TSTP Solution File: LCL546+1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : LCL546+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n020.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 13:47:29 EDT 2022
% Result : Theorem 6.03s 6.31s
% Output : Refutation 6.03s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.11 % Problem : LCL546+1 : TPTP v8.1.0. Released v3.3.0.
% 0.07/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.12/0.33 % Computer : n020.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 13:37:30 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.44/1.03 ============================== Prover9 ===============================
% 0.44/1.03 Prover9 (32) version 2009-11A, November 2009.
% 0.44/1.03 Process 12632 was started by sandbox2 on n020.cluster.edu,
% 0.44/1.03 Sun Jul 3 13:37:30 2022
% 0.44/1.03 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_12478_n020.cluster.edu".
% 0.44/1.03 ============================== end of head ===========================
% 0.44/1.03
% 0.44/1.03 ============================== INPUT =================================
% 0.44/1.03
% 0.44/1.03 % Reading from file /tmp/Prover9_12478_n020.cluster.edu
% 0.44/1.03
% 0.44/1.03 set(prolog_style_variables).
% 0.44/1.03 set(auto2).
% 0.44/1.03 % set(auto2) -> set(auto).
% 0.44/1.03 % set(auto) -> set(auto_inference).
% 0.44/1.03 % set(auto) -> set(auto_setup).
% 0.44/1.03 % set(auto_setup) -> set(predicate_elim).
% 0.44/1.03 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.44/1.03 % set(auto) -> set(auto_limits).
% 0.44/1.03 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.44/1.03 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.44/1.03 % set(auto) -> set(auto_denials).
% 0.44/1.03 % set(auto) -> set(auto_process).
% 0.44/1.03 % set(auto2) -> assign(new_constants, 1).
% 0.44/1.03 % set(auto2) -> assign(fold_denial_max, 3).
% 0.44/1.03 % set(auto2) -> assign(max_weight, "200.000").
% 0.44/1.03 % set(auto2) -> assign(max_hours, 1).
% 0.44/1.03 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.44/1.03 % set(auto2) -> assign(max_seconds, 0).
% 0.44/1.03 % set(auto2) -> assign(max_minutes, 5).
% 0.44/1.03 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.44/1.03 % set(auto2) -> set(sort_initial_sos).
% 0.44/1.03 % set(auto2) -> assign(sos_limit, -1).
% 0.44/1.03 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.44/1.03 % set(auto2) -> assign(max_megs, 400).
% 0.44/1.03 % set(auto2) -> assign(stats, some).
% 0.44/1.03 % set(auto2) -> clear(echo_input).
% 0.44/1.03 % set(auto2) -> set(quiet).
% 0.44/1.03 % set(auto2) -> clear(print_initial_clauses).
% 0.44/1.03 % set(auto2) -> clear(print_given).
% 0.44/1.03 assign(lrs_ticks,-1).
% 0.44/1.03 assign(sos_limit,10000).
% 0.44/1.03 assign(order,kbo).
% 0.44/1.03 set(lex_order_vars).
% 0.44/1.03 clear(print_given).
% 0.44/1.03
% 0.44/1.03 % formulas(sos). % not echoed (89 formulas)
% 0.44/1.03
% 0.44/1.03 ============================== end of input ==========================
% 0.44/1.03
% 0.44/1.03 % From the command line: assign(max_seconds, 300).
% 0.44/1.03
% 0.44/1.03 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.44/1.03
% 0.44/1.03 % Formulas that are not ordinary clauses:
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 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].
% 0.44/1.03 10 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 11 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 16 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 17 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 20 cn2 <-> (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) # label(cn2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 21 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 22 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 23 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 24 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 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].
% 0.44/1.03 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.03 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].
% 0.44/1.06 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].
% 0.44/1.06 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].
% 0.44/1.06 44 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption].
% 0.44/1.06 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].
% 0.44/1.06 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].
% 0.44/1.06 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].
% 0.77/1.06 48 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption].
% 0.77/1.06 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].
% 0.77/1.06 50 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption].
% 0.77/1.06 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].
% 0.77/1.06 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].
% 0.77/1.06 53 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption].
% 0.77/1.06 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption].
% 0.77/1.06 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 0.77/1.06 56 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption].
% 0.77/1.06 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].
% 0.77/1.06 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].
% 0.77/1.06
% 0.77/1.06 ============================== end of process non-clausal formulas ===
% 0.77/1.06
% 0.77/1.06 ============================== PROCESS INITIAL CLAUSES ===============
% 0.77/1.06
% 0.77/1.06 ============================== PREDICATE ELIMINATION =================
% 0.77/1.06
% 0.77/1.06 ============================== end predicate elimination =============
% 0.77/1.06
% 0.77/1.06 Auto_denials: (non-Horn, no changes).
