0.10/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.10/0.11 % Command : tptp2X_and_run_prover9 %d %s 0.11/0.32 % Computer : n022.cluster.edu 0.11/0.32 % Model : x86_64 x86_64 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.11/0.32 % Memory : 8042.1875MB 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64 0.11/0.32 % CPULimit : 960 0.11/0.32 % WCLimit : 120 0.11/0.32 % DateTime : Tue Aug 9 05:56:16 EDT 2022 0.11/0.32 % CPUTime : 0.80/1.08 ============================== Prover9 =============================== 0.80/1.08 Prover9 (32) version 2009-11A, November 2009. 0.80/1.08 Process 3106 was started by sandbox2 on n022.cluster.edu, 0.80/1.08 Tue Aug 9 05:56:17 2022 0.80/1.08 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_2952_n022.cluster.edu". 0.80/1.08 ============================== end of head =========================== 0.80/1.08 0.80/1.08 ============================== INPUT ================================= 0.80/1.08 0.80/1.08 % Reading from file /tmp/Prover9_2952_n022.cluster.edu 0.80/1.08 0.80/1.08 set(prolog_style_variables). 0.80/1.08 set(auto2). 0.80/1.08 % set(auto2) -> set(auto). 0.80/1.08 % set(auto) -> set(auto_inference). 0.80/1.08 % set(auto) -> set(auto_setup). 0.80/1.08 % set(auto_setup) -> set(predicate_elim). 0.80/1.08 % set(auto_setup) -> assign(eq_defs, unfold). 0.80/1.08 % set(auto) -> set(auto_limits). 0.80/1.08 % set(auto_limits) -> assign(max_weight, "100.000"). 0.80/1.08 % set(auto_limits) -> assign(sos_limit, 20000). 0.80/1.08 % set(auto) -> set(auto_denials). 0.80/1.08 % set(auto) -> set(auto_process). 0.80/1.08 % set(auto2) -> assign(new_constants, 1). 0.80/1.08 % set(auto2) -> assign(fold_denial_max, 3). 0.80/1.08 % set(auto2) -> assign(max_weight, "200.000"). 0.80/1.08 % set(auto2) -> assign(max_hours, 1). 0.80/1.08 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.80/1.08 % set(auto2) -> assign(max_seconds, 0). 0.80/1.08 % set(auto2) -> assign(max_minutes, 5). 0.80/1.08 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.80/1.08 % set(auto2) -> set(sort_initial_sos). 0.80/1.08 % set(auto2) -> assign(sos_limit, -1). 0.80/1.08 % set(auto2) -> assign(lrs_ticks, 3000). 0.80/1.08 % set(auto2) -> assign(max_megs, 400). 0.80/1.08 % set(auto2) -> assign(stats, some). 0.80/1.08 % set(auto2) -> clear(echo_input). 0.80/1.08 % set(auto2) -> set(quiet). 0.80/1.08 % set(auto2) -> clear(print_initial_clauses). 0.80/1.08 % set(auto2) -> clear(print_given). 0.80/1.08 assign(lrs_ticks,-1). 0.80/1.08 assign(sos_limit,10000). 0.80/1.08 assign(order,kbo). 0.80/1.08 set(lex_order_vars). 0.80/1.08 clear(print_given). 0.80/1.08 0.80/1.08 % formulas(sos). % not echoed (89 formulas) 0.80/1.08 0.80/1.08 ============================== end of input ========================== 0.80/1.08 0.80/1.08 % From the command line: assign(max_seconds, 960). 0.80/1.08 0.80/1.08 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.80/1.08 0.80/1.08 % Formulas that are not ordinary clauses: 0.80/1.08 1 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 2 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]. 0.80/1.08 3 (all X all Y is_a_theorem(implies(and(X,Y),Y))) <-> and_2 # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 4 (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]. 0.80/1.08 5 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]. 