0.04/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.04/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.34 % Computer : n005.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 1200 0.13/0.34 % DateTime : Tue Jul 13 16:15:33 EDT 2021 0.13/0.34 % CPUTime : 0.46/1.04 ============================== Prover9 =============================== 0.46/1.04 Prover9 (32) version 2009-11A, November 2009. 0.46/1.04 Process 22433 was started by sandbox on n005.cluster.edu, 0.46/1.04 Tue Jul 13 16:15:34 2021 0.46/1.04 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1200 -f /tmp/Prover9_22280_n005.cluster.edu". 0.46/1.04 ============================== end of head =========================== 0.46/1.04 0.46/1.04 ============================== INPUT ================================= 0.46/1.04 0.46/1.04 % Reading from file /tmp/Prover9_22280_n005.cluster.edu 0.46/1.04 0.46/1.04 set(prolog_style_variables). 0.46/1.04 set(auto2). 0.46/1.04 % set(auto2) -> set(auto). 0.46/1.04 % set(auto) -> set(auto_inference). 0.46/1.04 % set(auto) -> set(auto_setup). 0.46/1.04 % set(auto_setup) -> set(predicate_elim). 0.46/1.04 % set(auto_setup) -> assign(eq_defs, unfold). 0.46/1.04 % set(auto) -> set(auto_limits). 0.46/1.04 % set(auto_limits) -> assign(max_weight, "100.000"). 0.46/1.04 % set(auto_limits) -> assign(sos_limit, 20000). 0.46/1.04 % set(auto) -> set(auto_denials). 0.46/1.04 % set(auto) -> set(auto_process). 0.46/1.04 % set(auto2) -> assign(new_constants, 1). 0.46/1.04 % set(auto2) -> assign(fold_denial_max, 3). 0.46/1.04 % set(auto2) -> assign(max_weight, "200.000"). 0.46/1.04 % set(auto2) -> assign(max_hours, 1). 0.46/1.04 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.46/1.04 % set(auto2) -> assign(max_seconds, 0). 0.46/1.04 % set(auto2) -> assign(max_minutes, 5). 0.46/1.04 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.46/1.04 % set(auto2) -> set(sort_initial_sos). 0.46/1.04 % set(auto2) -> assign(sos_limit, -1). 0.46/1.04 % set(auto2) -> assign(lrs_ticks, 3000). 0.46/1.04 % set(auto2) -> assign(max_megs, 400). 0.46/1.04 % set(auto2) -> assign(stats, some). 0.46/1.04 % set(auto2) -> clear(echo_input). 0.46/1.04 % set(auto2) -> set(quiet). 0.46/1.04 % set(auto2) -> clear(print_initial_clauses). 0.46/1.04 % set(auto2) -> clear(print_given). 0.46/1.04 assign(lrs_ticks,-1). 0.46/1.04 assign(sos_limit,10000). 0.46/1.04 assign(order,kbo). 0.46/1.04 set(lex_order_vars). 0.46/1.04 clear(print_given). 0.46/1.04 0.46/1.04 % formulas(sos). % not echoed (88 formulas) 0.46/1.04 0.46/1.04 ============================== end of input ========================== 0.46/1.04 0.46/1.04 % From the command line: assign(max_seconds, 1200). 0.46/1.04 0.46/1.04 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.46/1.04 0.46/1.04 % Formulas that are not ordinary clauses: 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 6 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 10 (all P is_a_theorem(implies(or(P,P),P))) <-> r1 # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 11 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 12 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 15 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 17 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 20 (all P all Q is_a_theorem(implies(and(P,Q),P))) <-> kn2 # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 23 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 24 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 25 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 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]. 0.46/1.04 35 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 36 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.46/1.04 38 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 39 (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) <-> axiom_m10 # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 40 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.46/1.04 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]. 