% 0.77/1.06
% 0.77/1.06 Term ordering decisions:
% 0.77/1.06 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.
% 0.77/1.06
% 0.77/1.06 ============================== end of process initial clauses ========
% 0.77/1.06
% 0.77/1.06 ============================== CLAUSES FOR SEARCH ====================
% 6.03/6.31
% 6.03/6.31 ============================== end of clauses for search =============
% 6.03/6.31
% 6.03/6.31 ============================== SEARCH ================================
% 6.03/6.31
% 6.03/6.31 % Starting search at 0.05 seconds.
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=22.000, iters=3405
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=20.000, iters=3364
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=19.000, iters=3480
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=18.000, iters=3375
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=17.000, iters=3345
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=16.000, iters=3343
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=15.000, iters=3346
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=14.000, iters=3333
% 6.03/6.31
% 6.03/6.31 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 68 (0.00 of 1.53 sec).
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=13.000, iters=3336
% 6.03/6.31
% 6.03/6.31 Low Water (displace): id=5257, wt=29.000
% 6.03/6.31
% 6.03/6.31 Low Water (displace): id=5248, wt=28.000
% 6.03/6.31
% 6.03/6.31 Low Water (displace): id=5531, wt=27.000
% 6.03/6.31
% 6.03/6.31 Low Water (displace): id=10931, wt=9.000
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=11.000, iters=3385
% 6.03/6.31
% 6.03/6.31 Low Water (keep): wt=10.000, iters=3335
% 6.03/6.31
% 6.03/6.31 ============================== PROOF =================================
% 6.03/6.31 % SZS status Theorem
% 6.03/6.31 % SZS output start Refutation
% 6.03/6.31
% 6.03/6.31 % Proof 1 at 5.15 (+ 0.15) seconds.
% 6.03/6.31 % Length of proof is 81.
% 6.03/6.31 % Level of proof is 13.
% 6.03/6.31 % Maximum clause weight is 15.000.
% 6.03/6.31 % Given clauses 5016.
% 6.03/6.31
% 6.03/6.31 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].
% 6.03/6.31 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].
% 6.03/6.31 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].
% 6.03/6.31 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].
% 6.03/6.31 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 6.03/6.31 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].
% 6.03/6.31 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].
% 6.03/6.31 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].
% 6.03/6.31 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].
% 6.03/6.31 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].
% 6.03/6.31 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 6.03/6.31 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 6.03/6.31 50 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption].
% 6.03/6.31 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 6.03/6.31 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].
% 6.03/6.31 59 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(1)].
% 6.03/6.31 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(2)].
% 6.03/6.31 66 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(3)].
% 6.03/6.31 70 -implies_2 | is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) # label(implies_2) # label(axiom). [clausify(5)].
% 6.03/6.31 74 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(7)].
% 6.03/6.31 78 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(9)].
% 6.03/6.31 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)].
% 6.03/6.31 90 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(15)].
% 6.03/6.31 114 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(27)].
% 6.03/6.31 115 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(114),flip(b)].
% 6.03/6.31 118 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(29)].
% 6.03/6.31 121 op_or # label(hilbert_op_or) # label(axiom). [assumption].
% 6.03/6.31 122 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption].
% 6.03/6.31 124 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption].
% 6.03/6.31 125 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption].
% 6.03/6.31 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption].
% 6.03/6.31 129 and_1 # label(hilbert_and_1) # label(axiom). [assumption].
% 6.03/6.31 131 and_3 # label(hilbert_and_3) # label(axiom). [assumption].
% 6.03/6.31 134 or_3 # label(hilbert_or_3) # label(axiom). [assumption].
% 6.03/6.31 137 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption].
% 6.03/6.31 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption].
% 6.03/6.31 139 -necessitation | -is_a_theorem(A) | is_a_theorem(necessarily(A)) # label(necessitation) # label(axiom). [clausify(32)].
% 6.03/6.31 155 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(37)].
% 6.03/6.31 182 axiom_m6 | -is_a_theorem(strict_implies(c88,possibly(c88))) # label(axiom_m6) # label(axiom). [clausify(50)].
% 6.03/6.31 191 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(55)].
% 6.03/6.31 192 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(191),flip(b)].
% 6.03/6.31 195 -op_strict_implies | strict_implies(A,B) = necessarily(implies(A,B)) # label(op_strict_implies) # label(axiom). [clausify(57)].
% 6.03/6.31 196 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B). [copy(195),flip(b)].
% 6.03/6.31 199 op_possibly # label(km4b_op_possibly) # label(axiom). [assumption].
% 6.03/6.31 200 necessitation # label(km4b_necessitation) # label(axiom). [assumption].
