0.03/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.32 % Computer : n029.cluster.edu 0.12/0.32 % Model : x86_64 x86_64 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.32 % Memory : 8042.1875MB 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.32 % CPULimit : 1440 0.12/0.32 % WCLimit : 180 0.12/0.33 % DateTime : Mon Jul 3 10:14:12 EDT 2023 0.12/0.33 % CPUTime : 0.42/1.01 ============================== Prover9 =============================== 0.42/1.01 Prover9 (32) version 2009-11A, November 2009. 0.42/1.01 Process 25724 was started by sandbox2 on n029.cluster.edu, 0.42/1.01 Mon Jul 3 10:14:13 2023 0.42/1.01 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1440 -f /tmp/Prover9_25570_n029.cluster.edu". 0.42/1.01 ============================== end of head =========================== 0.42/1.01 0.42/1.01 ============================== INPUT ================================= 0.42/1.01 0.42/1.01 % Reading from file /tmp/Prover9_25570_n029.cluster.edu 0.42/1.01 0.42/1.01 set(prolog_style_variables). 0.42/1.01 set(auto2). 0.42/1.01 % set(auto2) -> set(auto). 0.42/1.01 % set(auto) -> set(auto_inference). 0.42/1.01 % set(auto) -> set(auto_setup). 0.42/1.01 % set(auto_setup) -> set(predicate_elim). 0.42/1.01 % set(auto_setup) -> assign(eq_defs, unfold). 0.42/1.01 % set(auto) -> set(auto_limits). 0.42/1.01 % set(auto_limits) -> assign(max_weight, "100.000"). 0.42/1.01 % set(auto_limits) -> assign(sos_limit, 20000). 0.42/1.01 % set(auto) -> set(auto_denials). 0.42/1.01 % set(auto) -> set(auto_process). 0.42/1.01 % set(auto2) -> assign(new_constants, 1). 0.42/1.01 % set(auto2) -> assign(fold_denial_max, 3). 0.42/1.01 % set(auto2) -> assign(max_weight, "200.000"). 0.42/1.01 % set(auto2) -> assign(max_hours, 1). 0.42/1.01 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.42/1.01 % set(auto2) -> assign(max_seconds, 0). 0.42/1.01 % set(auto2) -> assign(max_minutes, 5). 0.42/1.01 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.42/1.01 % set(auto2) -> set(sort_initial_sos). 0.42/1.01 % set(auto2) -> assign(sos_limit, -1). 0.42/1.01 % set(auto2) -> assign(lrs_ticks, 3000). 0.42/1.01 % set(auto2) -> assign(max_megs, 400). 0.42/1.01 % set(auto2) -> assign(stats, some). 0.42/1.01 % set(auto2) -> clear(echo_input). 0.42/1.01 % set(auto2) -> set(quiet). 0.42/1.01 % set(auto2) -> clear(print_initial_clauses). 0.42/1.01 % set(auto2) -> clear(print_given). 0.42/1.01 assign(lrs_ticks,-1). 0.42/1.01 assign(sos_limit,10000). 0.42/1.01 assign(order,kbo). 0.42/1.01 set(lex_order_vars). 0.42/1.01 clear(print_given). 0.42/1.01 0.42/1.01 % formulas(sos). % not echoed (77 formulas) 0.42/1.01 0.42/1.01 ============================== end of input ========================== 0.42/1.01 0.42/1.01 % From the command line: assign(max_seconds, 1440). 0.42/1.01 0.42/1.01 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.42/1.01 0.42/1.01 % Formulas that are not ordinary clauses: 0.42/1.01 1 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 2 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 4 substitution_of_equivalents <-> (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 5 (all X all Y all Z is_a_theorem(implies(implies(X,Z),implies(implies(Y,Z),implies(or(X,Y),Z))))) <-> or_3 # label(or_3) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 6 and_3 <-> (all X all Y is_a_theorem(implies(X,implies(Y,and(X,Y))))) # label(and_3) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 7 implies_3 <-> (all X all Y all Z is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) # label(implies_3) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 8 (all X all Y is_a_theorem(implies(implies(X,implies(X,Y)),implies(X,Y)))) <-> implies_2 # label(implies_2) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 9 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 10 equivalence_2 <-> (all X all Y is_a_theorem(implies(equiv(X,Y),implies(Y,X)))) # label(equivalence_2) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 11 cn1 <-> (all P all Q all R is_a_theorem(implies(implies(P,Q),implies(implies(Q,R),implies(P,R))))) # label(cn1) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 12 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 13 r4 <-> (all P all Q all R is_a_theorem(implies(or(P,or(Q,R)),or(Q,or(P,R))))) # label(r4) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 14 modus_ponens <-> (all X all Y (is_a_theorem(X) & is_a_theorem(implies(X,Y)) -> is_a_theorem(Y))) # label(modus_ponens) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 15 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 16 (all P all Q all R is_a_theorem(implies(implies(P,Q),implies(not(and(Q,R)),not(and(R,P)))))) <-> kn3 # label(kn3) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 17 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 18 modus_tollens <-> (all X all Y is_a_theorem(implies(implies(not(Y),not(X)),implies(X,Y)))) # label(modus_tollens) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 19 (all P is_a_theorem(implies(P,and(P,P)))) <-> kn1 # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 20 (all P is_a_theorem(implies(implies(not(P),P),P))) <-> cn3 # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 21 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 22 (all X all Y is_a_theorem(implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y))))) <-> equivalence_3 # label(equivalence_3) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 23 r5 <-> (all P all Q all R is_a_theorem(implies(implies(Q,R),implies(or(P,Q),or(P,R))))) # label(r5) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 24 (all X all Y is_a_theorem(implies(equiv(X,Y),implies(X,Y)))) <-> equivalence_1 # label(equivalence_1) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 25 (all X all Y is_a_theorem(implies(and(X,Y),Y))) <-> and_2 # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 26 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 27 op_equiv -> (all X all Y equiv(X,Y) = and(implies(X,Y),implies(Y,X))) # label(op_equiv) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 28 op_implies_and -> (all X all Y implies(X,Y) = not(and(X,not(Y)))) # label(op_implies_and) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 29 op_and -> (all X all Y not(or(not(X),not(Y))) = and(X,Y)) # label(op_and) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 30 op_implies_or -> (all X all Y or(not(X),Y) = implies(X,Y)) # label(op_implies_or) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 31 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 32 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 33 (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_4 # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 34 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 35 (all X all Y (is_a_theorem(strict_equiv(X,Y)) -> X = Y)) <-> substitution_strict_equiv # label(substitution_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 36 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 37 axiom_m8 <-> (all P all Q is_a_theorem(strict_implies(strict_implies(P,Q),strict_implies(possibly(P),possibly(Q))))) # label(axiom_m8) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 38 (all P all Q is_a_theorem(strict_implies(possibly(and(P,Q)),P))) <-> axiom_m7 # label(axiom_m7) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 39 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 40 (all X all Y all Z is_a_theorem(strict_implies(and(strict_implies(X,Y),strict_implies(Y,Z)),strict_implies(X,Z)))) <-> axiom_m5 # label(axiom_m5) # label(axiom) # label(non_clause). [assumption]. 0.42/1.01 41 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 42 axiom_m3 <-> (all X all Y all Z is_a_theorem(strict_implies(and(and(X,Y),Z),and(X,and(Y,Z))))) # label(axiom_m3) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 43 axiom_m1 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),and(Y,X)))) # label(axiom_m1) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 44 (all X all Y all Z is_a_theorem(implies(and(necessarily(implies(X,Y)),necessarily(implies(Y,Z))),necessarily(implies(X,Z))))) <-> axiom_s1 # label(axiom_s1) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 45 (all X all Y is_a_theorem(strict_implies(strict_implies(X,Y),strict_implies(not(possibly(Y)),not(possibly(X)))))) <-> axiom_s3 # label(axiom_s3) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 46 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 47 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 48 axiom_s2 <-> (all P all Q is_a_theorem(strict_implies(possibly(and(P,Q)),and(possibly(P),possibly(Q))))) # label(axiom_s2) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 49 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 50 (all X all Y (is_a_theorem(Y) & is_a_theorem(X) -> is_a_theorem(and(X,Y)))) <-> adjunction # label(adjunction) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 51 modus_ponens_strict_implies <-> (all X all Y (is_a_theorem(strict_implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) # label(modus_ponens_strict_implies) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 52 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 53 (all X all Y is_a_theorem(implies(necessarily(implies(X,Y)),implies(necessarily(X),necessarily(Y))))) <-> axiom_K # label(axiom_K) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 54 (all X is_a_theorem(implies(necessarily(X),X))) <-> axiom_M # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 55 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 56 op_strict_equiv -> (all X all Y strict_equiv(X,Y) = and(strict_implies(X,Y),strict_implies(Y,X))) # label(op_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 57 op_strict_implies -> (all X all Y necessarily(implies(X,Y)) = strict_implies(X,Y)) # label(op_strict_implies) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 58 op_possibly -> (all X not(necessarily(not(X))) = possibly(X)) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.74/1.04 0.74/1.04 ============================== end of process non-clausal formulas === 0.74/1.04 0.74/1.04 ============================== PROCESS INITIAL CLAUSES =============== 0.74/1.04 0.74/1.04 ============================== PREDICATE ELIMINATION ================= 0.74/1.04 0.74/1.04 ============================== end predicate elimination ============= 0.74/1.04 0.74/1.04 Auto_denials: (non-Horn, no changes). 