0.00/0.10 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.10 % Command : tptp2X_and_run_prover9 %d %s 0.09/0.30 % Computer : n021.cluster.edu 0.09/0.30 % Model : x86_64 x86_64 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.09/0.30 % Memory : 8042.1875MB 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64 0.09/0.30 % CPULimit : 1440 0.09/0.30 % WCLimit : 180 0.09/0.30 % DateTime : Mon Jul 3 08:58:37 EDT 2023 0.09/0.30 % CPUTime : 0.73/1.05 ============================== Prover9 =============================== 0.73/1.05 Prover9 (32) version 2009-11A, November 2009. 0.73/1.05 Process 26711 was started by sandbox2 on n021.cluster.edu, 0.73/1.05 Mon Jul 3 08:58:38 2023 0.73/1.05 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1440 -f /tmp/Prover9_26558_n021.cluster.edu". 0.73/1.05 ============================== end of head =========================== 0.73/1.05 0.73/1.05 ============================== INPUT ================================= 0.73/1.05 0.73/1.05 % Reading from file /tmp/Prover9_26558_n021.cluster.edu 0.73/1.05 0.73/1.05 set(prolog_style_variables). 0.73/1.05 set(auto2). 0.73/1.05 % set(auto2) -> set(auto). 0.73/1.05 % set(auto) -> set(auto_inference). 0.73/1.05 % set(auto) -> set(auto_setup). 0.73/1.05 % set(auto_setup) -> set(predicate_elim). 0.73/1.05 % set(auto_setup) -> assign(eq_defs, unfold). 0.73/1.05 % set(auto) -> set(auto_limits). 0.73/1.05 % set(auto_limits) -> assign(max_weight, "100.000"). 0.73/1.05 % set(auto_limits) -> assign(sos_limit, 20000). 0.73/1.05 % set(auto) -> set(auto_denials). 0.73/1.05 % set(auto) -> set(auto_process). 0.73/1.05 % set(auto2) -> assign(new_constants, 1). 0.73/1.05 % set(auto2) -> assign(fold_denial_max, 3). 0.73/1.05 % set(auto2) -> assign(max_weight, "200.000"). 0.73/1.05 % set(auto2) -> assign(max_hours, 1). 0.73/1.05 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.73/1.05 % set(auto2) -> assign(max_seconds, 0). 0.73/1.05 % set(auto2) -> assign(max_minutes, 5). 0.73/1.05 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.73/1.05 % set(auto2) -> set(sort_initial_sos). 0.73/1.05 % set(auto2) -> assign(sos_limit, -1). 0.73/1.05 % set(auto2) -> assign(lrs_ticks, 3000). 0.73/1.05 % set(auto2) -> assign(max_megs, 400). 0.73/1.05 % set(auto2) -> assign(stats, some). 0.73/1.05 % set(auto2) -> clear(echo_input). 0.73/1.05 % set(auto2) -> set(quiet). 0.73/1.05 % set(auto2) -> clear(print_initial_clauses). 0.73/1.05 % set(auto2) -> clear(print_given). 0.73/1.05 assign(lrs_ticks,-1). 0.73/1.05 assign(sos_limit,10000). 0.73/1.05 assign(order,kbo). 0.73/1.05 set(lex_order_vars). 0.73/1.05 clear(print_given). 0.73/1.05 0.73/1.05 % formulas(sos). % not echoed (89 formulas) 0.73/1.05 0.73/1.05 ============================== end of input ========================== 0.73/1.05 0.73/1.05 % From the command line: assign(max_seconds, 1440). 0.73/1.05 0.73/1.05 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.73/1.05 0.73/1.05 % Formulas that are not ordinary clauses: 0.73/1.05 1 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 2 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 9 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 12 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 15 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 17 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 19 (all P is_a_theorem(implies(P,and(P,P)))) <-> kn1 # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 20 (all P is_a_theorem(implies(implies(not(P),P),P))) <-> cn3 # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 21 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 25 (all X all Y is_a_theorem(implies(and(X,Y),Y))) <-> and_2 # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 26 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 32 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 33 (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_4 # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 34 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 36 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 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]. 