0.77/1.07 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]. 0.77/1.07 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]. 0.77/1.07 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]. 0.77/1.07 45 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.77/1.07 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]. 0.77/1.07 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]. 0.77/1.07 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]. 0.77/1.07 49 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.77/1.07 50 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.77/1.07 51 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.77/1.07 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]. 0.77/1.07 53 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.77/1.07 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]. 0.77/1.07 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]. 0.77/1.07 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]. 0.77/1.07 57 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.77/1.07 58 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.77/1.07 0.77/1.07 ============================== end of process non-clausal formulas === 0.77/1.07 0.77/1.07 ============================== PROCESS INITIAL CLAUSES =============== 0.77/1.07 0.77/1.07 ============================== PREDICATE ELIMINATION ================= 0.77/1.07 0.77/1.07 ============================== end predicate elimination ============= 0.77/1.07 0.77/1.07 Auto_denials: (non-Horn, no changes). 0.77/1.07 0.77/1.07 Term ordering decisions: 0.77/1.07 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. 0.77/1.07 0.77/1.07 ============================== end of process initial clauses ======== 0.77/1.07 0.77/1.07 ============================== CLAUSES FOR SEARCH ==================== 6.13/6.47 6.13/6.47 ============================== end of clauses for search ============= 6.13/6.47 6.13/6.47 ============================== SEARCH ================================ 6.13/6.47 6.13/6.47 % Starting search at 0.05 seconds. 6.13/6.47 6.13/6.47 Low Water (keep): wt=22.000, iters=3356 6.13/6.47 6.13/6.47 Low Water (keep): wt=20.000, iters=3346 6.13/6.47 6.13/6.47 Low Water (keep): wt=19.000, iters=3393 6.13/6.47 6.13/6.47 Low Water (keep): wt=18.000, iters=3350 6.13/6.47 6.13/6.47 Low Water (keep): wt=17.000, iters=3351 6.13/6.47 6.13/6.47 Low Water (keep): wt=16.000, iters=3411 6.13/6.47 6.13/6.47 Low Water (keep): wt=15.000, iters=3387 6.13/6.47 6.13/6.47 Low Water (keep): wt=14.000, iters=3426 6.13/6.47 6.13/6.47 Low Water (keep): wt=13.000, iters=3400 6.13/6.47 6.13/6.47 Low Water (keep): wt=12.000, iters=3340 6.13/6.47 6.13/6.47 Low Water (displace): id=15563, wt=32.000 6.13/6.47 6.13/6.47 Low Water (displace): id=20761, wt=11.000 6.13/6.47 6.13/6.47 Low Water (displace): id=20776, wt=10.000 6.13/6.47 6.13/6.47 Low Water (displace): id=20862, wt=9.000 6.13/6.47 6.13/6.47 Low Water (displace): id=24618, wt=8.000 6.13/6.47 6.13/6.47 Low Water (keep): wt=11.000, iters=3335 6.13/6.47 6.13/6.47 ============================== PROOF ================================= 6.13/6.47 % SZS status Theorem 6.13/6.47 % SZS output start Refutation 6.13/6.47 6.13/6.47 % Proof 1 at 5.38 (+ 0.07) seconds. 6.13/6.47 % Length of proof is 88. 6.13/6.47 % Level of proof is 14. 6.13/6.47 % Maximum clause weight is 14.000. 6.13/6.47 % Given clauses 1959. 6.13/6.47 6.13/6.47 3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 6.13/6.47 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]. 6.13/6.47 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]. 6.13/6.47 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]. 6.13/6.47 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]. 6.13/6.47 16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 6.13/6.47 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]. 6.13/6.47 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]. 6.13/6.47 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]. 6.13/6.47 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]. 