0.03/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.14 % Command : tptp2X_and_run_prover9 %d %s 0.15/0.35 % Computer : n013.cluster.edu 0.15/0.35 % Model : x86_64 x86_64 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.15/0.35 % Memory : 8042.1875MB 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64 0.15/0.35 % CPULimit : 1440 0.15/0.35 % WCLimit : 180 0.15/0.35 % DateTime : Mon Jul 3 08:09:47 EDT 2023 0.15/0.36 % CPUTime : 0.78/1.06 ============================== Prover9 =============================== 0.78/1.06 Prover9 (32) version 2009-11A, November 2009. 0.78/1.06 Process 6634 was started by sandbox on n013.cluster.edu, 0.78/1.06 Mon Jul 3 08:09:48 2023 0.78/1.06 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1440 -f /tmp/Prover9_6479_n013.cluster.edu". 0.78/1.06 ============================== end of head =========================== 0.78/1.06 0.78/1.06 ============================== INPUT ================================= 0.78/1.06 0.78/1.06 % Reading from file /tmp/Prover9_6479_n013.cluster.edu 0.78/1.06 0.78/1.06 set(prolog_style_variables). 0.78/1.06 set(auto2). 0.78/1.06 % set(auto2) -> set(auto). 0.78/1.06 % set(auto) -> set(auto_inference). 0.78/1.06 % set(auto) -> set(auto_setup). 0.78/1.06 % set(auto_setup) -> set(predicate_elim). 0.78/1.06 % set(auto_setup) -> assign(eq_defs, unfold). 0.78/1.06 % set(auto) -> set(auto_limits). 0.78/1.06 % set(auto_limits) -> assign(max_weight, "100.000"). 0.78/1.06 % set(auto_limits) -> assign(sos_limit, 20000). 0.78/1.06 % set(auto) -> set(auto_denials). 0.78/1.06 % set(auto) -> set(auto_process). 0.78/1.06 % set(auto2) -> assign(new_constants, 1). 0.78/1.06 % set(auto2) -> assign(fold_denial_max, 3). 0.78/1.06 % set(auto2) -> assign(max_weight, "200.000"). 0.78/1.06 % set(auto2) -> assign(max_hours, 1). 0.78/1.06 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.78/1.06 % set(auto2) -> assign(max_seconds, 0). 0.78/1.06 % set(auto2) -> assign(max_minutes, 5). 0.78/1.06 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.78/1.06 % set(auto2) -> set(sort_initial_sos). 0.78/1.06 % set(auto2) -> assign(sos_limit, -1). 0.78/1.06 % set(auto2) -> assign(lrs_ticks, 3000). 0.78/1.06 % set(auto2) -> assign(max_megs, 400). 0.78/1.06 % set(auto2) -> assign(stats, some). 0.78/1.06 % set(auto2) -> clear(echo_input). 0.78/1.06 % set(auto2) -> set(quiet). 0.78/1.06 % set(auto2) -> clear(print_initial_clauses). 0.78/1.06 % set(auto2) -> clear(print_given). 0.78/1.06 assign(lrs_ticks,-1). 0.78/1.06 assign(sos_limit,10000). 0.78/1.06 assign(order,kbo). 0.78/1.06 set(lex_order_vars). 0.78/1.06 clear(print_given). 0.78/1.06 0.78/1.06 % formulas(sos). % not echoed (83 formulas) 0.78/1.06 0.78/1.06 ============================== end of input ========================== 0.78/1.06 0.78/1.06 % From the command line: assign(max_seconds, 1440). 0.78/1.06 0.78/1.06 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.78/1.06 0.78/1.06 % Formulas that are not ordinary clauses: 0.78/1.06 1 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 2 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 9 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 12 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 15 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 17 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 19 (all P is_a_theorem(implies(P,and(P,P)))) <-> kn1 # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 20 (all P is_a_theorem(implies(implies(not(P),P),P))) <-> cn3 # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 21 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 25 (all X all Y is_a_theorem(implies(and(X,Y),Y))) <-> and_2 # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 26 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 32 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 33 (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_4 # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 34 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 36 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 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]. 