0.74/1.04 0.74/1.04 Term ordering decisions: 0.74/1.04 Function symbol KB weights: c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. c16=1. c17=1. c18=1. c19=1. c20=1. c21=1. c22=1. c23=1. c24=1. c25=1. c26=1. c27=1. c28=1. c29=1. c30=1. c31=1. c32=1. c33=1. c34=1. c35=1. c36=1. c37=1. c38=1. c39=1. c40=1. c41=1. c42=1. c43=1. c44=1. c45=1. c46=1. c47=1. c48=1. c49=1. c50=1. c51=1. c52=1. c53=1. c54=1. c55=1. c56=1. c57=1. c58=1. c59=1. c60=1. c61=1. c62=1. c63=1. c64=1. c65=1. c66=1. c67=1. c68=1. c69=1. c70=1. c71=1. c72=1. c73=1. c74=1. c75=1. c76=1. c77=1. c78=1. c79=1. c80=1. c81=1. c82=1. c83=1. c84=1. c85=1. c86=1. c87=1. c88=1. c89=1. c90=1. c91=1. c92=1. c93=1. c94=1. implies=1. and=1. strict_implies=1. or=1. equiv=1. strict_equiv=1. necessarily=1. possibly=1. not=1. 0.74/1.04 0.74/1.04 ============================== end of process initial clauses ======== 0.74/1.04 0.74/1.04 ============================== CLAUSES FOR SEARCH ==================== 168.93/169.21 168.93/169.21 ============================== end of clauses for search ============= 168.93/169.21 168.93/169.21 ============================== SEARCH ================================ 168.93/169.21 168.93/169.21 % Starting search at 0.05 seconds. 168.93/169.21 168.93/169.21 Low Water (keep): wt=27.000, iters=3370 168.93/169.21 168.93/169.21 Low Water (keep): wt=23.000, iters=3333 168.93/169.21 168.93/169.21 Low Water (keep): wt=22.000, iters=3377 168.93/169.21 168.93/169.21 Low Water (keep): wt=21.000, iters=3367 168.93/169.21 168.93/169.21 Low Water (keep): wt=20.000, iters=3355 168.93/169.21 168.93/169.21 Low Water (keep): wt=19.000, iters=3353 168.93/169.21 168.93/169.21 Low Water (keep): wt=18.000, iters=3346 168.93/169.21 168.93/169.21 Low Water (keep): wt=17.000, iters=3385 168.93/169.21 168.93/169.21 Low Water (keep): wt=16.000, iters=3334 168.93/169.21 168.93/169.21 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 411 (0.00 of 2.13 sec). 168.93/169.21 % back CAC tautology: 16333 equiv(not(A),B) = equiv(B,not(A)). [back_rewrite(282),rewrite([15822(4),281(4)])]. 168.93/169.21 % back CAC tautology: 16242 equiv(possibly(necessarily(not(A))),B) = equiv(B,possibly(necessarily(not(A)))). [back_rewrite(2544),rewrite([15822(8),2430(8)])]. 168.93/169.21 % back CAC tautology: 16240 equiv(implies(A,necessarily(not(B))),C) = equiv(C,implies(A,necessarily(not(B)))). [back_rewrite(2546),rewrite([15822(8),2433(8)])]. 168.93/169.21 % back CAC tautology: 16239 equiv(implies(A,and(B,not(C))),D) = equiv(D,implies(A,and(B,not(C)))). [back_rewrite(2547),rewrite([15822(8),2435(8)])]. 168.93/169.21 % back CAC tautology: 15975 equiv(necessarily(not(A)),possibly(A)) = equiv(possibly(A),necessarily(not(A))). [back_rewrite(11795),rewrite([15822(7),11794(7)]),flip(a)]. 168.93/169.21 168.93/169.21 Low Water (displace): id=19513, wt=48.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19504, wt=41.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19516, wt=40.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19512, wt=38.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19510, wt=37.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19543, wt=36.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19508, wt=34.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19750, wt=33.000 168.93/169.21 168.93/169.21 Low Water (displace): id=19935, wt=32.000 168.93/169.21 168.93/169.21 Low Water (displace): id=18278, wt=31.000 168.93/169.21 168.93/169.21 Low Water (displace): id=20050, wt=30.000 168.93/169.21 168.93/169.21 Low Water (displace): id=25556, wt=14.000 168.93/169.21 % back CAC tautology: 28497 or(or(A,B),not(C)) = or(not(C),or(A,B)). [para(28428(a,2),25147(a,1)),rewrite([25147(6)]),flip(a)]. 168.93/169.21 % back CAC tautology: 28489 or(and(A,possibly(B)),not(C)) = or(not(C),and(A,possibly(B))). [para(28428(a,2),2201(a,2)),rewrite([2201(5)])]. 168.93/169.21 168.93/169.21 Low Water (displace): id=30240, wt=13.000 168.93/169.21 168.93/169.21 Low Water (displace): id=30258, wt=12.000 168.93/169.21 168.93/169.21 Low Water (displace): id=30378, wt=11.000 168.93/169.21 168.93/169.21 Low Water (displace): id=30390, wt=10.000 168.93/169.21 168.93/169.21 Low Water (keep): wt=15.000, iters=3335 168.93/169.21 168.93/169.21 Low Water (keep): wt=14.000, iters=3369 168.93/169.21 168.93/169.21 Low Water (keep): wt=13.000, iters=3335 168.93/169.21 168.93/169.21 Low Water (keep): wt=12.000, iters=3757 168.93/169.21 168.93/169.21 ============================== PROOF ================================= 168.93/169.21 % SZS status Theorem 168.93/169.21 % SZS output start Refutation 168.93/169.21 168.93/169.21 % Proof 1 at 163.40 (+ 4.81) seconds. 168.93/169.21 % Length of proof is 231. 168.93/169.21 % Level of proof is 45. 168.93/169.21 % Maximum clause weight is 17.000. 168.93/169.21 % Given clauses 24647. 168.93/169.21 168.93/169.21 4 substitution_of_equivalents <-> (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 18 modus_tollens <-> (all X all Y is_a_theorem(implies(implies(not(Y),not(X)),implies(X,Y)))) # label(modus_tollens) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 27 op_equiv -> (all X all Y equiv(X,Y) = and(implies(X,Y),implies(Y,X))) # label(op_equiv) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 28 op_implies_and -> (all X all Y implies(X,Y) = not(and(X,not(Y)))) # label(op_implies_and) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 31 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 35 (all X all Y (is_a_theorem(strict_equiv(X,Y)) -> X = Y)) <-> substitution_strict_equiv # label(substitution_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 40 (all X all Y all Z