TSTP Solution File: LCL524+1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : LCL524+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n007.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 13:47:22 EDT 2022
% Result : Theorem 5.76s 6.03s
% Output : Refutation 5.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : LCL524+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 : n007.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 : Sun Jul 3 03:32:55 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.44/1.01 ============================== Prover9 ===============================
% 0.44/1.01 Prover9 (32) version 2009-11A, November 2009.
% 0.44/1.01 Process 25762 was started by sandbox2 on n007.cluster.edu,
% 0.44/1.01 Sun Jul 3 03:32:56 2022
% 0.44/1.01 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_25606_n007.cluster.edu".
% 0.44/1.01 ============================== end of head ===========================
% 0.44/1.01
% 0.44/1.01 ============================== INPUT =================================
% 0.44/1.01
% 0.44/1.01 % Reading from file /tmp/Prover9_25606_n007.cluster.edu
% 0.44/1.01
% 0.44/1.01 set(prolog_style_variables).
% 0.44/1.01 set(auto2).
% 0.44/1.01 % set(auto2) -> set(auto).
% 0.44/1.01 % set(auto) -> set(auto_inference).
% 0.44/1.01 % set(auto) -> set(auto_setup).
% 0.44/1.01 % set(auto_setup) -> set(predicate_elim).
% 0.44/1.01 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.44/1.01 % set(auto) -> set(auto_limits).
% 0.44/1.01 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.44/1.01 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.44/1.01 % set(auto) -> set(auto_denials).
% 0.44/1.01 % set(auto) -> set(auto_process).
% 0.44/1.01 % set(auto2) -> assign(new_constants, 1).
% 0.44/1.01 % set(auto2) -> assign(fold_denial_max, 3).
% 0.44/1.01 % set(auto2) -> assign(max_weight, "200.000").
% 0.44/1.01 % set(auto2) -> assign(max_hours, 1).
% 0.44/1.01 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.44/1.01 % set(auto2) -> assign(max_seconds, 0).
% 0.44/1.01 % set(auto2) -> assign(max_minutes, 5).
% 0.44/1.01 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.44/1.01 % set(auto2) -> set(sort_initial_sos).
% 0.44/1.01 % set(auto2) -> assign(sos_limit, -1).
% 0.44/1.01 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.44/1.01 % set(auto2) -> assign(max_megs, 400).
% 0.44/1.01 % set(auto2) -> assign(stats, some).
% 0.44/1.01 % set(auto2) -> clear(echo_input).
% 0.44/1.01 % set(auto2) -> set(quiet).
% 0.44/1.01 % set(auto2) -> clear(print_initial_clauses).
% 0.44/1.01 % set(auto2) -> clear(print_given).
% 0.44/1.01 assign(lrs_ticks,-1).
% 0.44/1.01 assign(sos_limit,10000).
% 0.44/1.01 assign(order,kbo).
% 0.44/1.01 set(lex_order_vars).
% 0.44/1.01 clear(print_given).
% 0.44/1.01
% 0.44/1.01 % formulas(sos). % not echoed (82 formulas)
% 0.44/1.01
% 0.44/1.01 ============================== end of input ==========================
% 0.44/1.01
% 0.44/1.01 % From the command line: assign(max_seconds, 300).
% 0.44/1.01
% 0.44/1.01 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.44/1.01
% 0.44/1.01 % Formulas that are not ordinary clauses:
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 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].
% 0.44/1.01 10 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 11 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 16 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 17 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 20 cn2 <-> (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) # label(cn2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 21 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 22 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 23 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 24 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 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].
% 0.44/1.01 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.01 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].
% 0.44/1.04 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].
% 0.44/1.04 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].
% 0.44/1.04 44 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.04 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].
% 0.44/1.04 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].
% 0.44/1.04 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].
% 0.44/1.04 48 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.04 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].
% 0.44/1.04 50 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.04 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].
% 0.44/1.04 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].
% 0.44/1.04 53 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.04 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.04 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.04 56 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.04 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].
% 0.44/1.04 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].
% 0.44/1.04
% 0.44/1.04 ============================== end of process non-clausal formulas ===
% 0.44/1.04
% 0.44/1.04 ============================== PROCESS INITIAL CLAUSES ===============
% 0.44/1.04
% 0.44/1.04 ============================== PREDICATE ELIMINATION =================
% 0.44/1.04
% 0.44/1.04 ============================== end predicate elimination =============
% 0.44/1.04
% 0.44/1.04 Auto_denials: (non-Horn, no changes).
% 0.44/1.04
% 0.44/1.04 Term ordering decisions:
% 0.44/1.04 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.