0.78/1.06 39 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.78/1.06 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]. 0.78/1.06 41 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.82/1.09 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]. 0.82/1.09 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]. 0.82/1.09 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]. 0.82/1.09 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]. 0.82/1.09 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]. 0.82/1.09 47 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.82/1.09 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]. 0.82/1.09 49 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.82/1.09 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]. 0.82/1.09 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]. 0.82/1.09 52 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.82/1.09 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]. 0.82/1.09 54 (all X is_a_theorem(implies(necessarily(X),X))) <-> axiom_M # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.82/1.09 55 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.82/1.09 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]. 0.82/1.09 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]. 0.82/1.09 58 op_possibly -> (all X not(necessarily(not(X))) = possibly(X)) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.82/1.09 0.82/1.09 ============================== end of process non-clausal formulas === 0.82/1.09 0.82/1.09 ============================== PROCESS INITIAL CLAUSES =============== 0.82/1.09 0.82/1.09 ============================== PREDICATE ELIMINATION ================= 0.82/1.09 0.82/1.09 ============================== end predicate elimination ============= 0.82/1.09 0.82/1.09 Auto_denials: (non-Horn, no changes). 0.82/1.09 0.82/1.09 Term ordering decisions: 0.82/1.09 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. 0.82/1.09 0.82/1.09 ============================== end of process initial clauses ======== 0.82/1.09 0.82/1.09 ============================== CLAUSES FOR SEARCH ==================== 66.49/66.79 66.49/66.79 ============================== end of clauses for search ============= 66.49/66.79 66.49/66.79 ============================== SEARCH ================================ 66.49/66.79 66.49/66.79 % Starting search at 0.05 seconds. 66.49/66.79 66.49/66.79 Low Water (keep): wt=22.000, iters=3359 66.49/66.79 66.49/66.79 Low Water (keep): wt=21.000, iters=3378 66.49/66.79 66.49/66.79 Low Water (keep): wt=20.000, iters=3374 66.49/66.79 66.49/66.79 Low Water (keep): wt=19.000, iters=3405 66.49/66.79 66.49/66.79 Low Water (keep): wt=18.000, iters=3334 66.49/66.79 66.49/66.79 Low Water (keep): wt=17.000, iters=3424 66.49/66.79 66.49/66.79 Low Water (keep): wt=16.000, iters=3347 66.49/66.79 66.49/66.79 Low Water (keep): wt=15.000, iters=3343 66.49/66.79 66.49/66.79 Low Water (keep): wt=14.000, iters=3336 66.49/66.79 66.49/66.79 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 64 (0.00 of 1.63 sec). 66.49/66.79 66.49/66.79 Low Water (keep): wt=13.000, iters=3348 66.49/66.79 66.49/66.79 Low Water (keep): wt=12.000, iters=3351 66.49/66.79 66.49/66.79 Low Water (keep): wt=11.000, iters=3333 66.49/66.79 66.49/66.79 Low Water (displace): id=17189, wt=10.000 66.49/66.79 66.49/66.79 Low Water (displace): id=17244, wt=9.000 66.49/66.79 66.49/66.79 Low Water (displace): id=18256, wt=8.000 66.49/66.79 66.49/66.79 Low Water (keep): wt=10.000, iters=3333 66.49/66.79 66.49/66.79 ============================== PROOF ================================= 66.49/66.79 % SZS status Theorem 66.49/66.79 % SZS output start Refutation 66.49/66.79 66.49/66.79 % Proof 1 at 62.03 (+ 3.72) seconds. 66.49/66.79 % Length of proof is 172. 66.49/66.79 % Level of proof is 25. 66.49/66.79 % Maximum clause weight is 20.000. 66.49/66.79 % Given clauses 15295. 