TSTP Solution File: LCL541+1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : LCL541+1 : TPTP v8.1.0. Bugfixed v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n027.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 13:47:28 EDT 2022
% Result : Theorem 73.61s 73.91s
% Output : Refutation 73.63s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : LCL541+1 : TPTP v8.1.0. Bugfixed v4.0.0.
% 0.11/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.11/0.33 % Computer : n027.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 600
% 0.11/0.33 % DateTime : Sat Jul 2 17:00:09 EDT 2022
% 0.11/0.33 % CPUTime :
% 0.65/0.92 ============================== Prover9 ===============================
% 0.65/0.92 Prover9 (32) version 2009-11A, November 2009.
% 0.65/0.92 Process 25249 was started by sandbox on n027.cluster.edu,
% 0.65/0.92 Sat Jul 2 17:00:09 2022
% 0.65/0.92 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_25096_n027.cluster.edu".
% 0.65/0.92 ============================== end of head ===========================
% 0.65/0.92
% 0.65/0.92 ============================== INPUT =================================
% 0.65/0.92
% 0.65/0.92 % Reading from file /tmp/Prover9_25096_n027.cluster.edu
% 0.65/0.92
% 0.65/0.92 set(prolog_style_variables).
% 0.65/0.92 set(auto2).
% 0.65/0.92 % set(auto2) -> set(auto).
% 0.65/0.92 % set(auto) -> set(auto_inference).
% 0.65/0.92 % set(auto) -> set(auto_setup).
% 0.65/0.92 % set(auto_setup) -> set(predicate_elim).
% 0.65/0.92 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.65/0.92 % set(auto) -> set(auto_limits).
% 0.65/0.92 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.65/0.92 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.65/0.92 % set(auto) -> set(auto_denials).
% 0.65/0.92 % set(auto) -> set(auto_process).
% 0.65/0.92 % set(auto2) -> assign(new_constants, 1).
% 0.65/0.92 % set(auto2) -> assign(fold_denial_max, 3).
% 0.65/0.92 % set(auto2) -> assign(max_weight, "200.000").
% 0.65/0.92 % set(auto2) -> assign(max_hours, 1).
% 0.65/0.92 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.65/0.92 % set(auto2) -> assign(max_seconds, 0).
% 0.65/0.92 % set(auto2) -> assign(max_minutes, 5).
% 0.65/0.92 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.65/0.92 % set(auto2) -> set(sort_initial_sos).
% 0.65/0.92 % set(auto2) -> assign(sos_limit, -1).
% 0.65/0.92 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.65/0.92 % set(auto2) -> assign(max_megs, 400).
% 0.65/0.92 % set(auto2) -> assign(stats, some).
% 0.65/0.92 % set(auto2) -> clear(echo_input).
% 0.65/0.92 % set(auto2) -> set(quiet).
% 0.65/0.92 % set(auto2) -> clear(print_initial_clauses).
% 0.65/0.92 % set(auto2) -> clear(print_given).
% 0.65/0.92 assign(lrs_ticks,-1).
% 0.65/0.92 assign(sos_limit,10000).
% 0.65/0.92 assign(order,kbo).
% 0.65/0.92 set(lex_order_vars).
% 0.65/0.92 clear(print_given).
% 0.65/0.92
% 0.65/0.92 % formulas(sos). % not echoed (89 formulas)
% 0.65/0.92
% 0.65/0.92 ============================== end of input ==========================
% 0.65/0.92
% 0.65/0.92 % From the command line: assign(max_seconds, 300).
% 0.65/0.92
% 0.65/0.92 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.65/0.92
% 0.65/0.92 % Formulas that are not ordinary clauses:
% 0.65/0.92 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].
% 0.65/0.92 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].
% 0.65/0.92 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].
% 0.65/0.92 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.92 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].
% 0.65/0.92 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].
% 0.65/0.92 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.92 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.92 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].
% 0.65/0.92 10 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.92 11 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.92 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].
% 0.65/0.92 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].
% 0.65/0.92 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].
% 0.65/0.93 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].
% 0.65/0.93 16 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 17 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 20 cn2 <-> (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) # label(cn2) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 21 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 22 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 23 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 24 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 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].
% 0.65/0.93 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.93 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].
% 0.65/0.96 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].
% 0.65/0.96 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].
% 0.65/0.96 44 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.96 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].
% 0.65/0.96 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].
% 0.65/0.96 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].
% 0.65/0.96 48 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.96 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].
% 0.65/0.96 50 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.96 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].
% 0.65/0.96 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].
% 0.65/0.96 53 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.96 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.96 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.96 56 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption].
% 0.65/0.96 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].