0.80/1.08 6 equivalence_3 <-> (all X all Y is_a_theorem(implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y))))) # label(equivalence_3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 7 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 8 kn3 <-> (all P all Q all R is_a_theorem(implies(implies(P,Q),implies(not(and(Q,R)),not(and(R,P)))))) # label(kn3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 9 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 10 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 11 (all P all Q all R is_a_theorem(implies(implies(P,Q),implies(implies(Q,R),implies(P,R))))) <-> cn1 # label(cn1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 12 equivalence_1 <-> (all X all Y is_a_theorem(implies(equiv(X,Y),implies(X,Y)))) # label(equivalence_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 13 substitution_of_equivalents <-> (all X all Y (is_a_theorem(equiv(X,Y)) -> Y = X)) # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 14 (all P all Q all R is_a_theorem(implies(implies(Q,R),implies(or(P,Q),or(P,R))))) <-> r5 # label(r5) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 15 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 17 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 18 (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]. 0.80/1.08 19 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 20 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 21 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]. 0.80/1.08 22 (all P is_a_theorem(implies(P,and(P,P)))) <-> kn1 # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 23 (all X all Y is_a_theorem(implies(equiv(X,Y),implies(Y,X)))) <-> equivalence_2 # label(equivalence_2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 24 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]. 0.80/1.08 25 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 26 (all X all Y all Z is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) <-> implies_3 # label(implies_3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 27 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]. 0.80/1.08 28 op_implies_or -> (all X all Y or(not(X),Y) = implies(X,Y)) # label(op_implies_or) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 29 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]. 0.80/1.08 30 op_or -> (all X all Y not(and(not(X),not(Y))) = or(X,Y)) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 31 op_and -> (all X all Y and(X,Y) = not(or(not(X),not(Y)))) # label(op_and) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 32 axiom_m7 <-> (all P all Q is_a_theorem(strict_implies(possibly(and(P,Q)),P))) # label(axiom_m7) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 33 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 34 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 35 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]. 0.80/1.08 36 (all X all Y is_a_theorem(strict_implies(and(X,Y),and(Y,X)))) <-> axiom_m1 # label(axiom_m1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 37 substitution_strict_equiv <-> (all X all Y (is_a_theorem(strict_equiv(X,Y)) -> Y = X)) # label(substitution_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 38 (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]. 0.80/1.08 39 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 41 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 42 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 43 (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) <-> axiom_m2 # label(axiom_m2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 44 (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]. 0.80/1.10 45 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 46 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]. 0.80/1.10 47 axiom_m5 <-> (all X all Y all Z is_a_theorem(strict_implies(and(strict_implies(X,Y),strict_implies(Y,Z)),strict_implies(X,Z)))) # label(axiom_m5) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 48 (all X all Y (is_a_theorem(strict_implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) <-> modus_ponens_strict_implies # label(modus_ponens_strict_implies) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 49 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]. 