0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.34 % Computer : n027.cluster.edu 0.12/0.34 % Model : x86_64 x86_64 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.34 % Memory : 8042.1875MB 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.34 % CPULimit : 1200 0.12/0.34 % DateTime : Tue Jul 13 16:16:10 EDT 2021 0.12/0.34 % CPUTime : 0.43/1.04 ============================== Prover9 =============================== 0.43/1.04 Prover9 (32) version 2009-11A, November 2009. 0.43/1.04 Process 6475 was started by sandbox on n027.cluster.edu, 0.43/1.04 Tue Jul 13 16:16:10 2021 0.43/1.04 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1200 -f /tmp/Prover9_6322_n027.cluster.edu". 0.43/1.04 ============================== end of head =========================== 0.43/1.04 0.43/1.04 ============================== INPUT ================================= 0.43/1.04 0.43/1.04 % Reading from file /tmp/Prover9_6322_n027.cluster.edu 0.43/1.04 0.43/1.04 set(prolog_style_variables). 0.43/1.04 set(auto2). 0.43/1.04 % set(auto2) -> set(auto). 0.43/1.04 % set(auto) -> set(auto_inference). 0.43/1.04 % set(auto) -> set(auto_setup). 0.43/1.04 % set(auto_setup) -> set(predicate_elim). 0.43/1.04 % set(auto_setup) -> assign(eq_defs, unfold). 0.43/1.04 % set(auto) -> set(auto_limits). 0.43/1.04 % set(auto_limits) -> assign(max_weight, "100.000"). 0.43/1.04 % set(auto_limits) -> assign(sos_limit, 20000). 0.43/1.04 % set(auto) -> set(auto_denials). 0.43/1.04 % set(auto) -> set(auto_process). 0.43/1.04 % set(auto2) -> assign(new_constants, 1). 0.43/1.04 % set(auto2) -> assign(fold_denial_max, 3). 0.43/1.04 % set(auto2) -> assign(max_weight, "200.000"). 0.43/1.04 % set(auto2) -> assign(max_hours, 1). 0.43/1.04 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.43/1.04 % set(auto2) -> assign(max_seconds, 0). 0.43/1.04 % set(auto2) -> assign(max_minutes, 5). 0.43/1.04 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.43/1.04 % set(auto2) -> set(sort_initial_sos). 0.43/1.04 % set(auto2) -> assign(sos_limit, -1). 0.43/1.04 % set(auto2) -> assign(lrs_ticks, 3000). 0.43/1.04 % set(auto2) -> assign(max_megs, 400). 0.43/1.04 % set(auto2) -> assign(stats, some). 0.43/1.04 % set(auto2) -> clear(echo_input). 0.43/1.04 % set(auto2) -> set(quiet). 0.43/1.04 % set(auto2) -> clear(print_initial_clauses). 0.43/1.04 % set(auto2) -> clear(print_given). 0.43/1.04 assign(lrs_ticks,-1). 0.43/1.04 assign(sos_limit,10000). 0.43/1.04 assign(order,kbo). 0.43/1.04 set(lex_order_vars). 0.43/1.04 clear(print_given). 0.43/1.04 0.43/1.04 % formulas(sos). % not echoed (83 formulas) 0.43/1.04 0.43/1.04 ============================== end of input ========================== 0.43/1.04 0.43/1.04 % From the command line: assign(max_seconds, 1200). 0.43/1.04 0.43/1.04 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.43/1.04 0.43/1.04 % Formulas that are not ordinary clauses: 0.43/1.04 1 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]. 0.43/1.04 2 (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]. 0.43/1.04 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 4 (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]. 0.43/1.04 5 (all X all Y (is_a_theorem(implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) <-> modus_ponens # label(modus_ponens) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 6 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 7 (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]. 0.43/1.04 8 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]. 0.43/1.04 9 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]. 0.43/1.04 10 (all P is_a_theorem(implies(or(P,P),P))) <-> r1 # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 11 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 12 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 13 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]. 0.43/1.04 14 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]. 