% 0.44/1.04
% 0.44/1.04 ============================== end of process initial clauses ========
% 0.44/1.04
% 0.44/1.04 ============================== CLAUSES FOR SEARCH ====================
% 5.76/6.03
% 5.76/6.03 ============================== end of clauses for search =============
% 5.76/6.03
% 5.76/6.03 ============================== SEARCH ================================
% 5.76/6.03
% 5.76/6.03 % Starting search at 0.05 seconds.
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=26.000, iters=3336
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=22.000, iters=3380
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=20.000, iters=3360
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=19.000, iters=3358
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=18.000, iters=3376
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=17.000, iters=3345
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=16.000, iters=3366
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=15.000, iters=3379
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=14.000, iters=3454
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=13.000, iters=3487
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=12.000, iters=3340
% 5.76/6.03
% 5.76/6.03 Low Water (displace): id=10136, wt=30.000
% 5.76/6.03
% 5.76/6.03 Low Water (displace): id=15713, wt=11.000
% 5.76/6.03
% 5.76/6.03 Low Water (displace): id=15724, wt=10.000
% 5.76/6.03
% 5.76/6.03 Low Water (displace): id=16516, wt=9.000
% 5.76/6.03
% 5.76/6.03 Low Water (keep): wt=11.000, iters=5701
% 5.76/6.03
% 5.76/6.03 ============================== PROOF =================================
% 5.76/6.03 % SZS status Theorem
% 5.76/6.03 % SZS output start Refutation
% 5.76/6.03
% 5.76/6.03 % Proof 1 at 4.96 (+ 0.07) seconds.
% 5.76/6.03 % Length of proof is 79.
% 5.76/6.03 % Level of proof is 12.
% 5.76/6.03 % Maximum clause weight is 15.000.
% 5.76/6.03 % Given clauses 2107.
% 5.76/6.03
% 5.76/6.03 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].
% 5.76/6.03 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].
% 5.76/6.03 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].
% 5.76/6.03 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].
% 5.76/6.03 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 5.76/6.03 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].
% 5.76/6.03 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].
% 5.76/6.03 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].
% 5.76/6.03 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].
% 5.76/6.03 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].
% 5.76/6.03 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 5.76/6.03 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 5.76/6.03 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 5.76/6.03 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 5.76/6.03 59 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(1)].
% 5.76/6.03 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(2)].
% 5.76/6.03 66 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(3)].
% 5.76/6.03 70 -implies_2 | is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) # label(implies_2) # label(axiom). [clausify(5)].
% 5.76/6.03 74 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(7)].
% 5.76/6.03 78 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(9)].
% 5.76/6.03 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)].
% 5.76/6.03 90 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(15)].
% 5.76/6.03 114 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(27)].
% 5.76/6.03 115 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(114),flip(b)].
% 5.76/6.03 118 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(29)].
% 5.76/6.03 121 op_or # label(hilbert_op_or) # label(axiom). [assumption].
% 5.76/6.03 122 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption].
% 5.76/6.03 124 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption].
% 5.76/6.03 125 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption].
% 5.76/6.03 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption].
% 5.76/6.03 129 and_1 # label(hilbert_and_1) # label(axiom). [assumption].
% 5.76/6.03 131 and_3 # label(hilbert_and_3) # label(axiom). [assumption].
% 5.76/6.03 134 or_3 # label(hilbert_or_3) # label(axiom). [assumption].
% 5.76/6.03 137 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption].
% 5.76/6.03 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption].
% 5.76/6.03 155 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(37)].
% 5.76/6.03 160 axiom_B | -is_a_theorem(implies(c67,necessarily(possibly(c67)))) # label(axiom_B) # label(axiom). [clausify(39)].
% 5.76/6.03 161 -axiom_5 | is_a_theorem(implies(possibly(A),necessarily(possibly(A)))) # label(axiom_5) # label(axiom). [clausify(40)].
% 5.76/6.03 191 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(55)].
% 5.76/6.03 192 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(191),flip(b)].
% 5.76/6.03 199 op_possibly # label(km5_op_possibly) # label(axiom). [assumption].
% 5.76/6.03 202 axiom_M # label(km5_axiom_M) # label(axiom). [assumption].
% 5.76/6.03 203 axiom_5 # label(km5_axiom_5) # label(axiom). [assumption].
% 5.76/6.03 204 -axiom_B # label(km4b_axiom_B) # label(negated_conjecture). [assumption].
% 5.76/6.03 205 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,121)].
% 5.76/6.03 206 not(and(A,not(B))) = implies(A,B). [back_unit_del(118),unit_del(a,122)].
% 5.76/6.03 208 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(59),unit_del(a,124)].
