): 424 x ^ (y -> x) = (1 v y) -> x. [para(29(a,1),33(a,1,1))]. given #55 (H,wt=9): 1135 x ^ (y -> x) = 1 -> x. [para(534(a,1),424(a,2,1))]. given #56 (H,wt=7): 1179 x ^ (y -> x) = x. [para(29(a,1),1135(a,2))]. given #57 (H,wt=7): 1205 (x -> y) ^ y = y. [para(1179(a,1),4(a,1)),flip(a)]. given #58 (H,wt=7): 1220 x -> (y -> x) = 1. [para(1179(a,1),874(a,1,1))]. given #59 (H,wt=9): 1215 (x -> y) v y = x -> y. [para(1179(a,1),60(a,1,2))]. given #60 (H,wt=9): 1309 x ^ (y -> (x v z)) = x. [para(1215(a,1),602(a,1,2))]. given #61 (H,wt=11): 765 x -> (x ^ (x -> y)) = x -> y. [para(32(a,1),725(a,2))]. given #62 (H,wt=11): 1267 (x -> (y -> x)) ^ (x -> 1) = 1. [para(1220(a,1),229(a,2))]. given #63 (H,wt=9): 1446 (x -> (y -> x)) ^ 1 = 1. [para(28(a,1),1267(a,1,2))]. given #64 (H,wt=12): 204 x ^ (y v ~ x) = x ^ (x -> y). [para(3(a,1),19(a,1,2))]. given #65 (H,wt=13): 290 (~ x -> ~ y) ^ (y -> x) = y => x. [para(27(a,1),4(a,1)),flip(a)]. given #66 (H,wt=13): 931 ((x ^ y) -> x) ^ 1 = (x ^ y) => x. [para(824(a,1),27(a,1,2))]. given #67 (H,wt=13): 1030 ~ (x ^ y) -> ~ x = ~ y -> ~ x. [para(12(a,1),1006(a,1,1))]. given #68 (H,wt=13): 1620 (x -> (y -> x)) ^ 1 = (x ^ y) => x. [para(21(a,1),931(a,1,1))]. given #69 (H,wt=7): 1808 (x ^ y) => x = 1. [para(1620(a,1),1446(a,1))]. given #70 (H,wt=7): 1819 (x ^ y) => y = 1 # label("(3.11)"). [para(4(a,1),1808(a,1,1))]. given #71 (H,wt=14): 296 (x -> ~ y) ^ (y -> ~ x) = x => ~ y. [para(14(a,1),27(a,1,2,1))]. given #72 (H,wt=14): 1525 x -> (x ^ (x -> y)) = x -> (y v ~ x). [para(204(a,1),725(a,1,2))]. given #73 (H,wt=10): 1985 x -> (y v ~ x) = x -> y. [para(1525(a,1),765(a,1))]. given #74 (H,wt=10): 1995 x -> (~ x v y) = x -> y. [para(3(a,1),1985(a,1,2))]. given #75 (H,wt=9): 2122 ~ x ^ (x -> y) = ~ x. [para(1995(a,1),1309(a,1,2))]. given #76 (H,wt=10): 2143 x ^ (~ x -> y) = ~ ~ x. [para(14(a,1),2122(a,1,1))]. given #77 (H,wt=8): 2180 x ^ (~ x -> y) = x. [para(14(a,1),2143(a,2))]. given #78 (H,wt=8): 2207 (~ x -> y) ^ x = x. [para(2180(a,1),4(a,1)),flip(a)]. given #79 (H,wt=14): 1548 (x -> ~ y) ^ (y -> ~ x) = y => ~ x. [para(14(a,1),290(a,1,1,1))]. given #80 (H,wt=9): 2295 x => ~ y = y => ~ x. [para(1548(a,1),296(a,1))]. given #81 (H,wt=9): 2296 ~ x => ~ y = y => x # label("(3.12)"). [para(14(a,1),2295(a,1,2)),flip(a)]. given #82 (H,wt=17): 288 (x -> y) ^ ((~ y -> ~ x) ^ z) = (x => y) ^ z. [para(27(a,1),2(a,1,1)),flip(a)]. given #83 (H,wt=11): 2417 (x -> y) ^ y = (x => y) ^ y. [para(2207(a,1),288(a,1,2))]. given #84 (H,wt=7): 2462 (x => y) ^ y = y # label("(3.13)"). [para(2417(a,1),1205(a,1))]. given #85 (H,wt=17): 752 (x -> y) ^ (~ (x ^ y) -> ~ x) = x => (x ^ y). [para(725(a,1),27(a,1,1))]. given #86 (H,wt=15): 2601 (x -> y) ^ (~ y -> ~ x) = x => (x ^ y). [para(1030(a,1),752(a,1,2))]. given #87 (H,wt=9): 2655 x => (x ^ y) = x => y # label("(3.14)"). [para(2601(a,1),27(a,1))]. -------- Proof 1 -------- "(Lemma 3.1, 5 parts)". ============================== PROOF ================================= % Proof 1 at 0.10 (+ 0.02) seconds: "(Lemma 3.1, 5 parts)". % Length of proof is 71. % Level of proof is 13. % Maximum clause weight is 39. % Given clauses 87. 1 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. 2 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. 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]. 6 x ^ (x v y) = x # label("(D6)"). [assumption]. 10 x ^ 1 = x # label("(D10)"). [assumption]. 12 ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. 14 ~ ~ x = x # label("(DN)"). [assumption]. 17 x -> x = 1 # label("(N2)"). [assumption]. 19 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. 20 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 21 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. 26 x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). [assumption]. 27 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(26),flip(a)]. 28 x -> 1 = 1 # label("(3.1)"). [assumption]. 29 1 -> x = x # label("(3.2)"). [assumption]. 32 x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. 33 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. 40 (A ^ B) -> B != 1 | (A ^ B) => B != 1 | ~ A => ~ B != B => A | (A => B) ^ B != B | A => (A ^ B) != A => B # answer("(Lemma 3.1, 5 parts)"). [assumption]. 60 x v (y ^ x) = x. [para(4(a,1),5(a,1,2))]. 65 x ^ (y v x) = x. [para(3(a,1),6(a,1,2))]. 113 1 ^ x = x. [para(10(a,1),4(a,1)),flip(a)]. 204 x ^ (y v ~ x) = x ^ (x -> y). [para(3(a,1),19(a,1,2))]. 229 (x -> y) ^ (x -> 1) = x -> y. [para(10(a,1),20(a,2,2))]. 234 1 ^ (x -> y) = x -> (x ^ y). [para(17(a,1),20(a,1,1))]. 288 (x -> y) ^ ((~ y -> ~ x) ^ z) = (x => y) ^ z. [para(27(a,1),2(a,1,1)),flip(a)]. 290 (~ x -> ~ y) ^ (y -> x) = y => x. [para(27(a,1),4(a,1)),flip(a)]. 296 (x -> ~ y) ^ (y -> ~ x) = x => ~ y. [para(14(a,1),27(a,1,2,1))]. 407 1 ^ (x -> y) = (y v x) -> y. [para(17(a,1),33(a,1,1))]. 424 x ^ (y -> x) = (1 v y) -> x. [para(29(a,1),33(a,1,1))]. 534 1 v x = 1. [para(113(a,1),5(a,1,2))]. 580 x ^ (y v (z v x)) = x. [para(1(a,1),65(a,1,2))]. 602 x ^ (y v (x v z)) = x. [para(3(a,1),580(a,1,2,2))]. 725 x -> (x ^ y) = x -> y. [para(234(a,1),113(a,1))]. 736 x -> (x v y) = x -> x. [para(6(a,1),725(a,1,2)),flip(a)]. 752 (x -> y) ^ (~ (x ^ y) -> ~ x) = x => (x ^ y). [para(725(a,1),27(a,1,1))]. 765 x -> (x ^ (x -> y)) = x -> y. [para(32(a,1),725(a,2))]. 789 x -> (x v y) = 1. [para(17(a,1),736(a,2))]. 822 x -> (y v x) = 1. [para(3(a,1),789(a,1,2))]. 