0.80/1.10 50 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 51 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 52 axiom_s1 <-> (all X all Y all Z is_a_theorem(implies(and(necessarily(implies(X,Y)),necessarily(implies(Y,Z))),necessarily(implies(X,Z))))) # label(axiom_s1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 53 adjunction <-> (all X all Y (is_a_theorem(X) & is_a_theorem(Y) -> is_a_theorem(and(X,Y)))) # label(adjunction) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 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]. 0.80/1.10 57 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 58 op_strict_implies -> (all X all Y strict_implies(X,Y) = necessarily(implies(X,Y))) # label(op_strict_implies) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 0.80/1.10 ============================== end of process non-clausal formulas === 0.80/1.10 0.80/1.10 ============================== PROCESS INITIAL CLAUSES =============== 0.80/1.10 0.80/1.10 ============================== PREDICATE ELIMINATION ================= 0.80/1.10 0.80/1.10 ============================== end predicate elimination ============= 0.80/1.10 0.80/1.10 Auto_denials: (non-Horn, no changes). 0.80/1.10 0.80/1.10 Term ordering decisions: 0.80/1.10 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. 0.80/1.10 0.80/1.10 ============================== end of process initial clauses ======== 0.80/1.10 0.80/1.10 ============================== CLAUSES FOR SEARCH ==================== 90.39/90.90 90.39/90.90 ============================== end of clauses for search ============= 90.39/90.90 90.39/90.90 ============================== SEARCH ================================ 90.39/90.90 90.39/90.90 % Starting search at 0.04 seconds. 90.39/90.90 90.39/90.90 Low Water (keep): wt=22.000, iters=3351 90.39/90.90 90.39/90.90 Low Water (keep): wt=21.000, iters=3405 90.39/90.90 90.39/90.90 Low Water (keep): wt=20.000, iters=3393 90.39/90.90 90.39/90.90 Low Water (keep): wt=19.000, iters=3367 90.39/90.90 90.39/90.90 Low Water (keep): wt=18.000, iters=3345 90.39/90.90 90.39/90.90 Low Water (keep): wt=17.000, iters=3344 90.39/90.90 90.39/90.90 Low Water (keep): wt=16.000, iters=3343 90.39/90.90 90.39/90.90 Low Water (keep): wt=15.000, iters=3338 90.39/90.90 90.39/90.90 Low Water (keep): wt=14.000, iters=3333 90.39/90.90 90.39/90.90 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 66 (0.00 of 2.61 sec). 90.39/90.90 90.39/90.90 Low Water (keep): wt=13.000, iters=3338 90.39/90.90 90.39/90.90 Low Water (displace): id=1203, wt=29.000 90.39/90.90 90.39/90.90 Low Water (displace): id=3213, wt=28.000 90.39/90.90 90.39/90.90 Low Water (displace): id=10924, wt=11.000 90.39/90.90 90.39/90.90 Low Water (displace): id=10925, wt=9.000 90.39/90.90 90.39/90.90 Low Water (keep): wt=12.000, iters=4370 90.39/90.90 90.39/90.90 Low Water (keep): wt=11.000, iters=3481 90.39/90.90 90.39/90.90 Low Water (keep): wt=10.000, iters=3333 90.39/90.90 90.39/90.90 ============================== PROOF ================================= 90.39/90.91 % SZS status Theorem 90.39/90.91 % SZS output start Refutation 90.39/90.91 90.39/90.91 % Proof 1 at 84.62 (+ 5.01) seconds. 90.39/90.91 % Length of proof is 157. 90.39/90.91 % Level of proof is 29. 90.39/90.91 % Maximum clause weight is 20.000. 90.39/90.91 % Given clauses 20545. 90.39/90.91 90.39/90.91 1 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 2 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]. 90.39/90.91 3 (all X all Y is_a_theorem(implies(and(X,Y),Y))) <-> and_2 # label(and_2) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 5 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]. 