0.43/1.04 15 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 17 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 18 (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]. 0.43/1.04 19 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]. 0.43/1.04 20 (all P all Q is_a_theorem(implies(and(P,Q),P))) <-> kn2 # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 21 (all X all Y is_a_theorem(implies(implies(not(Y),not(X)),implies(X,Y)))) <-> modus_tollens # label(modus_tollens) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 22 (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]. 0.43/1.04 23 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 24 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 25 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 26 (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) <-> substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 27 op_implies_or -> (all X all Y implies(X,Y) = or(not(X),Y)) # label(op_implies_or) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 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]. 0.43/1.04 29 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 30 op_equiv -> (all X all Y and(implies(X,Y),implies(Y,X)) = equiv(X,Y)) # label(op_equiv) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 31 op_and -> (all X all Y not(or(not(X),not(Y))) = and(X,Y)) # label(op_and) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 32 (all P all Q is_a_theorem(strict_implies(possibly(and(P,Q)),and(possibly(P),possibly(Q))))) <-> axiom_s2 # label(axiom_s2) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 33 (all X all Y all Z is_a_theorem(strict_implies(and(and(X,Y),Z),and(X,and(Y,Z))))) <-> axiom_m3 # label(axiom_m3) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 34 (all X all Y (is_a_theorem(X) & is_a_theorem(strict_implies(X,Y)) -> is_a_theorem(Y))) <-> modus_ponens_strict_implies # label(modus_ponens_strict_implies) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 35 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 36 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 37 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]. 0.43/1.04 38 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 39 (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) <-> axiom_m10 # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 40 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.43/1.04 41 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]. 0.76/1.07 42 adjunction <-> (all X all Y (is_a_theorem(Y) & is_a_theorem(X) -> is_a_theorem(and(X,Y)))) # label(adjunction) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 43 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 44 axiom_s3 <-> (all X all Y is_a_theorem(strict_implies(strict_implies(X,Y),strict_implies(not(possibly(Y)),not(possibly(X)))))) # label(axiom_s3) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 45 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 46 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]. 0.76/1.07 47 (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]. 0.76/1.07 48 (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]. 0.76/1.07 49 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 50 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 51 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 52 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]. 0.76/1.07 53 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 54 axiom_K <-> (all X all Y is_a_theorem(implies(necessarily(implies(X,Y)),implies(necessarily(X),necessarily(Y))))) # label(axiom_K) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 55 op_strict_equiv -> (all X all Y and(strict_implies(X,Y),strict_implies(Y,X)) = strict_equiv(X,Y)) # label(op_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 56 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]. 