66.49/66.79 66.49/66.79 1 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 26 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 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]. 66.49/66.79 32 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 33 (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_4 # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 47 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 54 (all X is_a_theorem(implies(necessarily(X),X))) <-> axiom_M # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 58 op_possibly -> (all X not(necessarily(not(X))) = possibly(X)) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 66.49/66.79 60 is_a_theorem(implies(A,implies(B,A))) | -implies_1 # label(implies_1) # label(axiom). [clausify(1)]. 66.49/66.79 64 is_a_theorem(implies(and(A,B),A)) | -and_1 # label(and_1) # label(axiom). [clausify(3)]. 66.49/66.79 65 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(4)]. 66.49/66.79 69 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))) | -or_3 # label(or_3) # label(axiom). [clausify(5)]. 66.49/66.79 70 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(6)]. 66.49/66.79 72 -implies_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) # label(implies_3) # label(axiom). [clausify(7)]. 66.49/66.79 75 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) | -implies_2 # label(implies_2) # label(axiom). [clausify(8)]. 66.49/66.79 78 -equivalence_2 | is_a_theorem(implies(equiv(A,B),implies(B,A))) # label(equivalence_2) # label(axiom). [clausify(10)]. 66.49/66.79 86 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(14)]. 66.49/66.79 96 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(18)]. 66.49/66.79 105 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) | -equivalence_3 # label(equivalence_3) # label(axiom). [clausify(22)]. 66.49/66.79 112 -or_1 | is_a_theorem(implies(A,or(A,B))) # label(or_1) # label(axiom). [clausify(26)]. 66.49/66.79 114 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(27)]. 66.49/66.79 115 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(114),flip(b)]. 66.49/66.79 116 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(28)]. 66.49/66.79 120 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(31)]. 66.49/66.79 121 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(120),flip(b)]. 66.49/66.79 122 and_3 # label(hilbert_and_3) # label(axiom). [assumption]. 66.49/66.79 123 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption]. 66.49/66.79 124 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption]. 66.49/66.79 125 op_or # label(hilbert_op_or) # label(axiom). [assumption]. 66.49/66.79 126 or_3 # label(hilbert_or_3) # label(axiom). [assumption]. 66.49/66.79 127 or_1 # label(hilbert_or_1) # label(axiom). [assumption]. 66.49/66.79 128 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption]. 66.49/66.79 131 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 66.49/66.79 133 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption]. 66.49/66.79 134 equivalence_2 # label(hilbert_equivalence_2) # label(axiom). [assumption]. 66.49/66.79 135 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption]. 66.49/66.79 136 and_1 # label(hilbert_and_1) # label(axiom). [assumption]. 66.49/66.79 137 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 66.49/66.79 138 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption]. 66.49/66.79 139 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption]. 66.49/66.79 141 axiom_5 | -is_a_theorem(implies(possibly(c56),necessarily(possibly(c56)))) # label(axiom_5) # label(axiom). [clausify(32)]. 66.49/66.79 143 is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))) | -axiom_4 # label(axiom_4) # label(axiom). [clausify(33)]. 66.49/66.79 172 is_a_theorem(implies(A,necessarily(possibly(A)))) | -axiom_B # label(axiom_B) # label(axiom). [clausify(47)]. 66.49/66.79 191 is_a_theorem(implies(necessarily(A),A)) | -axiom_M # label(axiom_M) # label(axiom). [clausify(54)]. 