0.73/1.05 39 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.73/1.05 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]. 0.73/1.05 41 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.73/1.07 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]. 0.73/1.07 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]. 0.73/1.07 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]. 0.73/1.07 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]. 0.73/1.07 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]. 0.73/1.07 47 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.73/1.07 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]. 0.73/1.07 49 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.73/1.07 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]. 0.73/1.07 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]. 0.73/1.07 52 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.73/1.07 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]. 0.73/1.07 54 (all X is_a_theorem(implies(necessarily(X),X))) <-> axiom_M # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.73/1.07 55 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.73/1.07 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]. 0.73/1.07 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]. 0.73/1.07 58 op_possibly -> (all X not(necessarily(not(X))) = possibly(X)) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.73/1.07 0.73/1.07 ============================== end of process non-clausal formulas === 0.73/1.07 0.73/1.07 ============================== PROCESS INITIAL CLAUSES =============== 0.73/1.07 0.73/1.07 ============================== PREDICATE ELIMINATION ================= 0.73/1.07 0.73/1.07 ============================== end predicate elimination ============= 0.73/1.07 0.73/1.07 Auto_denials: (non-Horn, no changes). 0.73/1.07 0.73/1.07 Term ordering decisions: 0.73/1.07 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. 0.73/1.07 0.73/1.07 ============================== end of process initial clauses ======== 0.73/1.07 0.73/1.07 ============================== CLAUSES FOR SEARCH ==================== 136.87/137.30 136.87/137.30 ============================== end of clauses for search ============= 136.87/137.30 136.87/137.30 ============================== SEARCH ================================ 136.87/137.30 136.87/137.30 % Starting search at 0.03 seconds. 136.87/137.30 136.87/137.30 Low Water (keep): wt=22.000, iters=3347 136.87/137.30 136.87/137.30 Low Water (keep): wt=20.000, iters=3350 136.87/137.30 136.87/137.30 Low Water (keep): wt=19.000, iters=3396 136.87/137.30 136.87/137.30 Low Water (keep): wt=18.000, iters=3341 136.87/137.30 136.87/137.30 Low Water (keep): wt=17.000, iters=3475 136.87/137.30 136.87/137.30 Low Water (keep): wt=16.000, iters=3395 136.87/137.30 136.87/137.30 Low Water (keep): wt=15.000, iters=3333 136.87/137.30 136.87/137.30 Low Water (keep): wt=14.000, iters=3338 136.87/137.30 136.87/137.30 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 52 (0.00 of 2.85 sec). 136.87/137.30 136.87/137.30 Low Water (keep): wt=13.000, iters=3342 136.87/137.30 136.87/137.30 Low Water (displace): id=4990, wt=29.000 136.87/137.30 136.87/137.30 Low Water (displace): id=4967, wt=28.000 136.87/137.30 136.87/137.30 Low Water (displace): id=5386, wt=27.000 136.87/137.30 136.87/137.30 Low Water (displace): id=10962, wt=10.000 136.87/137.30 136.87/137.30 Low Water (displace): id=10964, wt=9.000 136.87/137.30 136.87/137.30 Low Water (keep): wt=12.000, iters=3481 136.87/137.30 136.87/137.30 Low Water (displace): id=11442, wt=8.000 136.87/137.30 136.87/137.30 Low Water (keep): wt=11.000, iters=3494 136.87/137.30 136.87/137.30 Low Water (keep): wt=10.000, iters=3336 136.87/137.30 136.87/137.30 ============================== PROOF ================================= 136.87/137.31 % SZS status Theorem 136.87/137.31 % SZS output start Refutation 136.87/137.31 136.87/137.31 % Proof 1 at 129.48 (+ 6.63) seconds. 