0.76/1.07 57 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 58 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.76/1.07 0.76/1.07 ============================== end of process non-clausal formulas === 0.76/1.07 0.76/1.07 ============================== PROCESS INITIAL CLAUSES =============== 0.76/1.07 0.76/1.07 ============================== PREDICATE ELIMINATION ================= 0.76/1.07 0.76/1.07 ============================== end predicate elimination ============= 0.76/1.07 0.76/1.07 Auto_denials: (non-Horn, no changes). 0.76/1.07 0.76/1.07 Term ordering decisions: 0.76/1.07 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. 0.76/1.07 0.76/1.07 ============================== end of process initial clauses ======== 0.76/1.07 0.76/1.07 ============================== CLAUSES FOR SEARCH ==================== 57.93/58.22 57.93/58.22 ============================== end of clauses for search ============= 57.93/58.22 57.93/58.22 ============================== SEARCH ================================ 57.93/58.22 57.93/58.22 % Starting search at 0.04 seconds. 57.93/58.22 57.93/58.22 Low Water (keep): wt=22.000, iters=3338 57.93/58.22 57.93/58.22 Low Water (keep): wt=21.000, iters=3411 57.93/58.22 57.93/58.22 Low Water (keep): wt=20.000, iters=3395 57.93/58.22 57.93/58.22 Low Water (keep): wt=19.000, iters=3388 57.93/58.22 57.93/58.22 Low Water (keep): wt=18.000, iters=3394 57.93/58.22 57.93/58.22 Low Water (keep): wt=17.000, iters=3333 57.93/58.22 57.93/58.22 Low Water (keep): wt=16.000, iters=3350 57.93/58.22 57.93/58.22 Low Water (keep): wt=15.000, iters=3346 57.93/58.22 57.93/58.22 Low Water (keep): wt=14.000, iters=3344 57.93/58.22 57.93/58.22 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 63 (0.00 of 1.82 sec). 57.93/58.22 57.93/58.22 Low Water (keep): wt=13.000, iters=3984 57.93/58.22 57.93/58.22 Low Water (keep): wt=12.000, iters=3356 57.93/58.22 57.93/58.22 Low Water (displace): id=17288, wt=10.000 57.93/58.22 57.93/58.22 Low Water (displace): id=17326, wt=9.000 57.93/58.22 57.93/58.22 Low Water (keep): wt=11.000, iters=3334 57.93/58.22 57.93/58.22 Low Water (keep): wt=10.000, iters=3333 57.93/58.22 57.93/58.22 ============================== PROOF ================================= 57.93/58.22 % SZS status Theorem 57.93/58.22 % SZS output start Refutation 57.93/58.22 57.93/58.22 % Proof 1 at 53.72 (+ 3.47) seconds. 57.93/58.22 % Length of proof is 176. 57.93/58.22 % Level of proof is 24. 57.93/58.22 % Maximum clause weight is 16.000. 57.93/58.22 % Given clauses 14985. 57.93/58.22 57.93/58.22 1 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]. 57.93/58.22 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 4 (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]. 57.93/58.22 5 (all X all Y (is_a_theorem(implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) <-> modus_ponens # label(modus_ponens) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 8 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]. 57.93/58.22 13 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]. 57.93/58.22 16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 21 (all X all Y is_a_theorem(implies(implies(not(Y),not(X)),implies(X,Y)))) <-> modus_tollens # label(modus_tollens) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 22 (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]. 57.93/58.22 23 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 26 (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) <-> substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 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]. 57.93/58.22 29 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 30 op_equiv -> (all X all Y and(implies(X,Y),implies(Y,X)) = equiv(X,Y)) # label(op_equiv) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 35 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 38 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 42 adjunction <-> (all X all Y (is_a_theorem(Y) & is_a_theorem(X) -> is_a_theorem(and(X,Y)))) # label(adjunction) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 49 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 53 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 58 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 57.93/58.22 59 -implies_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) # label(implies_3) # label(axiom). [clausify(1)]. 