90.39/90.91 6 equivalence_3 <-> (all X all Y is_a_theorem(implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y))))) # label(equivalence_3) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 9 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 13 substitution_of_equivalents <-> (all X all Y (is_a_theorem(equiv(X,Y)) -> Y = X)) # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 18 (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]. 90.39/90.91 24 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]. 90.39/90.91 25 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 26 (all X all Y all Z is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) <-> implies_3 # label(implies_3) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 27 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]. 90.39/90.91 29 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]. 90.39/90.91 30 op_or -> (all X all Y not(and(not(X),not(Y))) = or(X,Y)) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 36 (all X all Y is_a_theorem(strict_implies(and(X,Y),and(Y,X)))) <-> axiom_m1 # label(axiom_m1) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 45 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 58 op_strict_implies -> (all X all Y strict_implies(X,Y) = necessarily(implies(X,Y))) # label(op_strict_implies) # label(axiom) # label(non_clause). [assumption]. 90.39/90.91 59 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(1)]. 90.39/90.91 61 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(2)]. 90.39/90.91 64 is_a_theorem(implies(and(A,B),B)) | -and_2 # label(and_2) # label(axiom). [clausify(3)]. 90.39/90.91 67 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(5)]. 90.39/90.91 69 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(6)]. 90.39/90.91 75 -or_1 | is_a_theorem(implies(A,or(A,B))) # label(or_1) # label(axiom). [clausify(9)]. 90.39/90.91 83 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(13)]. 90.39/90.91 95 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) | -implies_2 # label(implies_2) # label(axiom). [clausify(18)]. 90.39/90.91 106 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(24)]. 90.39/90.91 111 is_a_theorem(implies(A,implies(B,A))) | -implies_1 # label(implies_1) # label(axiom). [clausify(25)]. 90.39/90.91 113 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) | -implies_3 # label(implies_3) # label(axiom). [clausify(26)]. 90.39/90.91 114 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(27)]. 90.39/90.91 116 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(29)]. 90.39/90.91 117 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(116),flip(b)]. 90.39/90.91 118 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(30)]. 90.39/90.91 119 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(118),flip(b)]. 90.39/90.91 121 and_2 # label(hilbert_and_2) # label(axiom). [assumption]. 90.39/90.91 122 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption]. 90.39/90.91 123 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption]. 90.39/90.91 125 or_1 # label(hilbert_or_1) # label(axiom). [assumption]. 90.39/90.91 126 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption]. 90.39/90.91 127 op_or # label(hilbert_op_or) # label(axiom). [assumption]. 90.39/90.91 128 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 90.39/90.91 129 and_3 # label(hilbert_and_3) # label(axiom). [assumption]. 90.39/90.91 130 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption]. 90.39/90.91 134 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption]. 90.39/90.91 135 and_1 # label(hilbert_and_1) # label(axiom). [assumption]. 90.39/90.91 136 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption]. 90.39/90.91 137 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption]. 