TSTP Solution File: LCL535+1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : LCL535+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n028.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Sun Jul 17 13:47:25 EDT 2022
% Result : Theorem 4.60s 4.91s
% Output : Refutation 4.60s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : LCL535+1 : TPTP v8.1.0. Released v3.3.0.
% 0.06/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.12/0.33 % Computer : n028.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Mon Jul 4 13:33:19 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.73/1.00 ============================== Prover9 ===============================
% 0.73/1.00 Prover9 (32) version 2009-11A, November 2009.
% 0.73/1.00 Process 21537 was started by sandbox on n028.cluster.edu,
% 0.73/1.00 Mon Jul 4 13:33:20 2022
% 0.73/1.00 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_21384_n028.cluster.edu".
% 0.73/1.00 ============================== end of head ===========================
% 0.73/1.00
% 0.73/1.00 ============================== INPUT =================================
% 0.73/1.00
% 0.73/1.00 % Reading from file /tmp/Prover9_21384_n028.cluster.edu
% 0.73/1.00
% 0.73/1.00 set(prolog_style_variables).
% 0.73/1.00 set(auto2).
% 0.73/1.00 % set(auto2) -> set(auto).
% 0.73/1.00 % set(auto) -> set(auto_inference).
% 0.73/1.00 % set(auto) -> set(auto_setup).
% 0.73/1.00 % set(auto_setup) -> set(predicate_elim).
% 0.73/1.00 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.73/1.00 % set(auto) -> set(auto_limits).
% 0.73/1.00 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.73/1.00 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.73/1.00 % set(auto) -> set(auto_denials).
% 0.73/1.00 % set(auto) -> set(auto_process).
% 0.73/1.00 % set(auto2) -> assign(new_constants, 1).
% 0.73/1.00 % set(auto2) -> assign(fold_denial_max, 3).
% 0.73/1.00 % set(auto2) -> assign(max_weight, "200.000").
% 0.73/1.00 % set(auto2) -> assign(max_hours, 1).
% 0.73/1.00 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.73/1.00 % set(auto2) -> assign(max_seconds, 0).
% 0.73/1.00 % set(auto2) -> assign(max_minutes, 5).
% 0.73/1.00 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.73/1.00 % set(auto2) -> set(sort_initial_sos).
% 0.73/1.00 % set(auto2) -> assign(sos_limit, -1).
% 0.73/1.00 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.73/1.00 % set(auto2) -> assign(max_megs, 400).
% 0.73/1.00 % set(auto2) -> assign(stats, some).
% 0.73/1.00 % set(auto2) -> clear(echo_input).
% 0.73/1.00 % set(auto2) -> set(quiet).
% 0.73/1.00 % set(auto2) -> clear(print_initial_clauses).
% 0.73/1.00 % set(auto2) -> clear(print_given).
% 0.73/1.00 assign(lrs_ticks,-1).
% 0.73/1.00 assign(sos_limit,10000).
% 0.73/1.00 assign(order,kbo).
% 0.73/1.00 set(lex_order_vars).
% 0.73/1.00 clear(print_given).
% 0.73/1.00
% 0.73/1.00 % formulas(sos). % not echoed (88 formulas)
% 0.73/1.00
% 0.73/1.00 ============================== end of input ==========================
% 0.73/1.00
% 0.73/1.00 % From the command line: assign(max_seconds, 300).
% 0.73/1.00
% 0.73/1.00 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.73/1.00
% 0.73/1.00 % Formulas that are not ordinary clauses:
% 0.73/1.00 1 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].
% 0.73/1.00 2 substitution_of_equivalents <-> (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 3 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].
% 0.73/1.00 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 5 implies_2 <-> (all X all Y is_a_theorem(implies(implies(X,implies(X,Y)),implies(X,Y)))) # label(implies_2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 6 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].
% 0.73/1.00 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 9 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].
% 0.73/1.00 10 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 11 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 12 or_3 <-> (all X all Y all Z is_a_theorem(implies(implies(X,Z),implies(implies(Y,Z),implies(or(X,Y),Z))))) # label(or_3) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 13 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].
% 0.73/1.00 14 equivalence_2 <-> (all X all Y is_a_theorem(implies(equiv(X,Y),implies(Y,X)))) # label(equivalence_2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 15 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].
% 0.73/1.00 16 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 17 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 18 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].
% 0.73/1.00 19 cn1 <-> (all P all Q all R is_a_theorem(implies(implies(P,Q),implies(implies(Q,R),implies(P,R))))) # label(cn1) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 20 cn2 <-> (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) # label(cn2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 21 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 22 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 23 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 24 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 25 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].
% 0.73/1.00 26 r5 <-> (all P all Q all R is_a_theorem(implies(implies(Q,R),implies(or(P,Q),or(P,R))))) # label(r5) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 27 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 28 op_and -> (all X all Y and(X,Y) = not(or(not(X),not(Y)))) # label(op_and) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 29 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].
