0.11/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.11/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.34 % Computer : n019.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 : 960 0.12/0.34 % WCLimit : 120 0.12/0.34 % DateTime : Tue Aug 9 05:26:21 EDT 2022 0.12/0.34 % CPUTime : 0.45/1.03 ============================== Prover9 =============================== 0.45/1.03 Prover9 (32) version 2009-11A, November 2009. 0.45/1.03 Process 6836 was started by sandbox on n019.cluster.edu, 0.45/1.03 Tue Aug 9 05:26:22 2022 0.45/1.03 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 960 -f /tmp/Prover9_6610_n019.cluster.edu". 0.45/1.03 ============================== end of head =========================== 0.45/1.03 0.45/1.03 ============================== INPUT ================================= 0.45/1.03 0.45/1.03 % Reading from file /tmp/Prover9_6610_n019.cluster.edu 0.45/1.03 0.45/1.03 set(prolog_style_variables). 0.45/1.03 set(auto2). 0.45/1.03 % set(auto2) -> set(auto). 0.45/1.03 % set(auto) -> set(auto_inference). 0.45/1.03 % set(auto) -> set(auto_setup). 0.45/1.03 % set(auto_setup) -> set(predicate_elim). 0.45/1.03 % set(auto_setup) -> assign(eq_defs, unfold). 0.45/1.03 % set(auto) -> set(auto_limits). 0.45/1.03 % set(auto_limits) -> assign(max_weight, "100.000"). 0.45/1.03 % set(auto_limits) -> assign(sos_limit, 20000). 0.45/1.03 % set(auto) -> set(auto_denials). 0.45/1.03 % set(auto) -> set(auto_process). 0.45/1.03 % set(auto2) -> assign(new_constants, 1). 0.45/1.03 % set(auto2) -> assign(fold_denial_max, 3). 0.45/1.03 % set(auto2) -> assign(max_weight, "200.000"). 0.45/1.03 % set(auto2) -> assign(max_hours, 1). 0.45/1.03 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.45/1.03 % set(auto2) -> assign(max_seconds, 0). 0.45/1.03 % set(auto2) -> assign(max_minutes, 5). 0.45/1.03 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.45/1.03 % set(auto2) -> set(sort_initial_sos). 0.45/1.03 % set(auto2) -> assign(sos_limit, -1). 0.45/1.03 % set(auto2) -> assign(lrs_ticks, 3000). 0.45/1.03 % set(auto2) -> assign(max_megs, 400). 0.45/1.03 % set(auto2) -> assign(stats, some). 0.45/1.03 % set(auto2) -> clear(echo_input). 0.45/1.03 % set(auto2) -> set(quiet). 0.45/1.03 % set(auto2) -> clear(print_initial_clauses). 0.45/1.03 % set(auto2) -> clear(print_given). 0.45/1.03 assign(lrs_ticks,-1). 0.45/1.03 assign(sos_limit,10000). 0.45/1.03 assign(order,kbo). 0.45/1.03 set(lex_order_vars). 0.45/1.03 clear(print_given). 0.45/1.03 0.45/1.03 % formulas(sos). % not echoed (89 formulas) 0.45/1.03 0.45/1.03 ============================== end of input ========================== 0.45/1.03 0.45/1.03 % From the command line: assign(max_seconds, 960). 0.45/1.03 0.45/1.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.45/1.03 0.45/1.03 % Formulas that are not ordinary clauses: 0.45/1.03 1 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 3 (all X all Y is_a_theorem(implies(and(X,Y),Y))) <-> and_2 # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 7 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 9 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 10 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 15 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 17 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 19 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 20 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 22 (all P is_a_theorem(implies(P,and(P,P)))) <-> kn1 # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 25 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 33 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 34 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 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]. 0.45/1.03 39 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 41 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.45/1.03 42 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.45/1.05 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]. 0.45/1.05 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]. 0.45/1.05 45 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.45/1.05 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]. 