136.87/137.31 % Length of proof is 164. 136.87/137.31 % Level of proof is 29. 136.87/137.31 % Maximum clause weight is 20.000. 136.87/137.31 % Given clauses 21353. 136.87/137.31 136.87/137.31 1 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 136.87/137.31 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 26 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 49 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 136.87/137.31 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]. 136.87/137.31 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]. 136.87/137.31 60 is_a_theorem(implies(A,implies(B,A))) | -implies_1 # label(implies_1) # label(axiom). [clausify(1)]. 136.87/137.31 64 is_a_theorem(implies(and(A,B),A)) | -and_1 # label(and_1) # label(axiom). [clausify(3)]. 136.87/137.31 65 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(4)]. 136.87/137.31 70 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(6)]. 136.87/137.31 72 -implies_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) # label(implies_3) # label(axiom). [clausify(7)]. 136.87/137.31 75 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) | -implies_2 # label(implies_2) # label(axiom). [clausify(8)]. 136.87/137.31 86 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(14)]. 136.87/137.31 96 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(18)]. 136.87/137.31 105 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) | -equivalence_3 # label(equivalence_3) # label(axiom). [clausify(22)]. 136.87/137.31 112 -or_1 | is_a_theorem(implies(A,or(A,B))) # label(or_1) # label(axiom). [clausify(26)]. 136.87/137.31 114 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(27)]. 136.87/137.31 115 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(114),flip(b)]. 136.87/137.31 116 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(28)]. 136.87/137.31 120 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(31)]. 136.87/137.31 121 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(120),flip(b)]. 136.87/137.31 122 and_3 # label(hilbert_and_3) # label(axiom). [assumption]. 136.87/137.31 123 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption]. 136.87/137.31 124 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption]. 136.87/137.31 125 op_or # label(hilbert_op_or) # label(axiom). [assumption]. 136.87/137.31 127 or_1 # label(hilbert_or_1) # label(axiom). [assumption]. 136.87/137.31 128 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption]. 136.87/137.31 131 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 136.87/137.31 133 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption]. 136.87/137.31 135 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption]. 136.87/137.31 136 and_1 # label(hilbert_and_1) # label(axiom). [assumption]. 136.87/137.31 137 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 136.87/137.31 138 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption]. 136.87/137.31 139 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption]. 136.87/137.31 164 axiom_m1 | -is_a_theorem(strict_implies(and(c74,c75),and(c75,c74))) # label(axiom_m1) # label(axiom). [clausify(43)]. 136.87/137.31 175 -necessitation | -is_a_theorem(A) | is_a_theorem(necessarily(A)) # label(necessitation) # label(axiom). [clausify(49)]. 136.87/137.31 178 is_a_theorem(c88) | adjunction # label(adjunction) # label(axiom). [clausify(50)]. 136.87/137.31 179 is_a_theorem(c87) | adjunction # label(adjunction) # label(axiom). [clausify(50)]. 136.87/137.31 180 -is_a_theorem(and(c87,c88)) | adjunction # label(adjunction) # label(axiom). [clausify(50)]. 136.87/137.31 181 -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(B,A)) | -adjunction # label(adjunction) # label(axiom). [clausify(50)]. 