% 0.73/1.00 30 op_implies_or -> (all X all Y implies(X,Y) = or(not(X),Y)) # label(op_implies_or) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 31 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].
% 0.73/1.00 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 33 modus_ponens_strict_implies <-> (all X all Y (is_a_theorem(X) & is_a_theorem(strict_implies(X,Y)) -> is_a_theorem(Y))) # label(modus_ponens_strict_implies) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 34 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].
% 0.73/1.00 35 substitution_strict_equiv <-> (all X all Y (is_a_theorem(strict_equiv(X,Y)) -> X = Y)) # label(substitution_strict_equiv) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 36 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].
% 0.73/1.00 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.00 41 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].
% 0.73/1.03 42 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].
% 0.73/1.03 43 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].
% 0.73/1.03 44 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 45 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].
% 0.73/1.03 46 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 47 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].
% 0.73/1.03 48 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 49 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].
% 0.73/1.03 50 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 51 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].
% 0.73/1.03 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].
% 0.73/1.03 53 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 56 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption].
% 0.73/1.03 57 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].
% 0.73/1.03 58 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].
% 0.73/1.03
% 0.73/1.03 ============================== end of process non-clausal formulas ===
% 0.73/1.03
% 0.73/1.03 ============================== PROCESS INITIAL CLAUSES ===============
% 0.73/1.03
% 0.73/1.03 ============================== PREDICATE ELIMINATION =================
% 0.73/1.03
% 0.73/1.03 ============================== end predicate elimination =============
% 0.73/1.03
% 0.73/1.03 Auto_denials: (non-Horn, no changes).
% 0.73/1.03
% 0.73/1.03 Term ordering decisions:
% 0.73/1.03 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.
% 0.73/1.03
% 0.73/1.03 ============================== end of process initial clauses ========
% 0.73/1.03
% 0.73/1.03 ============================== CLAUSES FOR SEARCH ====================
% 4.60/4.91
% 4.60/4.91 ============================== end of clauses for search =============
% 4.60/4.91
% 4.60/4.91 ============================== SEARCH ================================
% 4.60/4.91
% 4.60/4.91 % Starting search at 0.05 seconds.
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=22.000, iters=3404
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=20.000, iters=3356
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=19.000, iters=3396
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=18.000, iters=3340
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=16.000, iters=3351
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=15.000, iters=3407
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=14.000, iters=3468
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=13.000, iters=3404
% 4.60/4.91
% 4.60/4.91 Low Water (keep): wt=12.000, iters=3336
% 4.60/4.91
% 4.60/4.91 Low Water (displace): id=15421, wt=34.000
% 4.60/4.91
% 4.60/4.91 Low Water (displace): id=14835, wt=32.000
% 4.60/4.91
% 4.60/4.91 Low Water (displace): id=15422, wt=29.000
% 4.60/4.91
% 4.60/4.91 Low Water (displace): id=15219, wt=28.000
% 4.60/4.91
% 4.60/4.91 Low Water (displace): id=20700, wt=10.000
% 4.60/4.91
% 4.60/4.91 Low Water (displace): id=20800, wt=9.000
% 4.60/4.91
% 4.60/4.91 ============================== PROOF =================================
% 4.60/4.91 % SZS status Theorem
% 4.60/4.91 % SZS output start Refutation
% 4.60/4.91
% 4.60/4.91 % Proof 1 at 3.88 (+ 0.05) seconds.
% 4.60/4.91 % Length of proof is 101.
% 4.60/4.91 % Level of proof is 12.
% 4.60/4.91 % Maximum clause weight is 15.000.
% 4.60/4.91 % Given clauses 1525.
% 4.60/4.91
% 4.60/4.91 1 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].
% 4.60/4.91 2 substitution_of_equivalents <-> (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) # label(substitution_of_equivalents) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 5 implies_2 <-> (all X all Y is_a_theorem(implies(implies(X,implies(X,Y)),implies(X,Y)))) # label(implies_2) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 6 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].
% 4.60/4.91 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 9 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].
% 4.60/4.91 12 or_3 <-> (all X all Y all Z is_a_theorem(implies(implies(X,Z),implies(implies(Y,Z),implies(or(X,Y),Z))))) # label(or_3) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 15 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].
% 4.60/4.91 27 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 29 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].
% 4.60/4.91 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 53 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 4.60/4.91 57 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].
% 4.60/4.91 59 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(1)].
% 4.60/4.91 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(2)].
% 4.60/4.91 68 -implies_1 | is_a_theorem(implies(A,implies(B,A))) # label(implies_1) # label(axiom). [clausify(4)].