6.13/6.47 36 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 6.13/6.47 38 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 6.13/6.47 50 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 6.13/6.47 53 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 6.13/6.47 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]. 6.13/6.47 58 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 6.13/6.47 64 is_a_theorem(implies(and(A,B),A)) | -and_1 # label(and_1) # label(axiom). [clausify(3)]. 6.13/6.47 66 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) | -implies_2 # label(implies_2) # label(axiom). [clausify(4)]. 6.13/6.47 70 -is_a_theorem(implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B) | -modus_ponens # label(modus_ponens) # label(axiom). [clausify(5)]. 6.13/6.47 75 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(8)]. 6.13/6.47 85 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(13)]. 6.13/6.47 91 -or_2 | is_a_theorem(implies(A,or(B,A))) # label(or_2) # label(axiom). [clausify(16)]. 6.13/6.47 102 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) | -modus_tollens # label(modus_tollens) # label(axiom). [clausify(21)]. 6.13/6.47 113 -is_a_theorem(equiv(A,B)) | B = A | -substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [clausify(26)]. 6.13/6.47 115 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(28)]. 6.13/6.47 116 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(29)]. 6.13/6.47 117 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(116),flip(b)]. 6.13/6.47 122 and_1 # label(hilbert_and_1) # label(axiom). [assumption]. 6.13/6.47 123 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption]. 6.13/6.47 125 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 6.13/6.47 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption]. 6.13/6.47 129 op_or # label(hilbert_op_or) # label(axiom). [assumption]. 6.13/6.47 130 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 6.13/6.47 132 or_2 # label(hilbert_or_2) # label(axiom). [assumption]. 6.13/6.47 133 and_3 # label(hilbert_and_3) # label(axiom). [assumption]. 6.13/6.47 134 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption]. 6.13/6.47 135 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption]. 6.13/6.47 149 -is_a_theorem(strict_implies(possibly(possibly(c64)),possibly(c64))) | axiom_m9 # label(axiom_m9) # label(axiom). [clausify(36)]. 6.13/6.47 154 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))) | -axiom_5 # label(axiom_5) # label(axiom). [clausify(38)]. 6.13/6.47 182 -is_a_theorem(A) | is_a_theorem(necessarily(A)) | -necessitation # label(necessitation) # label(axiom). [clausify(50)]. 6.13/6.47 187 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(53)]. 6.13/6.47 193 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B) # label(op_strict_implies) # label(axiom). [clausify(56)]. 6.13/6.47 196 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(58)]. 6.13/6.47 197 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(196),flip(b)]. 6.13/6.47 198 axiom_M # label(km5_axiom_M) # label(axiom). [assumption]. 6.13/6.47 199 axiom_5 # label(km5_axiom_5) # label(axiom). [assumption]. 6.13/6.47 200 op_possibly # label(km5_op_possibly) # label(axiom). [assumption]. 6.13/6.47 201 necessitation # label(km5_necessitation) # label(axiom). [assumption]. 6.13/6.47 205 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption]. 6.13/6.47 206 -axiom_m9 # label(s1_0_m6s3m9b_axiom_m9) # label(negated_conjecture). [assumption]. 6.13/6.47 208 is_a_theorem(implies(and(A,B),A)). [back_unit_del(64),unit_del(b,122)]. 6.13/6.47 209 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(75),unit_del(a,123)]. 6.13/6.47 211 not(and(A,not(B))) = implies(A,B). [back_unit_del(115),unit_del(a,125)]. 6.13/6.47 213 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(66),unit_del(b,127)]. 6.13/6.47 215 implies(not(A),B) = or(A,B). [back_unit_del(117),rewrite([211(5)]),unit_del(a,129)]. 6.13/6.47 216 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(113),unit_del(c,130)]. 