0.45/1.05 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]. 0.45/1.05 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]. 0.45/1.05 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]. 0.45/1.05 50 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.45/1.05 51 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.45/1.05 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]. 0.45/1.05 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]. 0.45/1.05 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.45/1.05 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.45/1.05 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]. 0.45/1.05 57 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.45/1.05 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]. 0.45/1.05 0.45/1.05 ============================== end of process non-clausal formulas === 0.45/1.05 0.45/1.05 ============================== PROCESS INITIAL CLAUSES =============== 0.45/1.05 0.45/1.05 ============================== PREDICATE ELIMINATION ================= 0.45/1.05 0.45/1.05 ============================== end predicate elimination ============= 0.45/1.05 0.45/1.05 Auto_denials: (non-Horn, no changes). 0.45/1.05 0.45/1.05 Term ordering decisions: 0.45/1.05 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. 0.45/1.05 0.45/1.05 ============================== end of process initial clauses ======== 0.45/1.05 0.45/1.05 ============================== CLAUSES FOR SEARCH ==================== 0.45/1.05 51.33/51.63 ============================== end of clauses for search ============= 51.33/51.63 51.33/51.63 ============================== SEARCH ================================ 51.33/51.63 51.33/51.63 % Starting search at 0.03 seconds. 51.33/51.63 51.33/51.63 Low Water (keep): wt=22.000, iters=3351 51.33/51.63 51.33/51.63 Low Water (keep): wt=21.000, iters=3405 51.33/51.63 51.33/51.63 Low Water (keep): wt=20.000, iters=3393 51.33/51.63 51.33/51.63 Low Water (keep): wt=19.000, iters=3367 51.33/51.63 51.33/51.63 Low Water (keep): wt=18.000, iters=3345 51.33/51.63 51.33/51.63 Low Water (keep): wt=17.000, iters=3344 51.33/51.63 51.33/51.63 Low Water (keep): wt=16.000, iters=3343 51.33/51.63 51.33/51.63 Low Water (keep): wt=15.000, iters=3338 51.33/51.63 51.33/51.63 Low Water (keep): wt=14.000, iters=3333 51.33/51.63 51.33/51.63 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 66 (0.00 of 1.34 sec). 51.33/51.63 51.33/51.63 Low Water (keep): wt=13.000, iters=3338 51.33/51.63 51.33/51.63 Low Water (displace): id=1203, wt=29.000 51.33/51.63 51.33/51.63 Low Water (displace): id=3213, wt=28.000 51.33/51.63 51.33/51.63 Low Water (displace): id=10924, wt=11.000 51.33/51.63 51.33/51.63 Low Water (displace): id=10925, wt=9.000 51.33/51.63 51.33/51.63 Low Water (keep): wt=12.000, iters=4370 51.33/51.63 51.33/51.63 Low Water (keep): wt=11.000, iters=3481 51.33/51.63 51.33/51.63 Low Water (keep): wt=10.000, iters=3333 51.33/51.63 51.33/51.63 ============================== PROOF ================================= 51.33/51.63 % SZS status Theorem 51.33/51.63 % SZS output start Refutation 51.33/51.63 51.33/51.63 % Proof 1 at 47.17 (+ 3.43) seconds. 51.33/51.63 % Length of proof is 138. 51.33/51.63 % Level of proof is 22. 51.33/51.63 % Maximum clause weight is 14.000. 51.33/51.63 % Given clauses 14785. 51.33/51.63 51.33/51.63 1 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 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]. 51.33/51.63 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]. 51.33/51.63 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]. 51.33/51.63 9 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 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]. 51.33/51.63 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]. 51.33/51.63 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]. 51.33/51.63 25 (all X all Y is_a_theorem(implies(X,implies(Y,X)))) <-> implies_1 # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 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]. 