90.39/90.91 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 90.39/90.91 147 -is_a_theorem(strict_implies(and(c63,c64),and(c64,c63))) | axiom_m1 # label(axiom_m1) # label(axiom). [clausify(36)]. 90.39/90.91 168 -is_a_theorem(A) | is_a_theorem(necessarily(A)) | -necessitation # label(necessitation) # label(axiom). [clausify(45)]. 90.39/90.91 197 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B) # label(op_strict_implies) # label(axiom). [clausify(58)]. 90.39/90.91 202 necessitation # label(km4b_necessitation) # label(axiom). [assumption]. 90.39/90.91 206 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption]. 90.39/90.91 207 -axiom_m1 # label(s1_0_axiom_m1) # label(negated_conjecture). [assumption]. 90.39/90.91 208 is_a_theorem(implies(and(A,B),B)). [back_unit_del(64),unit_del(b,121)]. 90.39/90.91 209 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(95),unit_del(b,122)]. 90.39/90.91 210 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(113),unit_del(b,123)]. 90.39/90.91 212 is_a_theorem(implies(A,or(A,B))). [back_unit_del(75),unit_del(a,125)]. 90.39/90.91 213 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(69),unit_del(a,126)]. 90.39/90.91 214 not(and(not(A),not(B))) = or(A,B). [back_unit_del(119),unit_del(a,127)]. 90.39/90.91 215 not(and(A,not(B))) = implies(A,B). [back_unit_del(114),unit_del(a,128)]. 90.39/90.91 216 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(67),unit_del(a,129)]. 90.39/90.91 217 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(106),unit_del(a,130)]. 90.39/90.91 221 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(61),unit_del(a,134)]. 90.39/90.91 222 is_a_theorem(implies(and(A,B),A)). [back_unit_del(59),unit_del(a,135)]. 90.39/90.91 223 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(117),unit_del(a,136)]. 90.39/90.91 224 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(111),unit_del(b,137)]. 90.39/90.91 225 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(83),unit_del(a,138)]. 90.39/90.91 231 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(168),unit_del(c,202)]. 90.39/90.91 234 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(197),unit_del(a,206)]. 90.39/90.91 235 -is_a_theorem(strict_implies(and(c63,c64),and(c64,c63))). [back_unit_del(147),unit_del(b,207)]. 90.39/90.91 237 implies(not(A),B) = or(A,B). [back_rewrite(214),rewrite([215(4)])]. 90.39/90.91 243 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(221),rewrite([237(3)])]. 90.39/90.91 271 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(217,b,213,a)]. 90.39/90.91 274 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(217,b,210,a)]. 90.39/90.91 275 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(217,b,209,a)]. 90.39/90.91 289 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(224,a,217,b)]. 90.39/90.91 353 is_a_theorem(or(A,or(not(A),B))). [para(237(a,1),212(a,1))]. 90.39/90.91 356 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(215(a,1),237(a,1,1)),flip(a)]. 90.39/90.91 378 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(243,a,217,b)]. 90.39/90.91 383 is_a_theorem(implies(or(A,not(not(B))),or(B,A))). [para(237(a,1),243(a,1,2))]. 90.39/90.91 2053 is_a_theorem(implies(implies(implies(A,B),implies(A,implies(A,B))),equiv(implies(A,implies(A,B)),implies(A,B)))). [resolve(271,a,209,a)]. 90.39/90.91 2252 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(274,a,224,a)]. 90.39/90.91 2329 is_a_theorem(implies(A,A)). [resolve(275,a,224,a)]. 90.39/90.91 2330 is_a_theorem(implies(A,and(A,A))). [resolve(275,a,216,a)]. 90.39/90.91 2336 is_a_theorem(implies(A,implies(B,B))). [resolve(2329,a,289,a)]. 90.39/90.91 2361 is_a_theorem(strict_implies(A,A)). [resolve(2329,a,231,a),rewrite([234(2)])]. 90.39/90.91 2367 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(2361,a,289,a)]. 