136.87/137.31 196 -op_strict_implies | strict_implies(A,B) = necessarily(implies(A,B)) # label(op_strict_implies) # label(axiom). [clausify(57)]. 136.87/137.31 197 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B). [copy(196),flip(b)]. 136.87/137.31 200 necessitation # label(km4b_necessitation) # label(axiom). [assumption]. 136.87/137.31 206 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption]. 136.87/137.31 209 -axiom_m1 # label(s1_0_axiom_m1) # label(negated_conjecture). [assumption]. 136.87/137.31 210 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(70),unit_del(a,122)]. 136.87/137.31 211 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(72),unit_del(a,123)]. 136.87/137.31 212 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(105),unit_del(b,124)]. 136.87/137.31 213 not(and(not(A),not(B))) = or(A,B). [back_unit_del(121),unit_del(a,125)]. 136.87/137.31 215 is_a_theorem(implies(A,or(A,B))). [back_unit_del(112),unit_del(a,127)]. 136.87/137.31 216 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(60),unit_del(b,128)]. 136.87/137.31 219 not(and(A,not(B))) = implies(A,B). [back_unit_del(116),unit_del(a,131)]. 136.87/137.31 221 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(86),unit_del(a,133)]. 136.87/137.31 223 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(115),unit_del(a,135)]. 136.87/137.31 224 is_a_theorem(implies(and(A,B),A)). [back_unit_del(64),unit_del(b,136)]. 136.87/137.31 225 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(65),unit_del(a,137)]. 136.87/137.31 226 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(75),unit_del(b,138)]. 136.87/137.31 227 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(96),unit_del(a,139)]. 136.87/137.31 229 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(175),unit_del(a,200)]. 136.87/137.31 235 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(197),unit_del(a,206)]. 136.87/137.31 237 -is_a_theorem(strict_implies(and(c74,c75),and(c75,c74))). [back_unit_del(164),unit_del(a,209)]. 136.87/137.31 240 implies(not(A),B) = or(A,B). [back_rewrite(213),rewrite([219(4)])]. 136.87/137.31 245 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(227),rewrite([240(3)])]. 136.87/137.31 279 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(221,b,216,a)]. 136.87/137.31 282 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(221,b,212,a)]. 136.87/137.31 283 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(221,b,211,a)]. 136.87/137.31 284 -is_a_theorem(A) | is_a_theorem(implies(B,and(A,B))). [resolve(221,b,210,a)]. 136.87/137.31 291 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(226,a,221,b)]. 136.87/137.31 358 is_a_theorem(or(A,or(not(A),B))). [para(240(a,1),215(a,1))]. 136.87/137.31 362 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(219(a,1),240(a,1,1)),flip(a)]. 136.87/137.31 380 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(245,a,221,b)]. 136.87/137.31 385 is_a_theorem(implies(or(A,not(not(B))),or(B,A))). [para(240(a,1),245(a,1,2))]. 136.87/137.31 2560 is_a_theorem(implies(implies(implies(A,B),implies(A,implies(A,B))),equiv(implies(A,implies(A,B)),implies(A,B)))). [resolve(282,a,226,a)]. 136.87/137.31 2571 is_a_theorem(implies(implies(implies(A,and(B,A)),B),equiv(B,implies(A,and(B,A))))). [resolve(282,a,210,a)]. 136.87/137.31 2653 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(283,a,216,a)]. 136.87/137.31 2927 is_a_theorem(implies(A,and(c87,A))) | adjunction. [resolve(284,a,179,a)]. 136.87/137.31 3553 is_a_theorem(implies(A,A)). [resolve(291,a,216,a)]. 136.87/137.31 3556 is_a_theorem(implies(A,and(A,A))). [resolve(291,a,210,a)]. 136.87/137.31 3567 is_a_theorem(implies(A,implies(B,B))). [resolve(3553,a,279,a)]. 136.87/137.31 3593 is_a_theorem(strict_implies(A,A)). [resolve(3553,a,229,a),rewrite([235(2)])]. 136.87/137.31 3605 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(3593,a,279,a)]. 