51.33/51.63 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]. 51.33/51.63 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]. 51.33/51.63 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]. 51.33/51.63 33 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 39 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 45 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 51 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 51.33/51.63 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]. 51.33/51.63 59 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(1)]. 51.33/51.63 61 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(2)]. 51.33/51.63 67 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(5)]. 51.33/51.63 69 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(6)]. 51.33/51.63 75 -or_1 | is_a_theorem(implies(A,or(A,B))) # label(or_1) # label(axiom). [clausify(9)]. 51.33/51.63 83 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(13)]. 51.33/51.63 95 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) | -implies_2 # label(implies_2) # label(axiom). [clausify(18)]. 51.33/51.63 106 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(24)]. 51.33/51.63 111 is_a_theorem(implies(A,implies(B,A))) | -implies_1 # label(implies_1) # label(axiom). [clausify(25)]. 51.33/51.63 113 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) | -implies_3 # label(implies_3) # label(axiom). [clausify(26)]. 51.33/51.63 114 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(27)]. 51.33/51.63 116 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(29)]. 51.33/51.63 117 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(116),flip(b)]. 51.33/51.63 118 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(30)]. 51.33/51.63 119 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(118),flip(b)]. 51.33/51.63 122 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption]. 51.33/51.63 123 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption]. 51.33/51.63 125 or_1 # label(hilbert_or_1) # label(axiom). [assumption]. 51.33/51.63 126 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption]. 51.33/51.63 127 op_or # label(hilbert_op_or) # label(axiom). [assumption]. 51.33/51.63 128 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 51.33/51.63 129 and_3 # label(hilbert_and_3) # label(axiom). [assumption]. 51.33/51.63 130 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption]. 51.33/51.63 134 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption]. 51.33/51.63 135 and_1 # label(hilbert_and_1) # label(axiom). [assumption]. 51.33/51.63 136 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption]. 51.33/51.63 137 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption]. 51.33/51.63 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 51.33/51.63 142 is_a_theorem(implies(A,necessarily(possibly(A)))) | -axiom_B # label(axiom_B) # label(axiom). [clausify(33)]. 51.33/51.63 154 -axiom_4 | is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))) # label(axiom_4) # label(axiom). [clausify(39)]. 51.33/51.63 168 -is_a_theorem(A) | is_a_theorem(necessarily(A)) | -necessitation # label(necessitation) # label(axiom). [clausify(45)]. 51.33/51.63 181 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(51)]. 51.33/51.63 190 axiom_m10 | -is_a_theorem(strict_implies(possibly(c94),necessarily(possibly(c94)))) # label(axiom_m10) # label(axiom). [clausify(54)]. 51.33/51.63 191 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(55)]. 51.33/51.63 192 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(191),flip(b)]. 51.33/51.63 197 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B) # label(op_strict_implies) # label(axiom). [clausify(58)]. 51.33/51.63 198 axiom_B # label(km4b_axiom_B) # label(axiom). [assumption]. 51.33/51.63 199 axiom_4 # label(km4b_axiom_4) # label(axiom). [assumption]. 51.33/51.63 200 op_possibly # label(km4b_op_possibly) # label(axiom). [assumption]. 51.33/51.63 201 axiom_M # label(km4b_axiom_M) # label(axiom). [assumption]. 51.33/51.63 202 necessitation # label(km4b_necessitation) # label(axiom). [assumption]. 