57.93/58.22 64 is_a_theorem(implies(and(A,B),A)) | -and_1 # label(and_1) # label(axiom). [clausify(3)]. 57.93/58.22 66 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) | -implies_2 # label(implies_2) # label(axiom). [clausify(4)]. 57.93/58.22 70 -is_a_theorem(implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B) | -modus_ponens # label(modus_ponens) # label(axiom). [clausify(5)]. 57.93/58.22 75 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(8)]. 57.93/58.22 85 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(13)]. 57.93/58.22 91 -or_2 | is_a_theorem(implies(A,or(B,A))) # label(or_2) # label(axiom). [clausify(16)]. 57.93/58.22 102 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) | -modus_tollens # label(modus_tollens) # label(axiom). [clausify(21)]. 57.93/58.22 104 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))) | -or_3 # label(or_3) # label(axiom). [clausify(22)]. 57.93/58.22 105 -implies_1 | is_a_theorem(implies(A,implies(B,A))) # label(implies_1) # label(axiom). [clausify(23)]. 57.93/58.22 113 -is_a_theorem(equiv(A,B)) | B = A | -substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [clausify(26)]. 57.93/58.22 115 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(28)]. 57.93/58.22 116 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(29)]. 57.93/58.22 117 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(116),flip(b)]. 57.93/58.22 118 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(30)]. 57.93/58.22 119 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(118),flip(b)]. 57.93/58.22 122 and_1 # label(hilbert_and_1) # label(axiom). [assumption]. 57.93/58.22 123 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption]. 57.93/58.22 124 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption]. 57.93/58.22 125 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 57.93/58.22 126 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption]. 57.93/58.22 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption]. 57.93/58.22 128 or_3 # label(hilbert_or_3) # label(axiom). [assumption]. 57.93/58.22 129 op_or # label(hilbert_op_or) # label(axiom). [assumption]. 57.93/58.22 130 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 57.93/58.22 132 or_2 # label(hilbert_or_2) # label(axiom). [assumption]. 57.93/58.22 133 and_3 # label(hilbert_and_3) # label(axiom). [assumption]. 57.93/58.22 134 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption]. 57.93/58.22 135 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption]. 57.93/58.22 137 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption]. 57.93/58.22 148 is_a_theorem(implies(A,necessarily(possibly(A)))) | -axiom_B # label(axiom_B) # label(axiom). [clausify(35)]. 57.93/58.22 153 -is_a_theorem(implies(possibly(c67),necessarily(possibly(c67)))) | axiom_5 # label(axiom_5) # label(axiom). [clausify(38)]. 57.93/58.22 161 -adjunction | -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(B,A)) # label(adjunction) # label(axiom). [clausify(42)]. 57.93/58.22 162 adjunction | is_a_theorem(c73) # label(adjunction) # label(axiom). [clausify(42)]. 57.93/58.22 163 adjunction | is_a_theorem(c72) # label(adjunction) # label(axiom). [clausify(42)]. 57.93/58.22 164 adjunction | -is_a_theorem(and(c72,c73)) # label(adjunction) # label(axiom). [clausify(42)]. 57.93/58.22 178 -axiom_4 | is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))) # label(axiom_4) # label(axiom). [clausify(49)]. 57.93/58.22 187 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(53)]. 57.93/58.22 196 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(58)]. 57.93/58.22 197 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(196),flip(b)]. 57.93/58.22 198 axiom_4 # label(km4b_axiom_4) # label(axiom). [assumption]. 