is_a_theorem(strict_implies(and(strict_implies(X,Y),strict_implies(Y,Z)),strict_implies(X,Z)))) <-> axiom_m5 # label(axiom_m5) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 42 axiom_m3 <-> (all X all Y all Z is_a_theorem(strict_implies(and(and(X,Y),Z),and(X,and(Y,Z))))) # label(axiom_m3) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 43 axiom_m1 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),and(Y,X)))) # label(axiom_m1) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 46 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 50 (all X all Y (is_a_theorem(Y) & is_a_theorem(X) -> is_a_theorem(and(X,Y)))) <-> adjunction # label(adjunction) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 51 modus_ponens_strict_implies <-> (all X all Y (is_a_theorem(strict_implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) # label(modus_ponens_strict_implies) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 52 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 56 op_strict_equiv -> (all X all Y strict_equiv(X,Y) = and(strict_implies(X,Y),strict_implies(Y,X))) # label(op_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 57 op_strict_implies -> (all X all Y necessarily(implies(X,Y)) = strict_implies(X,Y)) # label(op_strict_implies) # label(axiom) # label(non_clause). [assumption]. 168.93/169.21 65 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(4)]. 168.93/169.21 97 modus_tollens | -is_a_theorem(implies(implies(not(c40),not(c39)),implies(c39,c40))) # label(modus_tollens) # label(axiom). [clausify(18)]. 168.93/169.21 114 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(27)]. 168.93/169.21 115 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(114),flip(b)]. 168.93/169.21 116 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(28)]. 168.93/169.21 120 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(31)]. 168.93/169.21 121 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(120),flip(b)]. 168.93/169.21 130 -is_a_theorem(strict_equiv(A,B)) | B = A | -substitution_strict_equiv # label(substitution_strict_equiv) # label(axiom). [clausify(35)]. 168.93/169.21 140 is_a_theorem(strict_implies(and(strict_implies(A,B),strict_implies(B,C)),strict_implies(A,C))) | -axiom_m5 # label(axiom_m5) # label(axiom). [clausify(40)]. 168.93/169.21 143 -axiom_m3 | is_a_theorem(strict_implies(and(and(A,B),C),and(A,and(B,C)))) # label(axiom_m3) # label(axiom). [clausify(42)]. 168.93/169.21 145 -axiom_m1 | is_a_theorem(strict_implies(and(A,B),and(B,A))) # label(axiom_m1) # label(axiom). [clausify(43)]. 168.93/169.21 151 -axiom_m2 | is_a_theorem(strict_implies(and(A,B),A)) # label(axiom_m2) # label(axiom). [clausify(46)]. 168.93/169.21 163 -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(B,A)) | -adjunction # label(adjunction) # label(axiom). [clausify(50)]. 168.93/169.21 164 -modus_ponens_strict_implies | -is_a_theorem(strict_implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B) # label(modus_ponens_strict_implies) # label(axiom). [clausify(51)]. 168.93/169.21 169 is_a_theorem(strict_implies(A,and(A,A))) | -axiom_m4 # label(axiom_m4) # label(axiom). [clausify(52)]. 168.93/169.21 176 -op_strict_equiv | strict_equiv(A,B) = and(strict_implies(A,B),strict_implies(B,A)) # label(op_strict_equiv) # label(axiom). [clausify(56)]. 168.93/169.21 177 -op_strict_equiv | and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B). [copy(176),flip(b)]. 168.93/169.21 178 -op_strict_implies | strict_implies(A,B) = necessarily(implies(A,B)) # label(op_strict_implies) # label(axiom). [clausify(57)]. 168.93/169.21 179 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B). [copy(178),flip(b)]. 168.93/169.21 182 adjunction # label(s1_0_adjunction) # label(axiom). [assumption]. 168.93/169.21 184 op_strict_equiv # label(s1_0_op_strict_equiv) # label(axiom). [assumption]. 168.93/169.21 186 op_or # label(s1_0_op_or) # label(axiom). [assumption]. 168.93/169.21 187 axiom_m2 # label(s1_0_axiom_m2) # label(axiom). [assumption]. 168.93/169.21 188 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption]. 168.93/169.21 189 modus_ponens_strict_implies # label(s1_0_modus_ponens_strict_implies) # label(axiom). [assumption]. 168.93/169.21 190 op_equiv # label(s1_0_op_equiv) # label(axiom). [assumption]. 168.93/169.21 191 axiom_m1 # label(s1_0_axiom_m1) # label(axiom). [assumption]. 168.93/169.21 192 axiom_m5 # label(s1_0_axiom_m5) # label(axiom). [assumption]. 168.93/169.21 193 substitution_strict_equiv # label(s1_0_substitution_strict_equiv) # label(axiom). [assumption]. 168.93/169.21 194 axiom_m4 # label(s1_0_axiom_m4) # label(axiom). [assumption]. 168.93/169.21 195 axiom_m3 # label(s1_0_axiom_m3) # label(axiom). [assumption]. 168.93/169.21 196 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 168.93/169.21 197 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 168.93/169.21 198 -modus_tollens # label(hilbert_modus_tollens) # label(negated_conjecture). [assumption]. 168.93/169.21 200 -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(B,A)). [back_unit_del(163),unit_del(d,182)]. 168.93/169.21 201 and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B). [back_unit_del(177),unit_del(a,184)]. 168.93/169.21 203 not(and(not(A),not(B))) = or(A,B). [back_unit_del(121),unit_del(a,186)]. 168.93/169.21 204 is_a_theorem(strict_implies(and(A,B),A)). [back_unit_del(151),unit_del(a,187)]. 