% 0.65/0.96 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].
% 0.65/0.96
% 0.65/0.96 ============================== end of process non-clausal formulas ===
% 0.65/0.96
% 0.65/0.96 ============================== PROCESS INITIAL CLAUSES ===============
% 0.65/0.96
% 0.65/0.96 ============================== PREDICATE ELIMINATION =================
% 0.65/0.96
% 0.65/0.96 ============================== end predicate elimination =============
% 0.65/0.96
% 0.65/0.96 Auto_denials: (non-Horn, no changes).
% 0.65/0.96
% 0.65/0.96 Term ordering decisions:
% 0.65/0.96 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.
% 0.65/0.96
% 0.65/0.96 ============================== end of process initial clauses ========
% 0.65/0.96
% 0.65/0.96 ============================== CLAUSES FOR SEARCH ====================
% 73.61/73.91
% 73.61/73.91 ============================== end of clauses for search =============
% 73.61/73.91
% 73.61/73.91 ============================== SEARCH ================================
% 73.61/73.91
% 73.61/73.91 % Starting search at 0.05 seconds.
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=22.000, iters=3405
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=20.000, iters=3364
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=19.000, iters=3480
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=18.000, iters=3375
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=17.000, iters=3345
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=16.000, iters=3343
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=15.000, iters=3346
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=14.000, iters=3333
% 73.61/73.91
% 73.61/73.91 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 68 (0.00 of 1.50 sec).
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=13.000, iters=3336
% 73.61/73.91
% 73.61/73.91 Low Water (displace): id=5257, wt=29.000
% 73.61/73.91
% 73.61/73.91 Low Water (displace): id=5248, wt=28.000
% 73.61/73.91
% 73.61/73.91 Low Water (displace): id=5531, wt=27.000
% 73.61/73.91
% 73.61/73.91 Low Water (displace): id=10931, wt=9.000
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=11.000, iters=3385
% 73.61/73.91
% 73.61/73.91 Low Water (keep): wt=10.000, iters=3335
% 73.61/73.91
% 73.61/73.91 ============================== PROOF =================================
% 73.61/73.91 % SZS status Theorem
% 73.61/73.91 % SZS output start Refutation
% 73.61/73.91
% 73.61/73.91 % Proof 1 at 68.56 (+ 4.43) seconds.
% 73.61/73.91 % Length of proof is 156.
% 73.61/73.91 % Level of proof is 30.
% 73.61/73.91 % Maximum clause weight is 16.000.
% 73.61/73.91 % Given clauses 21724.
% 73.61/73.91
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 73.61/73.91 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].
% 73.61/73.91 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].
% 73.61/73.91 59 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(1)].
% 73.61/73.91 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(2)].
% 73.61/73.91 66 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(3)].
% 73.61/73.91 68 -implies_1 | is_a_theorem(implies(A,implies(B,A))) # label(implies_1) # label(axiom). [clausify(4)].
% 73.61/73.91 70 -implies_2 | is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) # label(implies_2) # label(axiom). [clausify(5)].
% 73.61/73.91 72 -implies_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) # label(implies_3) # label(axiom). [clausify(6)].
% 73.61/73.91 74 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(7)].
% 73.61/73.91 78 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(9)].
% 73.61/73.91 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)].
% 73.61/73.91 88 -equivalence_2 | is_a_theorem(implies(equiv(A,B),implies(B,A))) # label(equivalence_2) # label(axiom). [clausify(14)].
% 73.61/73.91 90 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(15)].
% 73.61/73.91 114 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(27)].
% 73.61/73.91 115 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(114),flip(b)].
% 73.61/73.91 118 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(29)].
% 73.61/73.91 120 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B) # label(op_equiv) # label(axiom). [clausify(31)].
% 73.61/73.91 121 op_or # label(hilbert_op_or) # label(axiom). [assumption].
% 73.61/73.91 122 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption].
% 73.61/73.91 123 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption].
% 73.61/73.91 124 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption].
% 73.61/73.91 125 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption].
% 73.61/73.91 126 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption].
% 73.61/73.91 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption].
% 73.61/73.91 128 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption].
% 73.61/73.91 129 and_1 # label(hilbert_and_1) # label(axiom). [assumption].
% 73.61/73.91 131 and_3 # label(hilbert_and_3) # label(axiom). [assumption].
% 73.61/73.91 134 or_3 # label(hilbert_or_3) # label(axiom). [assumption].
% 73.61/73.91 136 equivalence_2 # label(hilbert_equivalence_2) # label(axiom). [assumption].
% 73.61/73.91 137 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption].
% 73.61/73.91 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption].