57.93/58.22 199 op_possibly # label(km4b_op_possibly) # label(axiom). [assumption]. 57.93/58.22 201 axiom_M # label(km4b_axiom_M) # label(axiom). [assumption]. 57.93/58.22 203 axiom_B # label(km4b_axiom_B) # label(axiom). [assumption]. 57.93/58.22 204 -axiom_5 # label(km5_axiom_5) # label(negated_conjecture). [assumption]. 57.93/58.22 206 is_a_theorem(implies(and(A,B),A)). [back_unit_del(64),unit_del(b,122)]. 57.93/58.22 207 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(75),unit_del(a,123)]. 57.93/58.22 208 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(105),unit_del(a,124)]. 57.93/58.22 209 not(and(A,not(B))) = implies(A,B). [back_unit_del(115),unit_del(a,125)]. 57.93/58.22 210 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(59),unit_del(a,126)]. 57.93/58.22 211 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(66),unit_del(b,127)]. 57.93/58.22 212 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(104),unit_del(b,128)]. 57.93/58.22 213 implies(not(A),B) = or(A,B). [back_unit_del(117),rewrite([209(5)]),unit_del(a,129)]. 57.93/58.22 214 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(113),unit_del(c,130)]. 57.93/58.22 216 is_a_theorem(implies(A,or(B,A))). [back_unit_del(91),unit_del(a,132)]. 57.93/58.22 217 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(85),unit_del(a,133)]. 57.93/58.22 218 -is_a_theorem(implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B). [back_unit_del(70),unit_del(d,134)]. 57.93/58.22 219 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_unit_del(102),rewrite([213(3)]),unit_del(b,135)]. 57.93/58.22 221 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(119),unit_del(a,137)]. 57.93/58.22 224 is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))). [back_unit_del(178),unit_del(a,198)]. 57.93/58.22 225 not(necessarily(not(A))) = possibly(A). [back_unit_del(197),unit_del(a,199)]. 57.93/58.22 227 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(187),unit_del(a,201)]. 57.93/58.22 229 is_a_theorem(implies(A,necessarily(possibly(A)))). [back_unit_del(148),unit_del(b,203)]. 57.93/58.22 230 -is_a_theorem(implies(possibly(c67),necessarily(possibly(c67)))). [back_unit_del(153),unit_del(b,204)]. 57.93/58.22 252 -adjunction | -is_a_theorem(A) | is_a_theorem(and(A,implies(B,implies(C,B)))). [resolve(208,a,161,b)]. 57.93/58.22 263 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(209(a,1),213(a,1,1)),flip(a)]. 57.93/58.22 274 is_a_theorem(or(A,or(B,not(A)))). [para(213(a,1),216(a,1))]. 57.93/58.22 279 -is_a_theorem(A) | is_a_theorem(implies(B,and(A,B))). [resolve(218,a,217,a)]. 57.93/58.22 283 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(218,a,211,a)]. 57.93/58.22 284 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(218,a,210,a)]. 57.93/58.22 285 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(218,a,208,a)]. 57.93/58.22 286 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(218,a,207,a)]. 57.93/58.22 290 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(219,a,218,a)]. 57.93/58.22 326 or(necessarily(not(A)),B) = implies(possibly(A),B). [para(225(a,1),213(a,1,1)),flip(a)]. 57.93/58.22 328 not(necessarily(possibly(A))) = possibly(necessarily(not(A))). [para(225(a,1),225(a,1,1,1))]. 57.93/58.22 3913 is_a_theorem(implies(A,and(c72,A))) | adjunction. [resolve(279,a,163,b)]. 57.93/58.22 4190 is_a_theorem(implies(A,and(A,A))). [resolve(283,a,217,a)]. 57.93/58.22 4191 is_a_theorem(implies(implies(A,B),implies(or(A,A),B))). [resolve(283,a,212,a)]. 57.93/58.22 4193 is_a_theorem(implies(A,A)). [resolve(283,a,208,a)]. 57.93/58.22 4198 is_a_theorem(implies(A,implies(B,B))). [resolve(4193,a,285,a)]. 57.93/58.22 4302 is_a_theorem(implies(implies(necessarily(possibly(A)),B),implies(A,B))). [resolve(284,a,229,a)]. 57.93/58.22 4315 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(284,a,208,a)]. 57.93/58.22 4462 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(286,a,4190,a)]. 57.93/58.22 4499 is_a_theorem(implies(implies(necessarily(necessarily(A)),necessarily(A)),equiv(necessarily(A),necessarily(necessarily(A))))). [resolve(286,a,224,a)]. 