6.13/6.47 218 is_a_theorem(implies(A,or(B,A))). [back_unit_del(91),unit_del(a,132)]. 6.13/6.47 219 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(85),unit_del(a,133)]. 6.13/6.47 220 -is_a_theorem(implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B). [back_unit_del(70),unit_del(d,134)]. 6.13/6.47 221 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_unit_del(102),rewrite([215(3)]),unit_del(b,135)]. 6.13/6.47 226 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(187),unit_del(a,198)]. 6.13/6.47 227 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))). [back_unit_del(154),unit_del(b,199)]. 6.13/6.47 228 not(necessarily(not(A))) = possibly(A). [back_unit_del(197),unit_del(a,200)]. 6.13/6.47 229 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(182),unit_del(c,201)]. 6.13/6.47 232 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(193),unit_del(a,205)]. 6.13/6.47 233 -is_a_theorem(strict_implies(possibly(possibly(c64)),possibly(c64))). [back_unit_del(149),unit_del(b,206)]. 6.13/6.47 280 is_a_theorem(or(A,or(B,not(A)))). [para(215(a,1),218(a,1))]. 6.13/6.47 289 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(220,a,213,a)]. 6.13/6.47 292 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(220,a,209,a)]. 6.13/6.47 296 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(221,a,220,a)]. 6.13/6.47 338 not(necessarily(possibly(A))) = possibly(necessarily(not(A))). [para(228(a,1),228(a,1,1,1))]. 6.13/6.47 364 necessarily(or(A,B)) = strict_implies(not(A),B). [para(215(a,1),232(a,1,1))]. 6.13/6.47 4122 is_a_theorem(implies(A,and(A,A))). [resolve(289,a,219,a)]. 6.13/6.47 4356 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(292,a,4122,a)]. 6.13/6.47 4392 is_a_theorem(implies(implies(necessarily(possibly(A)),possibly(A)),equiv(possibly(A),necessarily(possibly(A))))). [resolve(292,a,227,a)]. 6.13/6.47 24876 is_a_theorem(equiv(A,and(A,A))). [resolve(4356,a,220,a),unit_del(a,208)]. 6.13/6.47 24915 and(A,A) = A. [resolve(24876,a,216,a)]. 6.13/6.47 24918 or(A,A) = not(not(A)). [para(24915(a,1),211(a,1,1)),rewrite([215(4)]),flip(a)]. 6.13/6.47 24974 is_a_theorem(or(A,not(not(not(A))))). [para(24918(a,1),280(a,1,2))]. 6.13/6.47 26112 is_a_theorem(or(not(A),A)). [resolve(24974,a,296,a),rewrite([215(3)])]. 6.13/6.47 26148 is_a_theorem(strict_implies(not(not(A)),A)). [resolve(26112,a,229,a),rewrite([364(3)])]. 6.13/6.47 26192 is_a_theorem(strict_implies(not(possibly(A)),necessarily(not(A)))). [para(228(a,1),26148(a,1,1,1))]. 6.13/6.47 26196 is_a_theorem(equiv(possibly(A),necessarily(possibly(A)))). [resolve(4392,a,220,a),unit_del(a,226)]. 6.13/6.47 26635 necessarily(possibly(A)) = possibly(A). [resolve(26196,a,216,a)]. 6.13/6.47 26666 possibly(necessarily(not(A))) = not(possibly(A)). [back_rewrite(338),rewrite([26635(2)]),flip(a)]. 6.13/6.47 27543 not(not(possibly(A))) = possibly(possibly(A)). [para(228(a,1),26666(a,1,1,1)),rewrite([26635(2),26666(5)]),flip(a)]. 6.13/6.47 27548 is_a_theorem(strict_implies(possibly(possibly(A)),possibly(A))). [para(26666(a,1),26192(a,1,1,1)),rewrite([27543(3),228(5),26635(4)])]. 6.13/6.47 27549 $F. [resolve(27548,a,233,a)]. 6.13/6.47 6.13/6.47 % SZS output end Refutation 6.13/6.47 ============================== end of proof ========================== 6.13/6.47 6.13/6.47 ============================== STATISTICS ============================ 6.13/6.47 6.13/6.47 Given=1959. Generated=96980. Kept=27485. proofs=1. 6.13/6.47 Usable=1366. Sos=8227. Demods=57. Limbo=6, Disabled=18031. Hints=0. 6.13/6.47 Megabytes=20.97. 6.13/6.47 User_CPU=5.38, System_CPU=0.07, Wall_clock=5. 6.13/6.47 6.13/6.47 ============================== end of statistics ===================== 6.13/6.47 6.13/6.47 ============================== end of search ========================= 6.13/6.47 6.13/6.47 THEOREM PROVED 6.13/6.47 % SZS status Theorem 6.13/6.47 6.13/6.47 Exiting with 1 proof. 6.13/6.47 6.13/6.47 Process 22433 exit (max_proofs) Tue Jul 13 16:15:39 2021 6.13/6.47 Prover9 interrupted 6.13/6.47 EOF