% 4.60/4.91 70 -implies_2 | is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) # label(implies_2) # label(axiom). [clausify(5)].
% 4.60/4.91 72 -implies_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) # label(implies_3) # label(axiom). [clausify(6)].
% 4.60/4.91 74 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(7)].
% 4.60/4.91 78 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(9)].
% 4.60/4.91 84 -or_3 | is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))) # label(or_3) # label(axiom). [clausify(12)].
% 4.60/4.91 90 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(15)].
% 4.60/4.91 114 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(27)].
% 4.60/4.91 115 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(114),flip(b)].
% 4.60/4.91 118 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(29)].
% 4.60/4.91 121 op_or # label(hilbert_op_or) # label(axiom). [assumption].
% 4.60/4.91 122 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption].
% 4.60/4.91 124 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption].
% 4.60/4.91 126 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption].
% 4.60/4.91 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption].
% 4.60/4.91 128 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption].
% 4.60/4.91 129 and_1 # label(hilbert_and_1) # label(axiom). [assumption].
% 4.60/4.91 131 and_3 # label(hilbert_and_3) # label(axiom). [assumption].
% 4.60/4.91 134 or_3 # label(hilbert_or_3) # label(axiom). [assumption].
% 4.60/4.91 137 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption].
% 4.60/4.91 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption].
% 4.60/4.91 139 -necessitation | -is_a_theorem(A) | is_a_theorem(necessarily(A)) # label(necessitation) # label(axiom). [clausify(32)].
% 4.60/4.91 155 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(37)].
% 4.60/4.91 161 -axiom_5 | is_a_theorem(implies(possibly(A),necessarily(possibly(A)))) # label(axiom_5) # label(axiom). [clausify(40)].
% 4.60/4.91 188 axiom_m9 | -is_a_theorem(strict_implies(possibly(possibly(c93)),possibly(c93))) # label(axiom_m9) # label(axiom). [clausify(53)].
% 4.60/4.91 191 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(55)].
% 4.60/4.91 192 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(191),flip(b)].
% 4.60/4.91 195 -op_strict_implies | strict_implies(A,B) = necessarily(implies(A,B)) # label(op_strict_implies) # label(axiom). [clausify(57)].
% 4.60/4.91 196 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B). [copy(195),flip(b)].
% 4.60/4.91 199 op_possibly # label(km5_op_possibly) # label(axiom). [assumption].
% 4.60/4.91 200 necessitation # label(km5_necessitation) # label(axiom). [assumption].
% 4.60/4.91 202 axiom_M # label(km5_axiom_M) # label(axiom). [assumption].
% 4.60/4.91 203 axiom_5 # label(km5_axiom_5) # label(axiom). [assumption].
% 4.60/4.91 205 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption].
% 4.60/4.91 207 -axiom_m9 # label(s1_0_m6s3m9b_axiom_m9) # label(negated_conjecture). [assumption].
% 4.60/4.91 208 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,121)].
% 4.60/4.91 209 not(and(A,not(B))) = implies(A,B). [back_unit_del(118),unit_del(a,122)].
% 4.60/4.91 211 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(59),unit_del(a,124)].
% 4.60/4.91 213 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(68),unit_del(a,126)].
% 4.60/4.91 214 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(70),unit_del(a,127)].
% 4.60/4.91 215 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(72),unit_del(a,128)].
% 4.60/4.91 216 is_a_theorem(implies(and(A,B),A)). [back_unit_del(74),unit_del(a,129)].
% 4.60/4.91 218 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(78),unit_del(a,131)].
% 4.60/4.91 221 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(84),unit_del(a,134)].
% 4.60/4.91 224 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(90),unit_del(a,137)].
% 4.60/4.91 225 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,138)].
% 4.60/4.91 227 not(necessarily(not(A))) = possibly(A). [back_unit_del(192),unit_del(a,199)].
% 4.60/4.91 228 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(139),unit_del(a,200)].
% 4.60/4.91 230 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(155),unit_del(a,202)].
% 4.60/4.91 231 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))). [back_unit_del(161),unit_del(a,203)].
% 4.60/4.91 232 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(196),unit_del(a,205)].
% 4.60/4.91 234 -is_a_theorem(strict_implies(possibly(possibly(c93)),possibly(c93))). [back_unit_del(188),unit_del(a,207)].
% 4.60/4.91 235 implies(not(A),B) = or(A,B). [back_rewrite(208),rewrite([209(4)])].
% 4.60/4.91 255 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(213,a,211,b)].
% 4.60/4.91 258 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(214,a,211,b)].
% 4.60/4.91 261 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(215,a,211,b)].
% 4.60/4.91 279 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(C,B),implies(or(A,C),B))). [resolve(221,a,211,b)].
% 4.60/4.91 288 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(224,a,211,b)].