% 73.61/73.91 139 -necessitation | -is_a_theorem(A) | is_a_theorem(necessarily(A)) # label(necessitation) # label(axiom). [clausify(32)].
% 73.61/73.91 172 axiom_m1 | -is_a_theorem(strict_implies(and(c77,c78),and(c78,c77))) # label(axiom_m1) # label(axiom). [clausify(45)].
% 73.61/73.91 195 -op_strict_implies | strict_implies(A,B) = necessarily(implies(A,B)) # label(op_strict_implies) # label(axiom). [clausify(57)].
% 73.61/73.91 196 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B). [copy(195),flip(b)].
% 73.61/73.91 200 necessitation # label(km4b_necessitation) # label(axiom). [assumption].
% 73.61/73.91 206 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption].
% 73.61/73.91 208 -axiom_m1 # label(s1_0_axiom_m1) # label(negated_conjecture). [assumption].
% 73.61/73.91 209 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,121)].
% 73.61/73.91 210 not(and(A,not(B))) = implies(A,B). [back_unit_del(118),unit_del(a,122)].
% 73.61/73.91 211 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(120),unit_del(a,123)].
% 73.61/73.91 212 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(59),unit_del(a,124)].
% 73.61/73.91 213 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(66),unit_del(a,125)].
% 73.61/73.91 214 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(68),unit_del(a,126)].
% 73.61/73.91 215 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(70),unit_del(a,127)].
% 73.61/73.91 216 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(72),unit_del(a,128)].
% 73.61/73.91 217 is_a_theorem(implies(and(A,B),A)). [back_unit_del(74),unit_del(a,129)].
% 73.61/73.91 219 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(78),unit_del(a,131)].
% 73.61/73.91 222 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(84),unit_del(a,134)].
% 73.61/73.91 224 is_a_theorem(implies(equiv(A,B),implies(B,A))). [back_unit_del(88),unit_del(a,136)].
% 73.61/73.91 225 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(90),unit_del(a,137)].
% 73.61/73.91 226 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,138)].
% 73.61/73.91 229 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(139),unit_del(a,200)].
% 73.61/73.91 234 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(196),unit_del(a,206)].
% 73.61/73.91 236 -is_a_theorem(strict_implies(and(c77,c78),and(c78,c77))). [back_unit_del(172),unit_del(a,208)].
% 73.61/73.91 237 implies(not(A),B) = or(A,B). [back_rewrite(209),rewrite([210(4)])].
% 73.61/73.91 244 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(213),rewrite([237(3)])].
% 73.61/73.91 257 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(214,a,212,b)].
% 73.61/73.91 260 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(215,a,212,b)].
% 73.61/73.91 263 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(216,a,212,b)].
% 73.61/73.91 281 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(C,B),implies(or(A,C),B))). [resolve(222,a,212,b)].
% 73.61/73.91 290 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(225,a,212,b)].
% 73.61/73.91 349 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(210(a,1),237(a,1,1)),flip(a)].
% 73.61/73.91 379 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(244,a,212,b)].
% 73.61/73.91 384 is_a_theorem(implies(or(A,not(not(B))),or(B,A))). [para(237(a,1),244(a,1,2))].
% 73.61/73.91 875 is_a_theorem(implies(A,and(A,A))). [resolve(260,a,219,a)].
% 73.61/73.91 877 is_a_theorem(implies(A,A)). [resolve(260,a,214,a)].
% 73.61/73.91 884 is_a_theorem(implies(A,implies(B,B))). [resolve(877,a,257,a)].
% 73.61/73.91 891 is_a_theorem(strict_implies(A,A)). [resolve(877,a,229,a),rewrite([234(2)])].
% 73.61/73.91 900 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(891,a,257,a)].
% 73.61/73.91 1102 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(263,a,214,a)].
% 73.61/73.91 1662 is_a_theorem(implies(implies(implies(A,A),B),implies(C,B))). [resolve(884,a,263,a)].
% 73.61/73.91 1817 is_a_theorem(implies(implies(strict_implies(A,A),B),implies(C,B))). [resolve(900,a,263,a)].
% 73.61/73.91 2549 is_a_theorem(implies(implies(A,B),implies(or(and(B,C),A),B))). [resolve(281,a,217,a)].
% 73.61/73.91 4056 is_a_theorem(implies(implies(strict_implies(A,A),B),equiv(B,strict_implies(A,A)))). [resolve(290,a,900,a)].
% 73.61/73.91 4057 is_a_theorem(implies(implies(implies(A,A),B),equiv(B,implies(A,A)))). [resolve(290,a,884,a)].
% 73.61/73.91 4058 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(290,a,875,a)].