824 ~ x -> ~ (x ^ y) = 1. [para(12(a,1),789(a,1,2))]. 874 (x ^ y) -> y = 1 # label("(3.10)"). [para(60(a,1),822(a,1,2))]. 931 ((x ^ y) -> x) ^ 1 = (x ^ y) => x. [para(824(a,1),27(a,1,2))]. 1006 (x v y) -> x = y -> x. [para(407(a,1),113(a,1))]. 1030 ~ (x ^ y) -> ~ x = ~ y -> ~ x. [para(12(a,1),1006(a,1,1))]. 1135 x ^ (y -> x) = 1 -> x. [para(534(a,1),424(a,2,1))]. 1179 x ^ (y -> x) = x. [para(29(a,1),1135(a,2))]. 1205 (x -> y) ^ y = y. [para(1179(a,1),4(a,1)),flip(a)]. 1215 (x -> y) v y = x -> y. [para(1179(a,1),60(a,1,2))]. 1220 x -> (y -> x) = 1. [para(1179(a,1),874(a,1,1))]. 1267 (x -> (y -> x)) ^ (x -> 1) = 1. [para(1220(a,1),229(a,2))]. 1309 x ^ (y -> (x v z)) = x. [para(1215(a,1),602(a,1,2))]. 1446 (x -> (y -> x)) ^ 1 = 1. [para(28(a,1),1267(a,1,2))]. 1525 x -> (x ^ (x -> y)) = x -> (y v ~ x). [para(204(a,1),725(a,1,2))]. 1548 (x -> ~ y) ^ (y -> ~ x) = y => ~ x. [para(14(a,1),290(a,1,1,1))]. 1620 (x -> (y -> x)) ^ 1 = (x ^ y) => x. [para(21(a,1),931(a,1,1))]. 1808 (x ^ y) => x = 1. [para(1620(a,1),1446(a,1))]. 1819 (x ^ y) => y = 1 # label("(3.11)"). [para(4(a,1),1808(a,1,1))]. 1985 x -> (y v ~ x) = x -> y. [para(1525(a,1),765(a,1))]. 1995 x -> (~ x v y) = x -> y. [para(3(a,1),1985(a,1,2))]. 2122 ~ x ^ (x -> y) = ~ x. [para(1995(a,1),1309(a,1,2))]. 2143 x ^ (~ x -> y) = ~ ~ x. [para(14(a,1),2122(a,1,1))]. 2180 x ^ (~ x -> y) = x. [para(14(a,1),2143(a,2))]. 2207 (~ x -> y) ^ x = x. [para(2180(a,1),4(a,1)),flip(a)]. 2295 x => ~ y = y => ~ x. [para(1548(a,1),296(a,1))]. 2296 ~ x => ~ y = y => x # label("(3.12)"). [para(14(a,1),2295(a,1,2)),flip(a)]. 2417 (x -> y) ^ y = (x => y) ^ y. [para(2207(a,1),288(a,1,2))]. 2462 (x => y) ^ y = y # label("(3.13)"). [para(2417(a,1),1205(a,1))]. 2601 (x -> y) ^ (~ y -> ~ x) = x => (x ^ y). [para(1030(a,1),752(a,1,2))]. 2655 x => (x ^ y) = x => y # label("(3.14)"). [para(2601(a,1),27(a,1))]. 2724 $F # answer("(Lemma 3.1, 5 parts)"). [hyper(40,a,874,a,b,1819,a,c,2296,a,d,2462,a,e,2655,a)]. ============================== end of proof ========================== ============================== STATISTICS ============================ Given=87. Generated=3570. Kept=2720. proofs=1. Usable=87. Sos=2603. Demods=0. Limbo=0, Disabled=67. Hints=61. Weight_deleted=0. Literals_deleted=0. Forward_subsumed=849. Back_subsumed=30. 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=2. Megabytes=3.46. User_CPU=0.10, System_CPU=0.02, Wall_clock=0. ============================== end of statistics ===================== ============================== end of search ========================= THEOREM PROVED THEOREM PROVED Exiting with 1 proof. ------ process 32327 exit (max_proofs) ------  Process 32327 exit (max_proofs) Mon Jan 21 11:14:29 2008