% 4.60/4.91 308 not(necessarily(possibly(A))) = possibly(necessarily(not(A))). [para(227(a,1),227(a,1,1,1))].
% 4.60/4.91 359 or(necessarily(not(A)),B) = implies(possibly(A),B). [para(227(a,1),235(a,1,1)),flip(a)].
% 4.60/4.91 856 is_a_theorem(implies(A,and(A,A))). [resolve(258,a,218,a)].
% 4.60/4.91 858 is_a_theorem(implies(A,A)). [resolve(258,a,213,a)].
% 4.60/4.91 865 is_a_theorem(implies(A,implies(B,B))). [resolve(858,a,255,a)].
% 4.60/4.91 1385 is_a_theorem(implies(implies(implies(A,A),B),implies(C,B))). [resolve(865,a,261,a)].
% 4.60/4.91 2785 is_a_theorem(implies(implies(A,B),implies(or(necessarily(B),A),B))). [resolve(279,a,230,a)].
% 4.60/4.91 3916 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(288,a,856,a)].
% 4.60/4.91 3943 is_a_theorem(implies(implies(necessarily(possibly(A)),possibly(A)),equiv(possibly(A),necessarily(possibly(A))))). [resolve(288,a,231,a)].
% 4.60/4.91 13778 or(necessarily(possibly(necessarily(not(A)))),B) = implies(possibly(necessarily(possibly(A))),B). [para(308(a,1),359(a,1,1,1))].
% 4.60/4.91 19308 is_a_theorem(implies(implies(implies(A,A),B),B)). [resolve(1385,a,258,a)].
% 4.60/4.91 19515 -is_a_theorem(implies(implies(A,A),B)) | is_a_theorem(B). [resolve(19308,a,211,b)].
% 4.60/4.91 21349 is_a_theorem(implies(or(necessarily(A),A),A)). [resolve(2785,a,19515,a)].
% 4.60/4.91 21960 is_a_theorem(equiv(A,and(A,A))). [resolve(3916,a,211,b),unit_del(a,216)].
% 4.60/4.91 21991 and(A,A) = A. [resolve(21960,a,225,a)].
% 4.60/4.91 21996 or(A,A) = not(not(A)). [para(21991(a,1),209(a,1,1)),rewrite([235(4)]),flip(a)].
% 4.60/4.91 23212 is_a_theorem(equiv(possibly(A),necessarily(possibly(A)))). [resolve(3943,a,211,b),unit_del(a,230)].
% 4.60/4.91 23518 necessarily(possibly(A)) = possibly(A). [resolve(23212,a,225,a)].
% 4.60/4.91 23543 or(possibly(necessarily(not(A))),B) = implies(possibly(possibly(A)),B). [back_rewrite(13778),rewrite([23518(4),23518(6)])].
% 4.60/4.91 23558 possibly(necessarily(not(A))) = not(possibly(A)). [back_rewrite(308),rewrite([23518(2)]),flip(a)].
% 4.60/4.91 23561 or(not(possibly(A)),B) = implies(possibly(possibly(A)),B). [back_rewrite(23543),rewrite([23558(3)])].
% 4.60/4.91 23592 is_a_theorem(implies(possibly(possibly(A)),possibly(A))). [para(23518(a,1),21349(a,1,1,1)),rewrite([21996(3),235(5),23561(4)])].
% 4.60/4.91 23666 is_a_theorem(strict_implies(possibly(possibly(A)),possibly(A))). [resolve(23592,a,228,a),rewrite([232(5)])].
% 4.60/4.91 23667 $F. [resolve(23666,a,234,a)].
% 4.60/4.91
% 4.60/4.91 % SZS output end Refutation
% 4.60/4.91 ============================== end of proof ==========================
% 4.60/4.91
% 4.60/4.91 ============================== STATISTICS ============================
% 4.60/4.91
% 4.60/4.91 Given=1525. Generated=66938. Kept=23602. proofs=1.
% 4.60/4.91 Usable=924. Sos=7291. Demods=54. Limbo=23, Disabled=15509. Hints=0.
% 4.60/4.91 Megabytes=19.55.
% 4.60/4.91 User_CPU=3.88, System_CPU=0.05, Wall_clock=4.
% 4.60/4.91
% 4.60/4.91 ============================== end of statistics =====================
% 4.60/4.91
% 4.60/4.91 ============================== end of search =========================
% 4.60/4.91
% 4.60/4.91 THEOREM PROVED
% 4.60/4.91 % SZS status Theorem
% 4.60/4.91
% 4.60/4.91 Exiting with 1 proof.
% 4.60/4.91
% 4.60/4.91 Process 21537 exit (max_proofs) Mon Jul 4 13:33:24 2022
% 4.60/4.91 Prover9 interrupted
%------------------------------------------------------------------------------