% 73.61/73.91 4071 is_a_theorem(implies(implies(implies(A,B),equiv(B,A)),equiv(equiv(B,A),implies(A,B)))). [resolve(290,a,224,a)].
% 73.61/73.91 4081 is_a_theorem(implies(implies(implies(A,B),B),equiv(B,implies(A,B)))). [resolve(290,a,214,a)].
% 73.61/73.91 16163 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(1102,a,212,b)].
% 73.61/73.91 16287 is_a_theorem(implies(implies(implies(A,A),B),B)). [resolve(1662,a,260,a)].
% 73.61/73.91 16892 is_a_theorem(implies(implies(strict_implies(A,A),B),B)). [resolve(1817,a,260,a)].
% 73.61/73.91 18255 -is_a_theorem(implies(implies(A,A),B)) | is_a_theorem(B). [resolve(16287,a,212,b)].
% 73.61/73.91 18266 -is_a_theorem(implies(strict_implies(A,A),B)) | is_a_theorem(B). [resolve(16892,a,212,b)].
% 73.61/73.91 18299 is_a_theorem(implies(or(and(A,B),A),A)). [resolve(18255,a,2549,a)].
% 73.61/73.91 18413 is_a_theorem(equiv(A,and(A,A))). [resolve(4058,a,212,b),unit_del(a,217)].
% 73.61/73.91 18430 and(A,A) = A. [resolve(18413,a,226,a)].
% 73.61/73.91 18433 or(A,A) = not(not(A)). [para(18430(a,1),210(a,1,1)),rewrite([237(4)]),flip(a)].
% 73.61/73.91 18434 equiv(A,A) = implies(A,A). [para(18430(a,1),211(a,1)),flip(a)].
% 73.61/73.91 18438 implies(or(A,A),B) = or(not(A),B). [para(18430(a,1),349(a,1,1)),rewrite([237(4)]),flip(a)].
% 73.61/73.91 18440 is_a_theorem(or(not(A),A)). [para(18430(a,1),18299(a,1,1,1)),rewrite([18438(2)])].
% 73.61/73.91 18474 is_a_theorem(implies(A,not(not(A)))). [resolve(18440,a,379,a)].
% 73.61/73.91 33607 is_a_theorem(implies(A,equiv(A,implies(B,A)))). [resolve(16163,a,4081,a)].
% 73.61/73.91 33612 is_a_theorem(implies(A,equiv(A,implies(B,B)))). [resolve(16163,a,4057,a)].
% 73.63/73.91 33613 is_a_theorem(implies(A,equiv(A,strict_implies(B,B)))). [resolve(16163,a,4056,a)].
% 73.63/73.91 33624 -is_a_theorem(A) | is_a_theorem(equiv(A,implies(B,A))). [resolve(33607,a,212,b)].
% 73.63/73.91 33663 is_a_theorem(equiv(strict_implies(A,A),implies(B,B))). [resolve(33612,a,18266,a)].
% 73.63/73.91 33681 is_a_theorem(equiv(implies(A,A),strict_implies(B,B))). [resolve(33613,a,18255,a)].
% 73.63/73.91 33684 -is_a_theorem(A) | is_a_theorem(equiv(A,strict_implies(B,B))). [resolve(33613,a,212,b)].
% 73.63/73.91 33745 strict_implies(A,A) = implies(B,B). [resolve(33663,a,226,a),flip(a)].
% 73.63/73.91 33747 strict_implies(A,A) = c_0. [new_symbol(33745)].
% 73.63/73.91 33749 implies(A,A) = c_0. [back_rewrite(33745),rewrite([33747(1)]),flip(a)].
% 73.63/73.91 33753 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(33684),rewrite([33747(2)])].
% 73.63/73.91 33756 is_a_theorem(c_0). [back_rewrite(33681),rewrite([33749(1),33747(2),18434(3),33749(3)])].
% 73.63/73.91 33767 is_a_theorem(implies(A,equiv(A,c_0))). [back_rewrite(33613),rewrite([33747(1)])].
% 73.63/73.91 34653 is_a_theorem(implies(implies(c_0,A),A)). [back_rewrite(16892),rewrite([33747(1)])].
% 73.63/73.91 35160 is_a_theorem(equiv(c_0,implies(A,c_0))). [resolve(33756,a,33624,a)].
% 73.63/73.91 36988 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(33753,a,18474,a)].
% 73.63/73.91 36995 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(33753,a,18440,a)].
% 73.63/73.91 37088 is_a_theorem(equiv(implies(A,implies(B,A)),c_0)). [resolve(33753,a,214,a)].