136.87/137.31 3633 -is_a_theorem(A) | is_a_theorem(and(strict_implies(B,B),A)) | -adjunction. [resolve(3593,a,181,b)]. 136.87/137.31 3785 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(3556,a,282,a)]. 136.87/137.31 4014 is_a_theorem(implies(implies(implies(A,A),B),equiv(B,implies(A,A)))). [resolve(3567,a,282,a)]. 136.87/137.31 4205 is_a_theorem(implies(implies(strict_implies(A,A),B),implies(C,B))). [resolve(3605,a,283,a)]. 136.87/137.31 4206 is_a_theorem(implies(implies(strict_implies(A,A),B),equiv(B,strict_implies(A,A)))). [resolve(3605,a,282,a)]. 136.87/137.31 9101 adjunction | -is_a_theorem(A) | is_a_theorem(and(c87,A)). [resolve(2927,a,221,b)]. 136.87/137.31 11598 adjunction | is_a_theorem(and(c87,c88)). [resolve(9101,b,178,a),merge(c)]. 136.87/137.31 11609 adjunction. [resolve(11598,b,180,a),merge(b)]. 136.87/137.31 14604 -is_a_theorem(A) | is_a_theorem(and(strict_implies(B,B),A)). [back_unit_del(3633),unit_del(c,11609)]. 136.87/137.31 18301 is_a_theorem(equiv(implies(A,implies(A,B)),implies(A,B))). [resolve(2560,a,221,b),unit_del(a,216)]. 136.87/137.31 18512 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(2653,a,221,b)]. 136.87/137.31 19913 is_a_theorem(equiv(A,and(A,A))). [resolve(3785,a,221,b),unit_del(a,224)]. 136.87/137.31 19930 and(A,A) = A. [resolve(19913,a,225,a)]. 136.87/137.31 19933 or(A,A) = not(not(A)). [para(19930(a,1),219(a,1,1)),rewrite([240(4)]),flip(a)]. 136.87/137.31 19934 equiv(A,A) = implies(A,A). [para(19930(a,1),223(a,1)),flip(a)]. 136.87/137.31 19938 implies(or(A,A),B) = or(not(A),B). [para(19930(a,1),362(a,1,1)),rewrite([240(4)]),flip(a)]. 136.87/137.31 20047 is_a_theorem(implies(A,not(not(A)))). [para(19933(a,1),215(a,1,2))]. 136.87/137.31 20051 is_a_theorem(or(A,not(not(not(A))))). [para(19933(a,1),358(a,1,2))]. 136.87/137.31 20076 implies(implies(A,B),and(A,not(B))) = not(implies(A,B)). [para(19933(a,1),362(a,1)),rewrite([219(3)]),flip(a)]. 136.87/137.31 20290 -is_a_theorem(A) | is_a_theorem(not(not(A))). [resolve(20047,a,221,b)]. 136.87/137.31 20751 is_a_theorem(not(not(implies(or(A,not(not(B))),or(B,A))))). [resolve(20290,a,385,a)]. 136.87/137.31 20775 is_a_theorem(or(not(A),A)). [resolve(20051,a,380,a),rewrite([240(3)])]. 136.87/137.31 21073 is_a_theorem(implies(implies(strict_implies(A,A),B),B)). [resolve(4205,a,291,a)]. 136.87/137.31 21902 -is_a_theorem(implies(strict_implies(A,A),B)) | is_a_theorem(B). [resolve(21073,a,221,b)]. 136.87/137.31 32287 implies(A,implies(A,B)) = implies(A,B). [resolve(18301,a,225,a),flip(a)]. 136.87/137.31 34289 is_a_theorem(implies(A,equiv(A,strict_implies(B,B)))). [resolve(18512,a,4206,a)]. 136.87/137.31 34290 is_a_theorem(implies(A,equiv(A,implies(B,B)))). [resolve(18512,a,4014,a)]. 136.87/137.31 34313 is_a_theorem(equiv(strict_implies(A,A),strict_implies(B,B))). [resolve(34289,a,21902,a)]. 136.87/137.31 34318 -is_a_theorem(A) | is_a_theorem(equiv(A,strict_implies(B,B))). [resolve(34289,a,221,b)]. 136.87/137.31 34330 is_a_theorem(equiv(strict_implies(A,A),implies(B,B))). [resolve(34290,a,21902,a)]. 136.87/137.31 34407 is_a_theorem(necessarily(equiv(strict_implies(A,A),strict_implies(B,B)))). [resolve(34313,a,229,a)]. 136.87/137.31 34408 strict_implies(A,A) = strict_implies(B,B). [resolve(34313,a,225,a)]. 136.87/137.31 34409 strict_implies(A,A) = c_0. [new_symbol(34408)]. 136.87/137.31 34410 is_a_theorem(c_0). [back_rewrite(34407),rewrite([34409(1),34409(2),19934(3),235(4),34409(3)])]. 136.87/137.31 34411 is_a_theorem(equiv(c_0,implies(A,A))). [back_rewrite(34330),rewrite([34409(1)])]. 136.87/137.31 34414 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(34318),rewrite([34409(2)])]. 136.87/137.31 35030 is_a_theorem(implies(implies(c_0,A),A)). [back_rewrite(21073),rewrite([34409(1)])]. 136.87/137.31 35532 -is_a_theorem(A) | is_a_theorem(and(c_0,A)). [back_rewrite(14604),rewrite([34409(2)])]. 136.87/137.31 37407 implies(A,A) = c_0. [resolve(34411,a,225,a)]. 136.87/137.31 37487 implies(A,c_0) = c_0. [para(37407(a,1),32287(a,1,2)),rewrite([37407(3)])]. 