51.33/51.63 206 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption]. 51.33/51.63 207 -axiom_m10 # label(s1_0_m10_axiom_m10) # label(negated_conjecture). [assumption]. 51.33/51.63 209 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(95),unit_del(b,122)]. 51.33/51.63 210 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(113),unit_del(b,123)]. 51.33/51.63 212 is_a_theorem(implies(A,or(A,B))). [back_unit_del(75),unit_del(a,125)]. 51.33/51.63 213 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(69),unit_del(a,126)]. 51.33/51.63 214 not(and(not(A),not(B))) = or(A,B). [back_unit_del(119),unit_del(a,127)]. 51.33/51.63 215 not(and(A,not(B))) = implies(A,B). [back_unit_del(114),unit_del(a,128)]. 51.33/51.63 216 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(67),unit_del(a,129)]. 51.33/51.63 217 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(106),unit_del(a,130)]. 51.33/51.63 221 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(61),unit_del(a,134)]. 51.33/51.63 222 is_a_theorem(implies(and(A,B),A)). [back_unit_del(59),unit_del(a,135)]. 51.33/51.63 223 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(117),unit_del(a,136)]. 51.33/51.63 224 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(111),unit_del(b,137)]. 51.33/51.63 225 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(83),unit_del(a,138)]. 51.33/51.63 227 is_a_theorem(implies(A,necessarily(possibly(A)))). [back_unit_del(142),unit_del(b,198)]. 51.33/51.63 228 is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))). [back_unit_del(154),unit_del(a,199)]. 51.33/51.63 229 not(necessarily(not(A))) = possibly(A). [back_unit_del(192),unit_del(a,200)]. 51.33/51.63 230 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(181),unit_del(a,201)]. 51.33/51.63 231 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(168),unit_del(c,202)]. 51.33/51.63 234 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(197),unit_del(a,206)]. 51.33/51.63 235 -is_a_theorem(strict_implies(possibly(c94),necessarily(possibly(c94)))). [back_unit_del(190),unit_del(a,207)]. 51.33/51.63 237 implies(not(A),B) = or(A,B). [back_rewrite(214),rewrite([215(4)])]. 51.33/51.63 243 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(221),rewrite([237(3)])]. 51.33/51.63 271 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(217,b,213,a)]. 51.33/51.63 274 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(217,b,210,a)]. 51.33/51.63 275 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(217,b,209,a)]. 51.33/51.63 289 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(224,a,217,b)]. 51.33/51.63 314 not(and(A,possibly(B))) = implies(A,necessarily(not(B))). [para(229(a,1),215(a,1,1,2))]. 51.33/51.63 315 possibly(and(A,not(B))) = not(strict_implies(A,B)). [para(215(a,1),229(a,1,1,1)),rewrite([234(2)]),flip(a)]. 51.33/51.63 323 is_a_theorem(strict_implies(A,necessarily(possibly(A)))). [resolve(231,a,227,a),rewrite([234(4)])]. 51.33/51.63 353 is_a_theorem(or(A,or(not(A),B))). [para(237(a,1),212(a,1))]. 51.33/51.63 368 necessarily(or(A,B)) = strict_implies(not(A),B). [para(237(a,1),234(a,1,1))]. 51.33/51.63 378 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(243,a,217,b)]. 51.33/51.63 2041 is_a_theorem(implies(implies(necessarily(necessarily(A)),necessarily(A)),equiv(necessarily(A),necessarily(necessarily(A))))). [resolve(271,a,228,a)]. 51.33/51.63 2252 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(274,a,224,a)]. 51.33/51.63 2329 is_a_theorem(implies(A,A)). [resolve(275,a,224,a)]. 51.33/51.63 2330 is_a_theorem(implies(A,and(A,A))). [resolve(275,a,216,a)]. 51.33/51.63 2361 is_a_theorem(strict_implies(A,A)). [resolve(2329,a,231,a),rewrite([234(2)])]. 51.33/51.63 2367 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(2361,a,289,a)]. 51.33/51.63 2514 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(2330,a,271,a)]. 51.33/51.63 2877 is_a_theorem(implies(implies(strict_implies(A,A),B),implies(C,B))). [resolve(2367,a,274,a)]. 51.33/51.63 2879 is_a_theorem(implies(implies(strict_implies(A,A),B),equiv(B,strict_implies(A,A)))). [resolve(2367,a,271,a)]. 