168.93/169.21 205 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(179),unit_del(a,188)]. 168.93/169.21 206 -is_a_theorem(strict_implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B). [back_unit_del(164),unit_del(a,189)]. 168.93/169.21 207 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(115),unit_del(a,190)]. 168.93/169.21 208 is_a_theorem(strict_implies(and(A,B),and(B,A))). [back_unit_del(145),unit_del(a,191)]. 168.93/169.21 209 is_a_theorem(strict_implies(and(strict_implies(A,B),strict_implies(B,C)),strict_implies(A,C))). [back_unit_del(140),unit_del(b,192)]. 168.93/169.21 210 -is_a_theorem(strict_equiv(A,B)) | B = A. [back_unit_del(130),unit_del(c,193)]. 168.93/169.21 211 is_a_theorem(strict_implies(A,and(A,A))). [back_unit_del(169),unit_del(b,194)]. 168.93/169.21 212 is_a_theorem(strict_implies(and(and(A,B),C),and(A,and(B,C)))). [back_unit_del(143),unit_del(a,195)]. 168.93/169.21 213 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(65),unit_del(a,196)]. 168.93/169.21 214 not(and(A,not(B))) = implies(A,B). [back_unit_del(116),unit_del(a,197)]. 168.93/169.21 215 -is_a_theorem(implies(implies(not(c40),not(c39)),implies(c39,c40))). [back_unit_del(97),unit_del(a,198)]. 168.93/169.21 220 implies(not(A),B) = or(A,B). [back_rewrite(203),rewrite([214(4)])]. 168.93/169.21 221 -is_a_theorem(implies(or(c40,not(c39)),implies(c39,c40))). [back_rewrite(215),rewrite([220(5)])]. 168.93/169.21 240 -is_a_theorem(A) | is_a_theorem(and(strict_implies(and(B,C),B),A)). [resolve(204,a,200,b)]. 168.93/169.21 241 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(and(B,C),B))). [resolve(204,a,200,a)]. 168.93/169.21 248 -is_a_theorem(A) | is_a_theorem(and(strict_implies(and(B,C),and(C,B)),A)). [resolve(208,a,200,b)]. 168.93/169.21 254 -is_a_theorem(and(strict_implies(A,B),strict_implies(B,C))) | is_a_theorem(strict_implies(A,C)). [resolve(209,a,206,a)]. 168.93/169.21 260 -is_a_theorem(A) | is_a_theorem(and(strict_implies(B,and(B,B)),A)). [resolve(211,a,200,b)]. 168.93/169.21 261 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(B,and(B,B)))). [resolve(211,a,200,a)]. 168.93/169.21 277 not(and(A,implies(B,C))) = implies(A,and(B,not(C))). [para(214(a,1),214(a,1,1,2))]. 168.93/169.21 280 necessarily(or(A,B)) = strict_implies(not(A),B). [para(220(a,1),205(a,1,1))]. 168.93/169.21 283 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(214(a,1),220(a,1,1)),flip(a)]. 168.93/169.21 463 is_a_theorem(and(strict_implies(and(A,B),A),strict_implies(C,and(C,C)))). [resolve(240,a,211,a)]. 168.93/169.21 520 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(and(B,C),B))). [resolve(241,a,211,a)]. 168.93/169.21 522 is_a_theorem(and(strict_implies(and(A,B),and(B,A)),strict_implies(and(C,D),C))). [resolve(241,a,208,a)]. 168.93/169.21 571 is_a_theorem(and(strict_implies(and(A,B),and(B,A)),strict_implies(and(C,D),and(D,C)))). [resolve(248,a,208,a)]. 168.93/169.21 1057 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(B,and(B,B)))). [resolve(260,a,211,a)]. 168.93/169.21 2316 implies(A,and(not(B),not(C))) = not(and(A,or(B,C))). [para(220(a,1),277(a,1,1,2)),flip(a)]. 168.93/169.21 9058 is_a_theorem(strict_equiv(and(A,A),A)). [para(201(a,1),463(a,1))]. 168.93/169.21 9098 and(A,A) = A. [resolve(9058,a,210,a),flip(a)]. 168.93/169.21 9590 is_a_theorem(and(strict_implies(A,A),strict_implies(B,B))). [back_rewrite(1057),rewrite([9098(1),9098(2)])]. 168.93/169.21 9605 is_a_theorem(and(strict_implies(A,A),strict_implies(and(B,C),B))). [back_rewrite(520),rewrite([9098(1)])]. 168.93/169.21 9608 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(B,B))). [back_rewrite(261),rewrite([9098(2)])]. 168.93/169.21 9609 -is_a_theorem(A) | is_a_theorem(and(strict_implies(B,B),A)). [back_rewrite(260),rewrite([9098(2)])]. 168.93/169.21 9610 is_a_theorem(strict_implies(A,A)). [back_rewrite(211),rewrite([9098(1)])]. 168.93/169.21 9647 strict_equiv(A,A) = strict_implies(A,A). [para(9098(a,1),201(a,1)),flip(a)]. 168.93/169.21 9649 is_a_theorem(strict_implies(and(A,B),and(A,and(A,B)))). [para(9098(a,1),212(a,1,1,1))]. 168.93/169.21 9652 or(A,A) = not(not(A)). [para(9098(a,1),214(a,1,1)),rewrite([220(4)]),flip(a)]. 168.93/169.21 9663 implies(implies(A,B),and(A,not(B))) = not(implies(A,B)). [para(9098(a,1),277(a,1,1)),flip(a)]. 168.93/169.21 9665 implies(or(A,A),B) = or(not(A),B). [para(9098(a,1),283(a,1,1)),rewrite([220(4)]),flip(a)]. 168.93/169.21 9799 strict_implies(or(A,A),B) = strict_implies(not(not(A)),B). [para(9652(a,2),280(a,2,1)),rewrite([280(3)]),flip(a)]. 168.93/169.21 9819 not(or(A,A)) = not(not(not(A))). [para(9652(a,2),9652(a,2,1)),rewrite([9652(3)]),flip(a)]. 168.93/169.21 9941 is_a_theorem(and(strict_implies(A,A),and(strict_implies(B,B),strict_implies(C,C)))). [resolve(9609,a,9590,a)]. 168.93/169.21 10734 implies(A,or(B,B)) = implies(A,not(not(B))). [para(9819(a,1),214(a,1,1,2)),rewrite([214(5)]),flip(a)]. 168.93/169.21 10735 or(or(A,A),B) = or(not(not(A)),B). [para(9819(a,1),220(a,1,1)),rewrite([220(4)]),flip(a)]. 168.93/169.21 10769 not(not(not(or(A,A)))) = not(not(not(not(not(A))))). [para(9819(a,1),9819(a,2,1,1)),rewrite([9652(3)])]. 168.93/169.21 14696 is_a_theorem(strict_implies(and(A,B),B)). [resolve(522,a,254,a)]. 168.93/169.21 14737 -is_a_theorem(A) | is_a_theorem(and(strict_implies(and(B,C),C),A)). [resolve(14696,a,200,b)]. 168.93/169.21 14738 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(and(B,C),C))). [resolve(14696,a,200,a)]. 