57.93/58.22 4503 is_a_theorem(implies(implies(implies(A,and(B,A)),B),equiv(B,implies(A,and(B,A))))). [resolve(286,a,217,a)]. 57.93/58.22 4583 is_a_theorem(implies(implies(implies(A,A),B),equiv(B,implies(A,A)))). [resolve(4198,a,286,a)]. 57.93/58.22 4585 is_a_theorem(implies(implies(implies(A,A),B),implies(C,B))). [resolve(4198,a,284,a)]. 57.93/58.22 4600 -adjunction | is_a_theorem(and(implies(A,implies(B,B)),implies(C,implies(D,C)))). [resolve(4198,a,252,b)]. 57.93/58.22 7973 adjunction | -is_a_theorem(A) | is_a_theorem(and(c72,A)). [resolve(3913,a,218,a)]. 57.93/58.22 9807 adjunction | is_a_theorem(and(c72,c73)). [resolve(7973,b,162,b),merge(c)]. 57.93/58.22 9818 adjunction. [resolve(9807,b,164,b),merge(b)]. 57.93/58.22 12715 is_a_theorem(and(implies(A,implies(B,B)),implies(C,implies(D,C)))). [back_unit_del(4600),unit_del(a,9818)]. 57.93/58.22 23008 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(or(A,A),B)). [resolve(4191,a,218,a)]. 57.93/58.22 23120 -is_a_theorem(implies(necessarily(possibly(A)),B)) | is_a_theorem(implies(A,B)). [resolve(4302,a,218,a)]. 57.93/58.22 23123 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(4315,a,218,a)]. 57.93/58.22 23146 is_a_theorem(equiv(A,and(A,A))). [resolve(4462,a,218,a),unit_del(a,206)]. 57.93/58.22 23165 and(A,A) = A. [resolve(23146,a,214,a)]. 57.93/58.22 23169 or(A,A) = not(not(A)). [para(23165(a,1),209(a,1,1)),rewrite([213(4)]),flip(a)]. 57.93/58.22 23170 equiv(A,A) = implies(A,A). [para(23165(a,1),221(a,1)),flip(a)]. 57.93/58.22 23171 implies(or(A,A),B) = or(not(A),B). [para(23165(a,1),263(a,1,1)),rewrite([213(4)]),flip(a)]. 57.93/58.22 23221 -is_a_theorem(implies(A,B)) | is_a_theorem(or(not(A),B)). [back_rewrite(23008),rewrite([23171(4)])]. 57.93/58.22 23229 is_a_theorem(implies(A,not(not(A)))). [para(23169(a,1),216(a,1,2))]. 57.93/58.22 23234 is_a_theorem(or(A,not(not(not(A))))). [para(23169(a,1),274(a,1,2))]. 57.93/58.22 24265 is_a_theorem(or(not(A),A)). [resolve(23234,a,290,a),rewrite([213(3)])]. 57.93/58.22 24290 is_a_theorem(or(possibly(necessarily(not(A))),necessarily(possibly(A)))). [para(328(a,1),24265(a,1,1))]. 57.93/58.22 24849 is_a_theorem(equiv(necessarily(A),necessarily(necessarily(A)))). [resolve(4499,a,218,a),unit_del(a,227)]. 57.93/58.22 24856 necessarily(necessarily(A)) = necessarily(A). [resolve(24849,a,214,a)]. 57.93/58.22 25203 is_a_theorem(implies(implies(implies(A,A),B),B)). [resolve(4585,a,283,a)]. 57.93/58.22 26043 -is_a_theorem(implies(implies(A,A),B)) | is_a_theorem(B). [resolve(25203,a,218,a)]. 57.93/58.22 27434 is_a_theorem(or(not(necessarily(A)),A)). [resolve(23221,a,227,a)]. 57.93/58.22 27451 is_a_theorem(implies(A,possibly(A))). [resolve(27434,a,290,a),rewrite([225(3)])]. 57.93/58.22 32897 is_a_theorem(equiv(implies(A,implies(B,B)),implies(B,B))). [para(221(a,1),12715(a,1))]. 57.93/58.22 33970 is_a_theorem(implies(A,not(possibly(necessarily(not(A)))))). [resolve(23120,a,23229,a),rewrite([328(3)])]. 57.93/58.22 34315 is_a_theorem(or(A,not(possibly(necessarily(or(A,A)))))). [para(23169(a,2),33970(a,1,2,1,1,1)),rewrite([213(6)])]. 57.93/58.22 34696 is_a_theorem(implies(possibly(necessarily(or(A,A))),A)). [resolve(34315,a,290,a)]. 57.93/58.22 35216 is_a_theorem(implies(A,equiv(A,implies(B,B)))). [resolve(23123,a,4583,a)]. 57.93/58.22 35276 is_a_theorem(equiv(implies(A,A),implies(B,B))). [resolve(35216,a,26043,a)]. 57.93/58.22 35282 -is_a_theorem(A) | is_a_theorem(equiv(A,implies(B,B))). [resolve(35216,a,218,a)]. 57.93/58.22 35426 implies(A,A) = implies(B,B). [resolve(35276,a,214,a)]. 57.93/58.22 35428 implies(A,A) = c_0. [new_symbol(35426)]. 57.93/58.22 35437 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(35282),rewrite([35428(2)])]. 57.93/58.22 35443 is_a_theorem(c_0). [back_rewrite(35276),rewrite([35428(1),35428(2),23170(3),35428(3)])]. 57.93/58.22 35566 is_a_theorem(equiv(implies(A,c_0),c_0)). [back_rewrite(32897),rewrite([35428(1),35428(3)])]. 