51.33/51.63 5393 not(strict_implies(A,necessarily(not(B)))) = possibly(and(A,possibly(B))). [para(314(a,1),229(a,1,1,1)),rewrite([234(4)])]. 51.33/51.63 17136 is_a_theorem(equiv(necessarily(A),necessarily(necessarily(A)))). [resolve(2041,a,217,b),unit_del(a,230)]. 51.33/51.63 17142 necessarily(necessarily(A)) = necessarily(A). [resolve(17136,a,225,a)]. 51.33/51.63 17454 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(2252,a,217,b)]. 51.33/51.63 18401 is_a_theorem(equiv(A,and(A,A))). [resolve(2514,a,217,b),unit_del(a,222)]. 51.33/51.63 18418 and(A,A) = A. [resolve(18401,a,225,a)]. 51.33/51.63 18421 or(A,A) = not(not(A)). [para(18418(a,1),215(a,1,1)),rewrite([237(4)]),flip(a)]. 51.33/51.63 18422 equiv(A,A) = implies(A,A). [para(18418(a,1),223(a,1)),flip(a)]. 51.33/51.63 18425 not(strict_implies(not(A),A)) = possibly(not(A)). [para(18418(a,1),315(a,1,1)),flip(a)]. 51.33/51.63 18612 is_a_theorem(implies(A,not(not(A)))). [para(18421(a,1),212(a,1,2))]. 51.33/51.63 18616 is_a_theorem(or(A,not(not(not(A))))). [para(18421(a,1),353(a,1,2))]. 51.33/51.63 18651 necessarily(not(not(A))) = strict_implies(not(A),A). [para(18421(a,1),368(a,1,1))]. 51.33/51.63 19175 is_a_theorem(or(not(A),A)). [resolve(18616,a,378,a),rewrite([237(3)])]. 51.33/51.63 19491 is_a_theorem(implies(implies(strict_implies(A,A),B),B)). [resolve(2877,a,275,a)]. 51.33/51.63 20703 -is_a_theorem(implies(strict_implies(A,A),B)) | is_a_theorem(B). [resolve(19491,a,217,b)]. 51.33/51.63 31620 is_a_theorem(implies(A,equiv(A,strict_implies(B,B)))). [resolve(17454,a,2879,a)]. 51.33/51.63 31650 is_a_theorem(equiv(strict_implies(A,A),strict_implies(B,B))). [resolve(31620,a,20703,a)]. 51.33/51.63 31655 -is_a_theorem(A) | is_a_theorem(equiv(A,strict_implies(B,B))). [resolve(31620,a,217,b)]. 51.33/51.63 31762 is_a_theorem(necessarily(equiv(strict_implies(A,A),strict_implies(B,B)))). [resolve(31650,a,231,a)]. 51.33/51.63 31763 strict_implies(A,A) = strict_implies(B,B). [resolve(31650,a,225,a)]. 51.33/51.63 31764 strict_implies(A,A) = c_0. [new_symbol(31763)]. 51.33/51.63 31765 is_a_theorem(c_0). [back_rewrite(31762),rewrite([31764(1),31764(2),18422(3),234(4),31764(3)])]. 51.33/51.63 31769 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(31655),rewrite([31764(2)])]. 51.33/51.63 35880 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(31769,a,19175,a)]. 51.33/51.63 35928 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(31769,a,18612,a)]. 51.33/51.63 37745 or(not(A),A) = c_0. [resolve(35880,a,225,a),flip(a)]. 51.33/51.63 38354 implies(A,not(not(A))) = c_0. [resolve(35928,a,225,a),flip(a)]. 51.33/51.63 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)]. 51.33/51.63 38362 not(not(A)) = A. [para(38356(a,1),225(a,1)),unit_del(a,31765)]. 51.33/51.63 38395 strict_implies(not(A),A) = necessarily(A). [back_rewrite(18651),rewrite([38362(2)]),flip(a)]. 51.33/51.63 38410 not(necessarily(A)) = possibly(not(A)). [back_rewrite(18425),rewrite([38395(2)])]. 51.33/51.63 39793 possibly(possibly(A)) = possibly(A). [para(38395(a,1),5393(a,1,1)),rewrite([17142(3),38410(3),38362(2),38410(4),38362(3),18418(4)]),flip(a)]. 51.33/51.63 39825 is_a_theorem(strict_implies(possibly(A),necessarily(possibly(A)))). [para(39793(a,1),323(a,1,2,1))]. 51.33/51.63 39826 $F. [resolve(39825,a,235,a)]. 51.33/51.63 51.33/51.63 % SZS output end Refutation 51.33/51.63 ============================== end of proof ========================== 51.33/51.63 51.33/51.63 ============================== STATISTICS ============================ 51.33/51.63 51.33/51.63 Given=14785. Generated=6665288. Kept=39762. proofs=1. 51.33/51.63 Usable=6541. Sos=5379. Demods=88. Limbo=1, Disabled=27987. Hints=0. 51.33/51.63 Megabytes=34.02. 51.33/51.63 User_CPU=47.17, System_CPU=3.43, Wall_clock=50. 51.33/51.63 51.33/51.63 ============================== end of statistics ===================== 51.33/51.63 51.33/51.63 ============================== end of search ========================= 51.33/51.63 51.33/51.63 THEOREM PROVED 51.33/51.63 % SZS status Theorem 51.33/51.63 51.33/51.63 Exiting with 1 proof. 51.33/51.63 51.33/51.63 Process 6836 exit (max_proofs) Tue Aug 9 05:27:12 2022 51.33/51.63 Prover9 interrupted 51.33/51.63 EOF