168.93/169.21 15094 is_a_theorem(and(strict_implies(and(A,B),B),strict_implies(and(C,D),and(C,and(C,D))))). [resolve(14737,a,9649,a)]. 168.93/169.21 15343 -is_a_theorem(strict_implies(not(not(A)),B)) | -is_a_theorem(or(A,A)) | is_a_theorem(B). [para(9799(a,1),206(a,1))]. 168.93/169.21 15344 -is_a_theorem(strict_implies(or(A,A),B)) | -is_a_theorem(not(not(A))) | is_a_theorem(B). [para(9799(a,2),206(a,1))]. 168.93/169.21 15793 is_a_theorem(strict_equiv(and(A,B),and(B,A))). [para(201(a,1),571(a,1))]. 168.93/169.21 15822 and(A,B) = and(B,A). [resolve(15793,a,210,a)]. 168.93/169.21 16344 -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(A,B)). [back_rewrite(200),rewrite([15822(3)])]. 168.93/169.21 16433 strict_equiv(A,B) = strict_equiv(B,A). [para(15822(a,1),201(a,1)),rewrite([201(3)])]. 168.93/169.21 16434 equiv(A,B) = equiv(B,A). [para(15822(a,1),207(a,1)),rewrite([207(3)])]. 168.93/169.21 16435 -is_a_theorem(and(strict_implies(A,B),strict_implies(C,A))) | is_a_theorem(strict_implies(C,B)). [para(15822(a,1),254(a,1))]. 168.93/169.21 16543 -is_a_theorem(strict_equiv(A,B)) | A = B. [para(16433(a,1),210(a,1))]. 168.93/169.21 16550 -is_a_theorem(equiv(A,B)) | A = B. [para(16434(a,1),213(a,1))]. 168.93/169.21 16953 strict_implies(A,or(B,B)) = strict_implies(A,not(not(B))). [para(10734(a,1),205(a,1,1)),rewrite([205(4)]),flip(a)]. 168.93/169.21 16954 or(A,or(B,B)) = or(A,not(not(B))). [para(10734(a,1),220(a,1)),rewrite([220(4)]),flip(a)]. 168.93/169.21 17926 -is_a_theorem(strict_implies(A,or(B,B))) | -is_a_theorem(A) | is_a_theorem(not(not(B))). [para(16953(a,2),206(a,1))]. 168.93/169.21 18112 or(or(A,A),not(not(B))) = or(not(not(A)),or(B,B)). [para(16954(a,1),10735(a,1))]. 168.93/169.21 18113 or(or(A,A),or(B,B)) = or(not(not(A)),not(not(B))). [para(16954(a,1),10735(a,2))]. 168.93/169.21 20016 -is_a_theorem(A) | is_a_theorem(and(A,and(strict_implies(B,B),and(strict_implies(C,C),strict_implies(D,D))))). [resolve(9941,a,16344,b)]. 168.93/169.21 20021 is_a_theorem(and(and(strict_implies(A,A),and(strict_implies(B,B),strict_implies(C,C))),strict_implies(D,D))). [resolve(9941,a,9608,a)]. 168.93/169.21 22656 not(and(not(A),or(B,C))) = or(A,and(not(B),not(C))). [para(2316(a,1),220(a,1))]. 168.93/169.21 22668 not(and(A,or(B,C))) = not(and(A,or(C,B))). [para(15822(a,1),2316(a,1,2)),rewrite([2316(4)])]. 168.93/169.21 23393 not(implies(A,or(B,B))) = not(implies(A,not(not(B)))). [para(10734(a,1),9663(a,2,1)),rewrite([9663(6)])]. 168.93/169.21 24575 not(and(or(A,B),or(B,A))) = not(or(A,B)). [para(9098(a,1),22668(a,1,1)),flip(a)]. 168.93/169.21 25141 not(or(A,B)) = not(or(B,A)). [para(15822(a,1),24575(a,1,1)),rewrite([24575(4)])]. 168.93/169.21 25146 implies(A,or(B,C)) = implies(A,or(C,B)). [para(25141(a,1),214(a,1,1,2)),rewrite([214(4)])]. 168.93/169.21 25170 strict_implies(A,or(B,C)) = strict_implies(A,or(C,B)). [para(25146(a,1),205(a,1,1)),rewrite([205(3)])]. 168.93/169.21 25413 and(strict_implies(A,or(B,C)),strict_implies(or(C,B),A)) = strict_equiv(A,or(C,B)). [para(25170(a,1),201(a,1,1))]. 168.93/169.21 28417 or(A,and(not(B),not(C))) = implies(or(B,C),A). [para(15822(a,1),22656(a,1,1)),rewrite([214(4)]),flip(a)]. 168.93/169.21 28428 or(not(A),B) = or(B,not(A)). [para(9098(a,1),28417(a,1,2)),rewrite([9665(4)]),flip(a)]. 168.93/169.21 28432 implies(or(A,B),C) = implies(or(B,A),C). [para(15822(a,1),28417(a,1,2)),rewrite([28417(4)])]. 168.93/169.21 28478 implies(or(A,A),B) = or(B,not(A)). [back_rewrite(9665),rewrite([28428(4)])]. 168.93/169.21 28484 strict_implies(not(not(A)),B) = strict_implies(not(B),not(A)). [para(28428(a,1),280(a,1,1)),rewrite([280(3)]),flip(a)]. 168.93/169.21 28575 -is_a_theorem(strict_implies(not(A),not(B))) | -is_a_theorem(or(B,B)) | is_a_theorem(A). [back_rewrite(15343),rewrite([28484(3)])]. 168.93/169.21 28577 strict_implies(or(A,A),B) = strict_implies(not(B),not(A)). [back_rewrite(9799),rewrite([28484(5)])]. 168.93/169.21 28630 strict_implies(or(A,B),C) = strict_implies(or(B,A),C). [para(28432(a,1),205(a,1,1)),rewrite([205(3)])]. 168.93/169.21 28632 strict_equiv(A,or(B,C)) = strict_equiv(A,or(C,B)). [back_rewrite(25413),rewrite([28630(4),201(5)])]. 168.93/169.21 28733 is_a_theorem(strict_implies(or(A,A),A)). [para(28577(a,2),9610(a,1))]. 168.93/169.21 28738 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(B,B))). [para(28577(a,2),9590(a,1,1))]. 168.93/169.21 28746 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(and(B,C),B))). [para(28577(a,2),9605(a,1,1))]. 168.93/169.21 29107 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(or(B,B),B))). [resolve(28733,a,16344,b)]. 168.93/169.21 29197 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(or(B,B),B))). [para(28577(a,2),28738(a,1,2))]. 168.93/169.21 29343 -is_a_theorem(strict_equiv(A,or(B,C))) | or(C,B) = A. [para(28632(a,1),210(a,1))]. 168.93/169.21 29345 strict_equiv(A,or(B,B)) = strict_equiv(A,not(not(B))). [para(9652(a,1),28632(a,1,2)),flip(a)]. 168.93/169.21 29374 or(A,B) = or(B,A). [para(9647(a,1),29343(a,1)),unit_del(a,9610)]. 168.93/169.21 29375 -is_a_theorem(strict_equiv(A,not(not(B)))) | or(B,B) = A. [para(9652(a,1),29343(a,1,2))]. 168.93/169.21 29401 strict_implies(not(A),B) = strict_implies(not(B),A). [para(29374(a,1),280(a,1,1)),rewrite([280(2)])]. 168.93/169.21 29405 or(or(A,A),or(B,B)) = or(not(not(B)),not(not(A))). [para(18113(a,2),29374(a,2)),flip(a)]. 