90.39/90.91 2514 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(2330,a,271,a)]. 90.39/90.91 2666 is_a_theorem(implies(implies(implies(A,A),B),equiv(B,implies(A,A)))). [resolve(2336,a,271,a)]. 90.39/90.91 2877 is_a_theorem(implies(implies(strict_implies(A,A),B),implies(C,B))). [resolve(2367,a,274,a)]. 90.39/90.91 2879 is_a_theorem(implies(implies(strict_implies(A,A),B),equiv(B,strict_implies(A,A)))). [resolve(2367,a,271,a)]. 90.39/90.91 17254 is_a_theorem(equiv(implies(A,implies(A,B)),implies(A,B))). [resolve(2053,a,217,b),unit_del(a,224)]. 90.39/90.91 17454 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(2252,a,217,b)]. 90.39/90.91 18401 is_a_theorem(equiv(A,and(A,A))). [resolve(2514,a,217,b),unit_del(a,222)]. 90.39/90.91 18418 and(A,A) = A. [resolve(18401,a,225,a)]. 90.39/90.91 18421 or(A,A) = not(not(A)). [para(18418(a,1),215(a,1,1)),rewrite([237(4)]),flip(a)]. 90.39/90.91 18422 equiv(A,A) = implies(A,A). [para(18418(a,1),223(a,1)),flip(a)]. 90.39/90.91 18426 implies(or(A,A),B) = or(not(A),B). [para(18418(a,1),356(a,1,1)),rewrite([237(4)]),flip(a)]. 90.39/90.91 18612 is_a_theorem(implies(A,not(not(A)))). [para(18421(a,1),212(a,1,2))]. 90.39/90.91 18616 is_a_theorem(or(A,not(not(not(A))))). [para(18421(a,1),353(a,1,2))]. 90.39/90.91 18638 implies(implies(A,B),and(A,not(B))) = not(implies(A,B)). [para(18421(a,1),356(a,1)),rewrite([215(3)]),flip(a)]. 90.39/90.91 18794 -is_a_theorem(A) | is_a_theorem(not(not(A))). [resolve(18612,a,217,b)]. 90.39/90.91 19149 is_a_theorem(not(not(implies(or(A,not(not(B))),or(B,A))))). [resolve(18794,a,383,a)]. 90.39/90.91 19175 is_a_theorem(or(not(A),A)). [resolve(18616,a,378,a),rewrite([237(3)])]. 90.39/90.91 19491 is_a_theorem(implies(implies(strict_implies(A,A),B),B)). [resolve(2877,a,275,a)]. 90.39/90.91 20703 -is_a_theorem(implies(strict_implies(A,A),B)) | is_a_theorem(B). [resolve(19491,a,217,b)]. 90.39/90.91 31154 implies(A,implies(A,B)) = implies(A,B). [resolve(17254,a,225,a),flip(a)]. 90.39/90.91 31620 is_a_theorem(implies(A,equiv(A,strict_implies(B,B)))). [resolve(17454,a,2879,a)]. 90.39/90.91 31621 is_a_theorem(implies(A,equiv(A,implies(B,B)))). [resolve(17454,a,2666,a)]. 90.39/90.91 31650 is_a_theorem(equiv(strict_implies(A,A),strict_implies(B,B))). [resolve(31620,a,20703,a)]. 90.39/90.91 31655 -is_a_theorem(A) | is_a_theorem(equiv(A,strict_implies(B,B))). [resolve(31620,a,217,b)]. 90.39/90.91 31667 is_a_theorem(equiv(strict_implies(A,A),implies(B,B))). [resolve(31621,a,20703,a)]. 90.39/90.91 31762 is_a_theorem(necessarily(equiv(strict_implies(A,A),strict_implies(B,B)))). [resolve(31650,a,231,a)]. 90.39/90.91 31763 strict_implies(A,A) = strict_implies(B,B). [resolve(31650,a,225,a)]. 90.39/90.91 31764 strict_implies(A,A) = c_0. [new_symbol(31763)]. 90.39/90.91 31765 is_a_theorem(c_0). [back_rewrite(31762),rewrite([31764(1),31764(2),18422(3),234(4),31764(3)])]. 90.39/90.91 31766 is_a_theorem(equiv(c_0,implies(A,A))). [back_rewrite(31667),rewrite([31764(1)])]. 90.39/90.91 31769 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(31655),rewrite([31764(2)])]. 90.39/90.91 31784 is_a_theorem(implies(A,equiv(A,c_0))). [back_rewrite(31620),rewrite([31764(1)])]. 90.39/90.91 32450 is_a_theorem(implies(implies(c_0,A),A)). [back_rewrite(19491),rewrite([31764(1)])]. 90.39/90.91 34444 implies(A,A) = c_0. [resolve(31766,a,225,a)]. 90.39/90.91 34578 implies(A,c_0) = c_0. [para(34444(a,1),31154(a,1,2)),rewrite([34444(3)])]. 90.39/90.91 34580 and(c_0,implies(c_0,A)) = equiv(A,c_0). [para(34578(a,1),223(a,1,1))]. 90.39/90.91 35880 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(31769,a,19175,a)]. 90.39/90.91 35928 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(31769,a,18612,a)]. 90.39/90.91 36141 is_a_theorem(equiv(implies(A,implies(B,A)),c_0)). [resolve(31769,a,224,a)]. 