% 5.76/6.03 209 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(66),unit_del(a,125)].
% 5.76/6.03 211 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(70),unit_del(a,127)].
% 5.76/6.03 213 is_a_theorem(implies(and(A,B),A)). [back_unit_del(74),unit_del(a,129)].
% 5.76/6.03 215 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(78),unit_del(a,131)].
% 5.76/6.03 218 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(84),unit_del(a,134)].
% 5.76/6.03 221 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(90),unit_del(a,137)].
% 5.76/6.03 222 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,138)].
% 5.76/6.03 224 not(necessarily(not(A))) = possibly(A). [back_unit_del(192),unit_del(a,199)].
% 5.76/6.03 227 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(155),unit_del(a,202)].
% 5.76/6.03 228 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))). [back_unit_del(161),unit_del(a,203)].
% 5.76/6.03 229 -is_a_theorem(implies(c67,necessarily(possibly(c67)))). [back_unit_del(160),unit_del(a,204)].
% 5.76/6.03 230 implies(not(A),B) = or(A,B). [back_rewrite(205),rewrite([206(4)])].
% 5.76/6.03 234 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(209),rewrite([230(3)])].
% 5.76/6.03 250 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(211,a,208,b)].
% 5.76/6.03 280 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(221,a,208,b)].
% 5.76/6.03 332 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(206(a,1),230(a,1,1)),flip(a)].
% 5.76/6.03 354 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(234,a,208,b)].
% 5.76/6.03 798 is_a_theorem(implies(implies(A,B),implies(or(A,A),B))). [resolve(250,a,218,a)].
% 5.76/6.03 799 is_a_theorem(implies(A,and(A,A))). [resolve(250,a,215,a)].
% 5.76/6.03 3709 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(280,a,799,a)].
% 5.76/6.03 3732 is_a_theorem(implies(implies(necessarily(possibly(A)),possibly(A)),equiv(possibly(A),necessarily(possibly(A))))). [resolve(280,a,228,a)].
% 5.76/6.03 14324 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(or(A,A),B)). [resolve(798,a,208,b)].
% 5.76/6.03 20006 is_a_theorem(equiv(A,and(A,A))). [resolve(3709,a,208,b),unit_del(a,213)].
% 5.76/6.03 20045 and(A,A) = A. [resolve(20006,a,222,a)].
% 5.76/6.03 20056 implies(or(A,A),B) = or(not(A),B). [para(20045(a,1),332(a,1,1)),rewrite([230(4)]),flip(a)].
% 5.76/6.03 20145 -is_a_theorem(implies(A,B)) | is_a_theorem(or(not(A),B)). [back_rewrite(14324),rewrite([20056(4)])].
% 5.76/6.03 21703 is_a_theorem(equiv(possibly(A),necessarily(possibly(A)))). [resolve(3732,a,208,b),unit_del(a,227)].
% 5.76/6.03 21718 necessarily(possibly(A)) = possibly(A). [resolve(21703,a,222,a)].
% 5.76/6.03 21758 -is_a_theorem(implies(c67,possibly(c67))). [back_rewrite(229),rewrite([21718(4)])].
% 5.76/6.03 23755 is_a_theorem(or(not(necessarily(A)),A)). [resolve(20145,a,227,a)].
% 5.76/6.03 23775 is_a_theorem(implies(A,possibly(A))). [resolve(23755,a,354,a),rewrite([224(3)])].
% 5.76/6.03 23776 $F. [resolve(23775,a,21758,a)].
% 5.76/6.03
% 5.76/6.03 % SZS output end Refutation
% 5.76/6.03 ============================== end of proof ==========================
% 5.76/6.03
% 5.76/6.03 ============================== STATISTICS ============================
% 5.76/6.03
% 5.76/6.03 Given=2107. Generated=107465. Kept=23711. proofs=1.
% 5.76/6.03 Usable=1615. Sos=9999. Demods=45. Limbo=10, Disabled=12226. Hints=0.
% 5.76/6.03 Megabytes=22.30.
% 5.76/6.03 User_CPU=4.96, System_CPU=0.07, Wall_clock=5.
% 5.76/6.03
% 5.76/6.03 ============================== end of statistics =====================
% 5.76/6.03
% 5.76/6.03 ============================== end of search =========================
% 5.76/6.03
% 5.76/6.03 THEOREM PROVED
% 5.76/6.03 % SZS status Theorem
% 5.76/6.03
% 5.76/6.03 Exiting with 1 proof.
% 5.76/6.03
% 5.76/6.03 Process 25762 exit (max_proofs) Sun Jul 3 03:33:01 2022
% 5.76/6.03 Prover9 interrupted
%------------------------------------------------------------------------------