168.93/169.21 29422 and(strict_implies(A,not(B)),strict_implies(not(A),B)) = strict_equiv(A,not(B)). [para(29401(a,1),201(a,1,1)),rewrite([15822(5),16433(7)])]. 168.93/169.21 29426 -is_a_theorem(and(strict_implies(not(A),B),strict_implies(A,C))) | is_a_theorem(strict_implies(not(B),C)). [para(29401(a,1),254(a,1,1))]. 168.93/169.21 29494 strict_equiv(A,not(or(B,B))) = strict_equiv(A,not(not(not(B)))). [para(9652(a,2),29345(a,2,2,1)),rewrite([9652(3)]),flip(a)]. 168.93/169.21 29541 is_a_theorem(strict_implies(not(A),not(and(A,B)))). [resolve(28746,a,254,a),rewrite([9652(3),29401(4)])]. 168.93/169.21 29624 -is_a_theorem(not(A)) | is_a_theorem(not(and(A,B))). [resolve(29541,a,206,a)]. 168.93/169.21 29630 is_a_theorem(strict_implies(not(A),implies(A,B))). [para(214(a,1),29541(a,1,2))]. 168.93/169.21 29634 is_a_theorem(strict_implies(or(A,A),implies(B,A))). [para(9652(a,2),29541(a,1,1)),rewrite([15822(3),214(4)])]. 168.93/169.21 29675 is_a_theorem(and(strict_implies(not(A),implies(A,B)),strict_implies(or(C,C),C))). [resolve(29630,a,29107,a)]. 168.93/169.21 29688 is_a_theorem(and(strict_implies(not(A),implies(A,B)),strict_implies(and(C,D),D))). [resolve(29630,a,14738,a)]. 168.93/169.21 29896 -is_a_theorem(not(not(not(or(A,A))))) | is_a_theorem(implies(B,not(not(not(A))))). [para(10769(a,2),29624(a,1)),rewrite([15822(10),214(11)])]. 168.93/169.21 29925 -is_a_theorem(not(not(A))) | is_a_theorem(implies(B,A)). [resolve(29634,a,15344,a)]. 168.93/169.21 30108 -is_a_theorem(not(not(implies(A,not(not(B)))))) | is_a_theorem(implies(C,implies(A,or(B,B)))). [para(23393(a,1),29925(a,1,1))]. 168.93/169.21 31322 is_a_theorem(strict_implies(not(A),not(or(A,A)))). [resolve(29197,a,16435,a),rewrite([9652(3),29401(4)])]. 168.93/169.21 31362 is_a_theorem(strict_implies(not(A),not(not(not(A))))). [para(9652(a,1),31322(a,1,2,1))]. 168.93/169.21 31508 is_a_theorem(and(strict_implies(not(A),not(not(not(A)))),strict_implies(B,B))). [resolve(31362,a,9608,a)]. 168.93/169.21 34566 -is_a_theorem(strict_equiv(A,not(not(not(B))))) | not(or(B,B)) = A. [para(29494(a,1),210(a,1))]. 168.93/169.21 34588 is_a_theorem(strict_implies(not(not(not(A))),implies(A,B))). [resolve(29675,a,16435,a),rewrite([9652(3)])]. 168.93/169.21 34600 -is_a_theorem(not(not(not(A)))) | is_a_theorem(implies(A,B)). [resolve(34588,a,206,a)]. 168.93/169.21 34640 -is_a_theorem(not(not(not(or(A,A))))) | is_a_theorem(or(B,not(A))). [para(9819(a,2),34600(a,1,1,1)),rewrite([220(8),29374(7)])]. 168.93/169.21 35375 is_a_theorem(strict_equiv(and(A,B),and(A,and(A,B)))). [para(201(a,1),15094(a,1)),rewrite([16433(4)])]. 168.93/169.21 35376 and(A,and(A,B)) = and(A,B). [resolve(35375,a,16543,a),flip(a)]. 168.93/169.21 37751 is_a_theorem(strict_equiv(not(A),not(not(not(A))))). [para(29422(a,1),31508(a,1))]. 168.93/169.21 38040 not(or(A,A)) = not(A). [resolve(37751,a,34566,a)]. 168.93/169.21 38042 not(not(not(A))) = not(A). [resolve(37751,a,29375,a),rewrite([9652(3)])]. 168.93/169.21 38130 -is_a_theorem(not(A)) | is_a_theorem(or(B,not(A))). [back_rewrite(34640),rewrite([38040(2),38042(3)])]. 168.93/169.21 38229 -is_a_theorem(not(A)) | is_a_theorem(implies(B,not(A))). [back_rewrite(29896),rewrite([38040(2),38042(3),38042(5)])]. 168.93/169.21 38645 implies(A,or(B,B)) = implies(A,B). [para(38040(a,1),214(a,1,1,2)),rewrite([214(3)]),flip(a)]. 168.93/169.21 38646 not(not(implies(A,B))) = implies(A,B). [para(214(a,1),38040(a,2)),rewrite([9652(5),214(3)])]. 168.93/169.21 38647 or(A,or(B,B)) = or(B,A). [para(38040(a,1),220(a,1,1)),rewrite([220(2),29374(3)]),flip(a)]. 168.93/169.21 38668 not(implies(A,not(not(B)))) = not(implies(A,B)). [para(23393(a,1),38040(a,2)),rewrite([38645(2),38645(3),9652(3),38646(3)]),flip(a)]. 168.93/169.21 38687 or(not(not(A)),not(not(B))) = or(B,not(not(A))). [para(38040(a,1),18112(a,2,1,1)),rewrite([9652(3),38040(2),38647(9)])]. 168.93/169.21 38778 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(C,implies(A,B))). [back_rewrite(30108),rewrite([38668(4),38646(3),38645(4)])]. 168.93/169.21 38831 implies(A,not(not(B))) = implies(A,B). [back_rewrite(10734),rewrite([38645(2)]),flip(a)]. 168.93/169.21 38892 or(A,not(not(B))) = or(B,A). [back_rewrite(29405),rewrite([38647(3),38647(2),29374(6),38687(6)]),flip(a)]. 168.93/169.21 39421 strict_implies(A,or(B,B)) = strict_implies(A,B). [para(38645(a,1),205(a,1,1)),rewrite([205(2)]),flip(a)]. 168.93/169.21 39484 -is_a_theorem(strict_implies(A,B)) | -is_a_theorem(A) | is_a_theorem(not(not(B))). [back_rewrite(17926),rewrite([39421(2)])]. 168.93/169.21 39485 strict_implies(A,not(not(B))) = strict_implies(A,B). [back_rewrite(16953),rewrite([39421(2)]),flip(a)]. 168.93/169.21 40447 -is_a_theorem(A) | is_a_theorem(not(not(A))). [resolve(39484,a,9610,a)]. 168.93/169.21 41058 is_a_theorem(not(not(strict_implies(A,A)))). [resolve(40447,a,9610,a)]. 168.93/169.21 41144 is_a_theorem(not(not(strict_implies(and(A,B),A)))). [resolve(40447,a,204,a)]. 168.93/169.21 41714 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(41058,a,38229,a),rewrite([38831(4)])]. 168.93/169.21 41720 is_a_theorem(or(A,strict_implies(B,B))). [resolve(41058,a,38130,a),rewrite([38892(4),29374(2)])]. 168.93/169.21 41864 is_a_theorem(implies(A,implies(B,strict_implies(C,C)))). [resolve(41714,a,38778,a)]. 168.93/169.21 41884 -is_a_theorem(A) | is_a_theorem(and(A,implies(B,strict_implies(C,C)))). [resolve(41714,a,16344,b)]. 