136.87/137.31 37490 and(implies(c_0,A),c_0) = equiv(c_0,A). [para(37487(a,1),223(a,1,2))]. 136.87/137.31 38970 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(34414,a,20775,a)]. 136.87/137.31 39023 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(34414,a,20047,a)]. 136.87/137.31 39250 is_a_theorem(equiv(implies(and(A,B),A),c_0)). [resolve(34414,a,224,a)]. 136.87/137.31 39255 is_a_theorem(equiv(implies(A,implies(B,A)),c_0)). [resolve(34414,a,216,a)]. 136.87/137.31 39510 is_a_theorem(and(c_0,implies(implies(c_0,A),A))). [resolve(35532,a,35030,a)]. 136.87/137.31 41186 or(not(A),A) = c_0. [resolve(38970,a,225,a),flip(a)]. 136.87/137.31 41964 implies(A,not(not(A))) = c_0. [resolve(39023,a,225,a),flip(a)]. 136.87/137.31 41966 equiv(A,not(not(A))) = c_0. [para(41964(a,1),223(a,1,1)),rewrite([240(4),41186(3),19930(3)]),flip(a)]. 136.87/137.31 41972 not(not(A)) = A. [para(41966(a,1),225(a,1)),unit_del(a,34410)]. 136.87/137.31 41991 is_a_theorem(implies(or(A,B),or(B,A))). [back_rewrite(20751),rewrite([41972(2),41972(5)])]. 136.87/137.31 41997 or(A,A) = A. [back_rewrite(19933),rewrite([41972(3)])]. 136.87/137.31 42031 or(not(A),B) = implies(A,B). [back_rewrite(19938),rewrite([41997(1)]),flip(a)]. 136.87/137.31 42403 not(implies(A,B)) = and(A,not(B)). [para(41997(a,1),362(a,1)),rewrite([20076(6)]),flip(a)]. 136.87/137.31 46107 implies(and(A,B),A) = c_0. [resolve(39250,a,225,a),flip(a)]. 136.87/137.31 46311 implies(A,implies(B,A)) = c_0. [resolve(39255,a,225,a),flip(a)]. 136.87/137.31 46315 and(c_0,implies(implies(A,B),B)) = equiv(B,implies(A,B)). [para(46311(a,1),223(a,1,1))]. 136.87/137.31 46358 is_a_theorem(equiv(A,implies(c_0,A))). [back_rewrite(39510),rewrite([46315(5)])]. 136.87/137.31 46437 implies(c_0,A) = A. [resolve(46358,a,225,a)]. 136.87/137.31 46496 equiv(c_0,A) = and(A,c_0). [back_rewrite(37490),rewrite([46437(2)]),flip(a)]. 136.87/137.31 47318 is_a_theorem(equiv(A,and(A,c_0))). [para(46437(a,1),2571(a,1,1,1)),rewrite([46107(3),46437(5),46437(5)])]. 136.87/137.31 47732 and(A,c_0) = A. [resolve(47318,a,225,a)]. 136.87/137.31 47734 equiv(c_0,A) = A. [back_rewrite(46496),rewrite([47732(4)])]. 136.87/137.31 47736 -is_a_theorem(A) | c_0 = A. [para(47734(a,1),225(a,1)),flip(b)]. 136.87/137.31 52475 implies(or(A,B),or(B,A)) = c_0. [resolve(41991,a,47736,a),flip(a)]. 136.87/137.31 52480 is_a_theorem(equiv(or(A,B),or(B,A))). [resolve(41991,a,282,a),rewrite([52475(3),46437(5)])]. 136.87/137.31 55418 or(A,B) = or(B,A). [resolve(52480,a,225,a)]. 136.87/137.31 55626 or(A,not(B)) = implies(B,A). [back_rewrite(42031),rewrite([55418(2)])]. 136.87/137.31 58389 and(not(A),not(B)) = not(or(A,B)). [para(240(a,1),42403(a,1,1)),flip(a)]. 136.87/137.31 66757 and(not(A),B) = and(B,not(A)). [para(41972(a,1),58389(a,1,2)),rewrite([55626(4),42403(4)])]. 136.87/137.31 66789 and(A,B) = and(B,A). [para(41972(a,1),66757(a,1,1)),rewrite([41972(3)])]. 136.87/137.31 66875 $F. [back_rewrite(237),rewrite([66789(6),34409(7)]),unit_del(a,34410)]. 136.87/137.31 136.87/137.31 % SZS output end Refutation 136.87/137.31 ============================== end of proof ========================== 136.87/137.31 136.87/137.31 ============================== STATISTICS ============================ 136.87/137.31 136.87/137.31 Given=21353. Generated=9094808. Kept=66809. proofs=1. 136.87/137.31 Usable=4704. Sos=7248. Demods=1078. Limbo=86, Disabled=54918. Hints=0. 136.87/137.31 Megabytes=50.88. 136.87/137.31 User_CPU=129.48, System_CPU=6.63, Wall_clock=136. 136.87/137.31 136.87/137.31 ============================== end of statistics ===================== 136.87/137.31 136.87/137.31 ============================== end of search ========================= 136.87/137.31 136.87/137.31 THEOREM PROVED 136.87/137.31 % SZS status Theorem 136.87/137.31 136.87/137.31 Exiting with 1 proof. 136.87/137.31 136.87/137.31 Process 26711 exit (max_proofs) Mon Jul 3 09:00:54 2023 136.87/137.31 Prover9 interrupted 136.87/137.31 EOF