66.49/66.79 198 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(58)]. 66.49/66.79 199 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(198),flip(b)]. 66.49/66.79 201 axiom_4 # label(km4b_axiom_4) # label(axiom). [assumption]. 66.49/66.79 202 axiom_M # label(km4b_axiom_M) # label(axiom). [assumption]. 66.49/66.79 203 axiom_B # label(km4b_axiom_B) # label(axiom). [assumption]. 66.49/66.79 205 op_possibly # label(km4b_op_possibly) # label(axiom). [assumption]. 66.49/66.79 206 -axiom_5 # label(km5_axiom_5) # label(negated_conjecture). [assumption]. 66.49/66.79 207 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(70),unit_del(a,122)]. 66.49/66.79 208 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(72),unit_del(a,123)]. 66.49/66.79 209 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(105),unit_del(b,124)]. 66.49/66.79 210 not(and(not(A),not(B))) = or(A,B). [back_unit_del(121),unit_del(a,125)]. 66.49/66.79 211 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(69),unit_del(b,126)]. 66.49/66.79 212 is_a_theorem(implies(A,or(A,B))). [back_unit_del(112),unit_del(a,127)]. 66.49/66.79 213 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(60),unit_del(b,128)]. 66.49/66.79 216 not(and(A,not(B))) = implies(A,B). [back_unit_del(116),unit_del(a,131)]. 66.49/66.79 218 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(86),unit_del(a,133)]. 66.49/66.79 219 is_a_theorem(implies(equiv(A,B),implies(B,A))). [back_unit_del(78),unit_del(a,134)]. 66.49/66.79 220 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(115),unit_del(a,135)]. 66.49/66.79 221 is_a_theorem(implies(and(A,B),A)). [back_unit_del(64),unit_del(b,136)]. 66.49/66.79 222 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(65),unit_del(a,137)]. 66.49/66.79 223 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(75),unit_del(b,138)]. 66.49/66.79 224 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(96),unit_del(a,139)]. 66.49/66.79 227 is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))). [back_unit_del(143),unit_del(b,201)]. 66.49/66.79 228 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(191),unit_del(b,202)]. 66.49/66.79 229 is_a_theorem(implies(A,necessarily(possibly(A)))). [back_unit_del(172),unit_del(b,203)]. 66.49/66.79 231 not(necessarily(not(A))) = possibly(A). [back_unit_del(199),unit_del(a,205)]. 66.49/66.79 232 -is_a_theorem(implies(possibly(c56),necessarily(possibly(c56)))). [back_unit_del(141),unit_del(a,206)]. 66.49/66.79 235 implies(not(A),B) = or(A,B). [back_rewrite(210),rewrite([216(4)])]. 66.49/66.79 237 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(224),rewrite([235(3)])]. 66.49/66.79 271 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(218,b,213,a)]. 66.49/66.79 274 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(218,b,209,a)]. 66.49/66.79 275 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(218,b,208,a)]. 66.49/66.79 283 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(223,a,218,b)]. 66.49/66.79 339 not(necessarily(possibly(A))) = possibly(necessarily(not(A))). [para(231(a,1),231(a,1,1,1))]. 66.49/66.79 348 is_a_theorem(or(A,or(not(A),B))). [para(235(a,1),212(a,1))]. 66.49/66.79 352 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(216(a,1),235(a,1,1)),flip(a)]. 66.49/66.79 360 or(necessarily(not(A)),B) = implies(possibly(A),B). [para(231(a,1),235(a,1,1)),flip(a)]. 66.49/66.79 363 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(237,a,218,b)]. 66.49/66.79 2612 is_a_theorem(implies(implies(necessarily(necessarily(A)),necessarily(A)),equiv(necessarily(A),necessarily(necessarily(A))))). [resolve(274,a,227,a)]. 66.49/66.79 2613 is_a_theorem(implies(implies(implies(A,B),implies(A,implies(A,B))),equiv(implies(A,implies(A,B)),implies(A,B)))). [resolve(274,a,223,a)]. 66.49/66.79 2615 is_a_theorem(implies(implies(implies(A,B),equiv(B,A)),equiv(equiv(B,A),implies(A,B)))). [resolve(274,a,219,a)]. 66.49/66.79 2666 is_a_theorem(implies(implies(necessarily(possibly(A)),B),implies(A,B))). [resolve(275,a,229,a)]. 