168.93/169.21 41922 -is_a_theorem(strict_implies(not(A),not(strict_implies(B,B)))) | is_a_theorem(A). [resolve(41720,a,28575,b)]. 168.93/169.21 42706 is_a_theorem(implies(A,strict_implies(and(B,C),B))). [resolve(41144,a,38229,a),rewrite([38831(5)])]. 168.93/169.21 43394 -is_a_theorem(A) | is_a_theorem(and(A,implies(B,strict_implies(and(C,D),C)))). [resolve(42706,a,16344,b)]. 168.93/169.21 48823 is_a_theorem(and(implies(A,strict_implies(B,B)),implies(C,strict_implies(D,D)))). [resolve(41884,a,41714,a)]. 168.93/169.21 48857 -is_a_theorem(strict_implies(or(A,A),not(strict_implies(B,B)))) | is_a_theorem(not(A)). [para(9652(a,2),41922(a,1,1))]. 168.93/169.21 58953 is_a_theorem(strict_implies(not(A),implies(and(B,A),C))). [resolve(29426,a,29688,a),rewrite([29401(4)])]. 168.93/169.21 61720 is_a_theorem(and(strict_implies(A,A),implies(B,strict_implies(and(C,D),C)))). [resolve(43394,a,9610,a)]. 168.93/169.21 62873 is_a_theorem(equiv(strict_implies(A,A),strict_implies(B,B))). [para(207(a,1),48823(a,1))]. 168.93/169.21 62875 strict_implies(A,A) = strict_implies(B,B). [resolve(62873,a,16550,a)]. 168.93/169.21 62883 strict_implies(A,A) = c_0. [new_symbol(62875)]. 168.93/169.21 62944 is_a_theorem(and(c_0,implies(A,strict_implies(and(B,C),B)))). [back_rewrite(61720),rewrite([62883(1)])]. 168.93/169.21 63791 -is_a_theorem(strict_implies(or(A,A),not(c_0))) | is_a_theorem(not(A)). [back_rewrite(48857),rewrite([62883(2)])]. 168.93/169.21 64450 -is_a_theorem(A) | is_a_theorem(and(A,implies(B,c_0))). [back_rewrite(41884),rewrite([62883(2)])]. 168.93/169.21 64468 is_a_theorem(implies(A,implies(B,c_0))). [back_rewrite(41864),rewrite([62883(1)])]. 168.93/169.21 64762 is_a_theorem(c_0). [back_rewrite(20021),rewrite([62883(1),62883(2),62883(3),9098(4),9098(3),62883(2),9098(3)])]. 168.93/169.21 64763 -is_a_theorem(A) | is_a_theorem(and(A,c_0)). [back_rewrite(20016),rewrite([62883(2),62883(3),62883(4),9098(5),9098(4)])]. 168.93/169.21 65006 strict_equiv(A,not(not(A))) = c_0. [para(62883(a,1),29422(a,1,2)),rewrite([39485(3),62883(1),9098(3)]),flip(a)]. 168.93/169.21 65211 is_a_theorem(and(c_0,strict_implies(not(A),implies(and(A,B),C)))). [resolve(64763,a,58953,a),rewrite([15822(2),15822(6)])]. 168.93/169.21 65803 not(not(A)) = A. [para(65006(a,1),210(a,1)),unit_del(a,64762)]. 168.93/169.21 65808 or(A,A) = A. [para(65006(a,1),29375(a,1)),unit_del(a,64762)]. 168.93/169.21 66008 -is_a_theorem(strict_implies(A,not(c_0))) | is_a_theorem(not(A)). [back_rewrite(63791),rewrite([65808(1)])]. 168.93/169.21 66049 or(A,not(B)) = implies(B,A). [back_rewrite(28478),rewrite([65808(1)]),flip(a)]. 168.93/169.21 66195 -is_a_theorem(implies(implies(c39,c40),implies(c39,c40))). [back_rewrite(221),rewrite([66049(4)])]. 168.93/169.21 66202 not(and(A,B)) = implies(A,not(B)). [para(65803(a,1),214(a,1,1,2))]. 168.93/169.21 68896 is_a_theorem(and(implies(A,implies(B,c_0)),implies(C,c_0))). [resolve(64450,a,64468,a)]. 168.93/169.21 70494 implies(A,implies(A,not(B))) = implies(A,not(B)). [para(35376(a,1),66202(a,1,1)),rewrite([66202(2),66202(4)]),flip(a)]. 168.93/169.21 72321 is_a_theorem(equiv(c_0,implies(A,c_0))). [para(207(a,1),68896(a,1))]. 168.93/169.21 72349 implies(A,c_0) = c_0. [resolve(72321,a,16550,a),flip(a)]. 168.93/169.21 72356 and(c_0,implies(c_0,A)) = equiv(A,c_0). [para(72349(a,1),207(a,1,1))]. 168.93/169.21 72861 is_a_theorem(equiv(c_0,strict_implies(and(A,B),A))). [para(72356(a,1),62944(a,1)),rewrite([16434(4)])]. 168.93/169.21 73172 strict_implies(and(A,B),A) = c_0. [resolve(72861,a,16550,a),flip(a)]. 168.93/169.21 73182 strict_implies(and(A,B),B) = c_0. [para(15822(a,1),73172(a,1,1))]. 168.93/169.21 73191 -is_a_theorem(and(c_0,strict_implies(A,B))) | is_a_theorem(strict_implies(and(A,C),B)). [para(73182(a,1),254(a,1,1)),rewrite([15822(5)])]. 168.93/169.21 75938 implies(A,implies(A,B)) = implies(A,B). [para(65803(a,1),70494(a,1,2,2)),rewrite([65803(4)])]. 168.93/169.21 75948 strict_implies(A,implies(A,B)) = strict_implies(A,B). [para(75938(a,1),205(a,1,1)),rewrite([205(2)]),flip(a)]. 168.93/169.21 83116 is_a_theorem(strict_implies(and(A,not(B)),implies(and(B,C),D))). [resolve(73191,a,65211,a),rewrite([15822(2)])]. 168.93/169.21 83212 is_a_theorem(strict_implies(and(A,not(A)),B)). [para(75948(a,1),83116(a,1))]. 168.93/169.21 83230 is_a_theorem(implies(A,A)). [resolve(83212,a,66008,a),rewrite([66202(3),65803(2)])]. 168.93/169.21 83231 $F. [resolve(83230,a,66195,a)]. 168.93/169.21 168.93/169.21 % SZS output end Refutation 168.93/169.21 ============================== end of proof ========================== 168.93/169.21 168.93/169.21 ============================== STATISTICS ============================ 168.93/169.21 168.93/169.21 Given=24647. Generated=8829724. Kept=83165. proofs=1. 168.93/169.21 Usable=12709. Sos=5832. Demods=466. Limbo=17, Disabled=64741. Hints=0. 168.93/169.21 Megabytes=88.19. 168.93/169.21 User_CPU=163.40, System_CPU=4.81, Wall_clock=168. 168.93/169.21 168.93/169.21 ============================== end of statistics ===================== 168.93/169.21 168.93/169.21 ============================== end of search ========================= 168.93/169.21 168.93/169.21 THEOREM PROVED 168.93/169.21 % SZS status Theorem 168.93/169.21 168.93/169.21 Exiting with 1 proof. 168.93/169.21 168.93/169.21 Process 25724 exit (max_proofs) Mon Jul 3 10:17:01 2023 168.93/169.21 Prover9 interrupted 168.93/169.21 EOF