57.93/58.22 35923 is_a_theorem(implies(implies(c_0,A),A)). [back_rewrite(25203),rewrite([35428(1)])]. 57.93/58.22 38668 is_a_theorem(equiv(implies(A,possibly(A)),c_0)). [resolve(35437,a,27451,a)]. 57.93/58.22 38721 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(35437,a,24265,a)]. 57.93/58.22 38741 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(35437,a,23229,a)]. 57.93/58.22 38879 is_a_theorem(equiv(implies(A,implies(B,A)),c_0)). [resolve(35437,a,208,a)]. 57.93/58.22 38880 is_a_theorem(equiv(implies(and(A,B),A),c_0)). [resolve(35437,a,206,a)]. 57.93/58.22 39018 implies(A,c_0) = c_0. [resolve(35566,a,214,a),flip(a)]. 57.93/58.22 39020 and(implies(c_0,A),c_0) = equiv(c_0,A). [para(39018(a,1),221(a,1,2))]. 57.93/58.22 39162 is_a_theorem(equiv(implies(implies(c_0,A),A),c_0)). [resolve(35923,a,35437,a)]. 57.93/58.22 41304 implies(A,possibly(A)) = c_0. [resolve(38668,a,214,a),flip(a)]. 57.93/58.22 41321 or(not(A),A) = c_0. [resolve(38721,a,214,a),flip(a)]. 57.93/58.22 42262 implies(A,not(not(A))) = c_0. [resolve(38741,a,214,a),flip(a)]. 57.93/58.22 42264 equiv(A,not(not(A))) = c_0. [para(42262(a,1),221(a,1,1)),rewrite([213(4),41321(3),23165(3)]),flip(a)]. 57.93/58.22 42269 not(not(A)) = A. [para(42264(a,1),214(a,1)),unit_del(a,35443)]. 57.93/58.22 42303 or(A,A) = A. [back_rewrite(23169),rewrite([42269(3)])]. 57.93/58.22 42309 is_a_theorem(implies(possibly(necessarily(A)),A)). [back_rewrite(34696),rewrite([42303(1)])]. 57.93/58.22 42600 is_a_theorem(implies(possibly(necessarily(A)),necessarily(A))). [para(24856(a,1),42309(a,1,1,1))]. 57.93/58.22 42773 is_a_theorem(implies(c_0,equiv(possibly(necessarily(A)),necessarily(A)))). [resolve(42600,a,286,a),rewrite([41304(4)])]. 57.93/58.22 42881 implies(A,implies(B,A)) = c_0. [resolve(38879,a,214,a),flip(a)]. 57.93/58.22 42903 implies(and(A,B),A) = c_0. [resolve(38880,a,214,a),flip(a)]. 57.93/58.22 43172 implies(implies(c_0,A),A) = c_0. [resolve(39162,a,214,a),flip(a)]. 57.93/58.22 43173 equiv(implies(c_0,A),A) = c_0. [para(43172(a,1),221(a,1,1)),rewrite([42881(4),23165(3)]),flip(a)]. 57.93/58.22 43179 implies(c_0,A) = A. [para(43173(a,1),214(a,1)),flip(b),unit_del(a,35443)]. 57.93/58.22 43194 is_a_theorem(equiv(possibly(necessarily(A)),necessarily(A))). [back_rewrite(42773),rewrite([43179(6)])]. 57.93/58.22 43196 equiv(c_0,A) = and(A,c_0). [back_rewrite(39020),rewrite([43179(2)]),flip(a)]. 57.93/58.22 43432 is_a_theorem(equiv(A,and(A,c_0))). [para(43179(a,1),4503(a,1,1,1)),rewrite([42903(3),43179(5),43179(5)])]. 57.93/58.22 44431 and(A,c_0) = A. [resolve(43432,a,214,a)]. 57.93/58.22 44437 equiv(c_0,A) = A. [back_rewrite(43196),rewrite([44431(4)])]. 57.93/58.22 44438 -is_a_theorem(A) | c_0 = A. [para(44437(a,1),214(a,1)),flip(b)]. 57.93/58.22 46492 or(possibly(necessarily(not(A))),necessarily(possibly(A))) = c_0. [resolve(44438,a,24290,a),flip(a)]. 57.93/58.22 49521 possibly(necessarily(A)) = necessarily(A). [resolve(43194,a,214,a),flip(a)]. 57.93/58.22 49524 implies(possibly(A),necessarily(possibly(A))) = c_0. [back_rewrite(46492),rewrite([49521(3),326(5)])]. 57.93/58.22 49533 $F. [back_rewrite(230),rewrite([49524(6)]),unit_del(a,35443)]. 57.93/58.22 57.93/58.22 % SZS output end Refutation 57.93/58.22 ============================== end of proof ========================== 57.93/58.22 57.93/58.22 ============================== STATISTICS ============================ 57.93/58.22 57.93/58.22 Given=14985. Generated=5984254. Kept=49469. proofs=1. 57.93/58.22 Usable=4487. Sos=5563. Demods=531. Limbo=9, Disabled=39551. Hints=0. 57.93/58.22 Megabytes=41.48. 57.93/58.22 User_CPU=53.72, System_CPU=3.47, Wall_clock=58. 57.93/58.22 57.93/58.22 ============================== end of statistics ===================== 57.93/58.22 57.93/58.22 ============================== end of search ========================= 57.93/58.22 57.93/58.22 THEOREM PROVED 57.93/58.22 % SZS status Theorem 57.93/58.22 57.93/58.22 Exiting with 1 proof. 57.93/58.22 57.93/58.22 Process 6475 exit (max_proofs) Tue Jul 13 16:17:08 2021 57.93/58.22 Prover9 interrupted 57.93/58.22 EOF