66.49/66.79 2675 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(275,a,213,a)]. 66.49/66.79 3567 is_a_theorem(implies(A,A)). [resolve(283,a,213,a)]. 66.49/66.79 3568 is_a_theorem(implies(implies(A,B),implies(or(A,A),B))). [resolve(283,a,211,a)]. 66.49/66.79 3570 is_a_theorem(implies(A,and(A,A))). [resolve(283,a,207,a)]. 66.49/66.79 3581 is_a_theorem(implies(A,implies(B,B))). [resolve(3567,a,271,a)]. 66.49/66.79 3768 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(3570,a,274,a)]. 66.49/66.79 3994 is_a_theorem(implies(implies(implies(A,A),B),implies(C,B))). [resolve(3581,a,275,a)]. 66.49/66.79 3995 is_a_theorem(implies(implies(implies(A,A),B),equiv(B,implies(A,A)))). [resolve(3581,a,274,a)]. 66.49/66.79 19630 is_a_theorem(equiv(necessarily(A),necessarily(necessarily(A)))). [resolve(2612,a,218,b),unit_del(a,228)]. 66.49/66.79 19638 necessarily(necessarily(A)) = necessarily(A). [resolve(19630,a,222,a)]. 66.49/66.79 19702 is_a_theorem(equiv(implies(A,implies(A,B)),implies(A,B))). [resolve(2613,a,218,b),unit_del(a,213)]. 66.49/66.79 20490 -is_a_theorem(implies(necessarily(possibly(A)),B)) | is_a_theorem(implies(A,B)). [resolve(2666,a,218,b)]. 66.49/66.79 20538 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(2675,a,218,b)]. 66.49/66.79 20991 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(or(A,A),B)). [resolve(3568,a,218,b)]. 66.49/66.79 21031 is_a_theorem(equiv(A,and(A,A))). [resolve(3768,a,218,b),unit_del(a,221)]. 66.49/66.79 21050 and(A,A) = A. [resolve(21031,a,222,a)]. 66.49/66.79 21057 or(A,A) = not(not(A)). [para(21050(a,1),216(a,1,1)),rewrite([235(4)]),flip(a)]. 66.49/66.79 21058 equiv(A,A) = implies(A,A). [para(21050(a,1),220(a,1)),flip(a)]. 66.49/66.79 21062 implies(or(A,A),B) = or(not(A),B). [para(21050(a,1),352(a,1,1)),rewrite([235(4)]),flip(a)]. 66.49/66.79 21079 -is_a_theorem(implies(A,B)) | is_a_theorem(or(not(A),B)). [back_rewrite(20991),rewrite([21062(4)])]. 66.49/66.79 21093 is_a_theorem(implies(A,not(not(A)))). [para(21057(a,1),212(a,1,2))]. 66.49/66.79 21098 is_a_theorem(or(A,not(not(not(A))))). [para(21057(a,1),348(a,1,2))]. 66.49/66.79 22274 is_a_theorem(implies(implies(implies(A,A),B),B)). [resolve(3994,a,283,a)]. 66.49/66.79 22280 is_a_theorem(or(not(A),A)). [resolve(21098,a,363,a),rewrite([235(3)])]. 66.49/66.79 22305 is_a_theorem(or(possibly(necessarily(not(A))),necessarily(possibly(A)))). [para(339(a,1),22280(a,1,1))]. 66.49/66.79 23370 -is_a_theorem(implies(implies(A,A),B)) | is_a_theorem(B). [resolve(22274,a,218,b)]. 66.49/66.79 25034 is_a_theorem(or(not(necessarily(A)),A)). [resolve(21079,a,228,a)]. 66.49/66.79 25049 is_a_theorem(implies(A,possibly(A))). [resolve(25034,a,363,a),rewrite([231(3)])]. 66.49/66.79 30664 implies(A,implies(A,B)) = implies(A,B). [resolve(19702,a,222,a),flip(a)]. 66.49/66.79 31186 is_a_theorem(implies(A,not(possibly(necessarily(not(A)))))). [resolve(20490,a,21093,a),rewrite([339(3)])]. 66.49/66.79 31541 is_a_theorem(or(A,not(possibly(necessarily(or(A,A)))))). [para(21057(a,2),31186(a,1,2,1,1,1)),rewrite([235(6)])]. 66.49/66.79 32012 is_a_theorem(implies(possibly(necessarily(or(A,A))),A)). [resolve(31541,a,363,a)]. 66.49/66.79 33159 is_a_theorem(implies(A,equiv(A,implies(B,B)))). [resolve(20538,a,3995,a)]. 66.49/66.79 33220 is_a_theorem(equiv(implies(A,A),implies(B,B))). [resolve(33159,a,23370,a)]. 66.49/66.79 33226 -is_a_theorem(A) | is_a_theorem(equiv(A,implies(B,B))). [resolve(33159,a,218,b)]. 66.49/66.79 33373 implies(A,A) = implies(B,B). [resolve(33220,a,222,a)]. 66.49/66.79 33375 implies(A,A) = c_0. [new_symbol(33373)]. 66.49/66.79 33384 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(33226),rewrite([33375(2)])]. 66.49/66.79 33390 is_a_theorem(c_0). [back_rewrite(33220),rewrite([33375(1),33375(2),21058(3),33375(3)])]. 66.49/66.79 33404 is_a_theorem(implies(A,equiv(A,c_0))). [back_rewrite(33159),rewrite([33375(1)])]. 66.49/66.79 34029 is_a_theorem(implies(implies(c_0,A),A)). [back_rewrite(22274),rewrite([33375(1)])]. 66.49/66.79 35735 implies(A,c_0) = c_0. [para(33375(a,1),30664(a,1,2)),rewrite([33375(3)])]. 