% 6.03/6.31 202 axiom_M # label(km4b_axiom_M) # label(axiom). [assumption].
% 6.03/6.31 206 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption].
% 6.03/6.31 208 -axiom_m6 # label(s1_0_m6s3m9b_axiom_m6) # label(negated_conjecture). [assumption].
% 6.03/6.31 209 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,121)].
% 6.03/6.31 210 not(and(A,not(B))) = implies(A,B). [back_unit_del(118),unit_del(a,122)].
% 6.03/6.31 212 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(59),unit_del(a,124)].
% 6.03/6.31 213 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(66),unit_del(a,125)].
% 6.03/6.31 215 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(70),unit_del(a,127)].
% 6.03/6.31 217 is_a_theorem(implies(and(A,B),A)). [back_unit_del(74),unit_del(a,129)].
% 6.03/6.31 219 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(78),unit_del(a,131)].
% 6.03/6.31 222 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(84),unit_del(a,134)].
% 6.03/6.31 225 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(90),unit_del(a,137)].
% 6.03/6.31 226 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,138)].
% 6.03/6.31 228 not(necessarily(not(A))) = possibly(A). [back_unit_del(192),unit_del(a,199)].
% 6.03/6.31 229 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(139),unit_del(a,200)].
% 6.03/6.31 231 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(155),unit_del(a,202)].
% 6.03/6.31 234 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(196),unit_del(a,206)].
% 6.03/6.31 236 -is_a_theorem(strict_implies(c88,possibly(c88))). [back_unit_del(182),unit_del(a,208)].
% 6.03/6.31 237 implies(not(A),B) = or(A,B). [back_rewrite(209),rewrite([210(4)])].
% 6.03/6.31 244 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(213),rewrite([237(3)])].
% 6.03/6.31 260 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(215,a,212,b)].
% 6.03/6.31 290 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(225,a,212,b)].
% 6.03/6.31 349 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(210(a,1),237(a,1,1)),flip(a)].
% 6.03/6.31 379 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(244,a,212,b)].
% 6.03/6.31 874 is_a_theorem(implies(implies(A,B),implies(or(A,A),B))). [resolve(260,a,222,a)].
% 6.03/6.31 875 is_a_theorem(implies(A,and(A,A))). [resolve(260,a,219,a)].
% 6.03/6.31 4058 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(290,a,875,a)].
% 6.03/6.31 15807 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(or(A,A),B)). [resolve(874,a,212,b)].
% 6.03/6.31 18413 is_a_theorem(equiv(A,and(A,A))). [resolve(4058,a,212,b),unit_del(a,217)].
% 6.03/6.31 18430 and(A,A) = A. [resolve(18413,a,226,a)].
% 6.03/6.31 18438 implies(or(A,A),B) = or(not(A),B). [para(18430(a,1),349(a,1,1)),rewrite([237(4)]),flip(a)].
% 6.03/6.31 18465 -is_a_theorem(implies(A,B)) | is_a_theorem(or(not(A),B)). [back_rewrite(15807),rewrite([18438(4)])].
% 6.03/6.31 24056 is_a_theorem(or(not(necessarily(A)),A)). [resolve(18465,a,231,a)].
% 6.03/6.31 24128 is_a_theorem(implies(A,possibly(A))). [resolve(24056,a,379,a),rewrite([228(3)])].
% 6.03/6.31 24175 is_a_theorem(strict_implies(A,possibly(A))). [resolve(24128,a,229,a),rewrite([234(3)])].
% 6.03/6.31 24176 $F. [resolve(24175,a,236,a)].
% 6.03/6.31
% 6.03/6.31 % SZS output end Refutation
% 6.03/6.31 ============================== end of proof ==========================
% 6.03/6.31
% 6.03/6.31 ============================== STATISTICS ============================
% 6.03/6.31
% 6.03/6.31 Given=5016. Generated=329688. Kept=24111. proofs=1.
% 6.03/6.31 Usable=4020. Sos=8519. Demods=37. Limbo=37, Disabled=11681. Hints=0.
% 6.03/6.31 Megabytes=23.85.
% 6.03/6.31 User_CPU=5.15, System_CPU=0.15, Wall_clock=6.
% 6.03/6.31
% 6.03/6.31 ============================== end of statistics =====================
% 6.03/6.31
% 6.03/6.31 ============================== end of search =========================
% 6.03/6.31
% 6.03/6.31 THEOREM PROVED
% 6.03/6.31 % SZS status Theorem
% 6.03/6.31
% 6.03/6.31 Exiting with 1 proof.
% 6.03/6.31
% 6.03/6.31 Process 12632 exit (max_proofs) Sun Jul 3 13:37:36 2022
% 6.03/6.31 Prover9 interrupted
%------------------------------------------------------------------------------