90.39/90.91 36148 is_a_theorem(equiv(implies(and(A,B),B),c_0)). [resolve(31769,a,208,a)]. 90.39/90.91 36176 is_a_theorem(equiv(implies(A,equiv(A,c_0)),c_0)). [resolve(31784,a,31769,a)]. 90.39/90.91 36315 is_a_theorem(equiv(implies(implies(c_0,A),A),c_0)). [resolve(32450,a,31769,a)]. 90.39/90.91 37745 or(not(A),A) = c_0. [resolve(35880,a,225,a),flip(a)]. 90.39/90.91 38354 implies(A,not(not(A))) = c_0. [resolve(35928,a,225,a),flip(a)]. 90.39/90.91 38356 equiv(A,not(not(A))) = c_0. [para(38354(a,1),223(a,1,1)),rewrite([237(4),37745(3),18418(3)]),flip(a)]. 90.39/90.91 38362 not(not(A)) = A. [para(38356(a,1),225(a,1)),unit_del(a,31765)]. 90.39/90.91 38392 is_a_theorem(implies(or(A,B),or(B,A))). [back_rewrite(19149),rewrite([38362(2),38362(5)])]. 90.39/90.91 38398 or(A,A) = A. [back_rewrite(18421),rewrite([38362(3)])]. 90.39/90.91 38434 or(not(A),B) = implies(A,B). [back_rewrite(18426),rewrite([38398(1)]),flip(a)]. 90.39/90.91 38894 not(implies(A,B)) = and(A,not(B)). [para(38398(a,1),356(a,1)),rewrite([18638(6)]),flip(a)]. 90.39/90.91 42546 implies(A,implies(B,A)) = c_0. [resolve(36141,a,225,a),flip(a)]. 90.39/90.91 42966 implies(and(A,B),B) = c_0. [resolve(36148,a,225,a),flip(a)]. 90.39/90.91 43136 implies(A,equiv(A,c_0)) = c_0. [resolve(36176,a,225,a),flip(a)]. 90.39/90.91 43140 is_a_theorem(implies(implies(equiv(A,c_0),A),equiv(A,equiv(A,c_0)))). [para(43136(a,1),271(a,1)),unit_del(a,31765)]. 90.39/90.91 43392 implies(implies(c_0,A),A) = c_0. [resolve(36315,a,225,a),flip(a)]. 90.39/90.91 43398 equiv(implies(c_0,A),A) = c_0. [para(43392(a,1),223(a,1,1)),rewrite([42546(4),18418(3)]),flip(a)]. 90.39/90.91 43430 implies(c_0,A) = A. [para(43398(a,1),225(a,1)),flip(b),unit_del(a,31765)]. 90.39/90.91 43554 equiv(A,c_0) = and(c_0,A). [back_rewrite(34580),rewrite([43430(3)]),flip(a)]. 90.39/90.91 43615 is_a_theorem(equiv(A,and(c_0,A))). [back_rewrite(43140),rewrite([43554(2),42966(3),43554(3),43430(5)])]. 90.39/90.91 44706 and(c_0,A) = A. [resolve(43615,a,225,a)]. 90.39/90.91 44766 equiv(A,c_0) = A. [back_rewrite(43554),rewrite([44706(4)])]. 90.39/90.91 44860 -is_a_theorem(A) | c_0 = A. [para(44766(a,1),225(a,1))]. 90.39/90.91 50429 implies(or(A,B),or(B,A)) = c_0. [resolve(38392,a,44860,a),flip(a)]. 90.39/90.91 50431 is_a_theorem(equiv(or(A,B),or(B,A))). [resolve(38392,a,271,a),rewrite([50429(3),43430(5)])]. 90.39/90.91 52943 or(A,B) = or(B,A). [resolve(50431,a,225,a)]. 90.39/90.91 53264 or(A,not(B)) = implies(B,A). [back_rewrite(38434),rewrite([52943(2)])]. 90.39/90.91 55611 and(not(A),not(B)) = not(or(A,B)). [para(237(a,1),38894(a,1,1)),flip(a)]. 90.39/90.91 63877 and(not(A),B) = and(B,not(A)). [para(38362(a,1),55611(a,1,2)),rewrite([53264(4),38894(4)])]. 90.39/90.91 63918 and(A,B) = and(B,A). [para(38362(a,1),63877(a,1,1)),rewrite([38362(3)])]. 90.39/90.91 64109 $F. [back_rewrite(235),rewrite([63918(6),31764(7)]),unit_del(a,31765)]. 90.39/90.91 90.39/90.91 % SZS output end Refutation 90.39/90.91 ============================== end of proof ========================== 90.39/90.91 90.39/90.91 ============================== STATISTICS ============================ 90.39/90.91 90.39/90.91 Given=20545. Generated=7885235. Kept=64045. proofs=1. 90.39/90.91 Usable=5023. Sos=6794. Demods=1052. Limbo=191, Disabled=52184. Hints=0. 90.39/90.91 Megabytes=47.80. 90.39/90.91 User_CPU=84.62, System_CPU=5.01, Wall_clock=90. 90.39/90.91 90.39/90.91 ============================== end of statistics ===================== 90.39/90.91 90.39/90.91 ============================== end of search ========================= 90.39/90.91 90.39/90.91 THEOREM PROVED 90.39/90.91 % SZS status Theorem 90.39/90.91 90.39/90.91 Exiting with 1 proof. 90.39/90.91 90.39/90.91 Process 3106 exit (max_proofs) Tue Aug 9 05:57:47 2022 90.39/90.91 Prover9 interrupted 90.39/90.91 EOF