66.49/66.79 36329 and(c_0,implies(c_0,A)) = equiv(A,c_0). [para(35735(a,1),220(a,1,1))]. 66.49/66.79 36507 is_a_theorem(equiv(implies(A,possibly(A)),c_0)). [resolve(33384,a,25049,a)]. 66.49/66.79 36559 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(33384,a,22280,a)]. 66.49/66.79 36579 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(33384,a,21093,a)]. 66.49/66.79 36719 is_a_theorem(equiv(implies(A,implies(B,A)),c_0)). [resolve(33384,a,213,a)]. 66.49/66.79 36777 is_a_theorem(equiv(implies(A,equiv(A,c_0)),c_0)). [resolve(33404,a,33384,a)]. 66.49/66.79 36977 is_a_theorem(equiv(implies(implies(c_0,A),A),c_0)). [resolve(34029,a,33384,a)]. 66.49/66.79 39112 implies(A,possibly(A)) = c_0. [resolve(36507,a,222,a),flip(a)]. 66.49/66.79 39127 or(not(A),A) = c_0. [resolve(36559,a,222,a),flip(a)]. 66.49/66.79 40104 implies(A,not(not(A))) = c_0. [resolve(36579,a,222,a),flip(a)]. 66.49/66.79 40106 equiv(A,not(not(A))) = c_0. [para(40104(a,1),220(a,1,1)),rewrite([235(4),39127(3),21050(3)]),flip(a)]. 66.49/66.79 40112 not(not(A)) = A. [para(40106(a,1),222(a,1)),unit_del(a,33390)]. 66.49/66.79 40132 or(A,A) = A. [back_rewrite(21057),rewrite([40112(3)])]. 66.49/66.79 40145 is_a_theorem(implies(possibly(necessarily(A)),A)). [back_rewrite(32012),rewrite([40132(1)])]. 66.49/66.79 40398 is_a_theorem(implies(possibly(necessarily(A)),necessarily(A))). [para(19638(a,1),40145(a,1,1,1))]. 66.49/66.79 40507 is_a_theorem(implies(c_0,equiv(possibly(necessarily(A)),necessarily(A)))). [resolve(40398,a,274,a),rewrite([39112(4)])]. 66.49/66.79 40607 implies(A,implies(B,A)) = c_0. [resolve(36719,a,222,a),flip(a)]. 66.49/66.79 40631 implies(A,equiv(A,c_0)) = c_0. [resolve(36777,a,222,a),flip(a)]. 66.49/66.79 40725 implies(implies(c_0,A),A) = c_0. [resolve(36977,a,222,a),flip(a)]. 66.49/66.79 40728 equiv(implies(c_0,A),A) = c_0. [para(40725(a,1),220(a,1,1)),rewrite([40607(4),21050(3)]),flip(a)]. 66.49/66.79 40734 implies(c_0,A) = A. [para(40728(a,1),222(a,1)),flip(b),unit_del(a,33390)]. 66.49/66.79 40739 is_a_theorem(equiv(possibly(necessarily(A)),necessarily(A))). [back_rewrite(40507),rewrite([40734(6)])]. 66.49/66.79 40741 equiv(A,c_0) = and(c_0,A). [back_rewrite(36329),rewrite([40734(3)]),flip(a)]. 66.49/66.79 40787 implies(A,and(c_0,A)) = c_0. [back_rewrite(40631),rewrite([40741(2)])]. 66.49/66.79 40826 is_a_theorem(equiv(and(c_0,A),A)). [para(40734(a,1),2615(a,1,1,1)),rewrite([40741(2),40787(3),40741(3),40734(5),40734(5)])]. 66.49/66.79 41496 and(c_0,A) = A. [resolve(40826,a,222,a),flip(a)]. 66.49/66.79 41512 equiv(A,c_0) = A. [back_rewrite(40741),rewrite([41496(4)])]. 66.49/66.79 41829 -is_a_theorem(A) | c_0 = A. [para(41512(a,1),222(a,1))]. 66.49/66.79 44161 or(possibly(necessarily(not(A))),necessarily(possibly(A))) = c_0. [resolve(41829,a,22305,a),flip(a)]. 66.49/66.79 47042 possibly(necessarily(A)) = necessarily(A). [resolve(40739,a,222,a),flip(a)]. 66.49/66.79 47045 implies(possibly(A),necessarily(possibly(A))) = c_0. [back_rewrite(44161),rewrite([47042(3),360(5)])]. 66.49/66.79 47055 $F. [back_rewrite(232),rewrite([47045(6)]),unit_del(a,33390)]. 66.49/66.79 66.49/66.79 % SZS output end Refutation 66.49/66.79 ============================== end of proof ========================== 66.49/66.79 66.49/66.79 ============================== STATISTICS ============================ 66.49/66.79 66.49/66.79 Given=15295. Generated=6582592. Kept=46989. proofs=1. 66.49/66.79 Usable=4903. Sos=5443. Demods=601. Limbo=10, Disabled=36774. Hints=0. 66.49/66.79 Megabytes=39.82. 66.49/66.79 User_CPU=62.03, System_CPU=3.72, Wall_clock=66. 66.49/66.79 66.49/66.79 ============================== end of statistics ===================== 66.49/66.79 66.49/66.79 ============================== end of search ========================= 66.49/66.79 66.49/66.79 THEOREM PROVED 66.49/66.79 % SZS status Theorem 66.49/66.79 66.49/66.79 Exiting with 1 proof. 66.49/66.79 66.49/66.79 Process 6634 exit (max_proofs) Mon Jul 3 08:10:54 2023 66.49/66.79 Prover9 interrupted 66.49/66.79 EOF