% 73.63/73.91 37115 is_a_theorem(equiv(implies(A,equiv(A,c_0)),c_0)). [resolve(33767,a,33753,a)].
% 73.63/73.91 37584 is_a_theorem(equiv(implies(implies(c_0,A),A),c_0)). [resolve(34653,a,33753,a)].
% 73.63/73.91 38608 implies(A,c_0) = c_0. [resolve(35160,a,226,a)].
% 73.63/73.91 38609 and(c_0,implies(c_0,A)) = equiv(A,c_0). [para(38608(a,1),211(a,1,1))].
% 73.63/73.91 39610 or(not(A),A) = c_0. [resolve(36995,a,226,a),flip(a)].
% 73.63/73.91 40357 implies(A,not(not(A))) = c_0. [resolve(36988,a,226,a),flip(a)].
% 73.63/73.91 40358 equiv(A,not(not(A))) = c_0. [para(40357(a,1),211(a,1,1)),rewrite([237(4),39610(3),18430(3)]),flip(a)].
% 73.63/73.91 40364 not(not(A)) = A. [para(40358(a,1),226(a,1)),unit_del(a,33756)].
% 73.63/73.91 40386 or(A,A) = A. [back_rewrite(18433),rewrite([40364(3)])].
% 73.63/73.91 40393 is_a_theorem(implies(or(A,B),or(B,A))). [back_rewrite(384),rewrite([40364(2)])].
% 73.63/73.91 40422 or(not(A),B) = implies(A,B). [back_rewrite(18438),rewrite([40386(1)]),flip(a)].
% 73.63/73.91 40552 not(implies(A,B)) = and(A,not(B)). [para(210(a,1),40364(a,1,1))].
% 73.63/73.91 45060 implies(A,implies(B,A)) = c_0. [resolve(37088,a,226,a),flip(a)].
% 73.63/73.91 45141 implies(A,equiv(A,c_0)) = c_0. [resolve(37115,a,226,a),flip(a)].
% 73.63/73.91 45660 implies(implies(c_0,A),A) = c_0. [resolve(37584,a,226,a),flip(a)].
% 73.63/73.91 45661 equiv(implies(c_0,A),A) = c_0. [para(45660(a,1),211(a,1,1)),rewrite([45060(4),18430(3)]),flip(a)].
% 73.63/73.91 45682 implies(c_0,A) = A. [para(45661(a,1),226(a,1)),flip(b),unit_del(a,33756)].
% 73.63/73.91 45700 equiv(A,c_0) = and(c_0,A). [back_rewrite(38609),rewrite([45682(3)]),flip(a)].
% 73.63/73.91 45820 implies(A,and(c_0,A)) = c_0. [back_rewrite(45141),rewrite([45700(2)])].
% 73.63/73.91 45911 is_a_theorem(equiv(and(c_0,A),A)). [para(45682(a,1),4071(a,1,1,1)),rewrite([45700(2),45820(3),45700(3),45682(5),45682(5)])].
% 73.63/73.91 46359 and(c_0,A) = A. [resolve(45911,a,226,a),flip(a)].
% 73.63/73.91 46362 equiv(A,c_0) = A. [back_rewrite(45700),rewrite([46359(4)])].
% 73.63/73.91 46384 -is_a_theorem(A) | c_0 = A. [para(46362(a,1),226(a,1))].
% 73.63/73.91 52626 implies(or(A,B),or(B,A)) = c_0. [resolve(40393,a,46384,a),flip(a)].
% 73.63/73.91 52629 is_a_theorem(equiv(or(A,B),or(B,A))). [resolve(40393,a,290,a),rewrite([52626(3),45682(5)])].
% 73.63/73.91 54899 or(A,B) = or(B,A). [resolve(52629,a,226,a)].
% 73.63/73.91 55148 or(A,not(B)) = implies(B,A). [back_rewrite(40422),rewrite([54899(2)])].
% 73.63/73.91 57381 and(not(A),not(B)) = not(or(A,B)). [para(237(a,1),40552(a,1,1)),flip(a)].
% 73.63/73.91 64759 and(not(A),B) = and(B,not(A)). [para(40364(a,1),57381(a,1,2)),rewrite([55148(4),40552(4)])].
% 73.63/73.91 64805 and(A,B) = and(B,A). [para(40364(a,1),64759(a,1,1)),rewrite([40364(3)])].
% 73.63/73.91 65102 $F. [back_rewrite(236),rewrite([64805(6),33747(7)]),unit_del(a,33756)].
% 73.63/73.91
% 73.63/73.91 % SZS output end Refutation
% 73.63/73.91 ============================== end of proof ==========================
% 73.63/73.91
% 73.63/73.91 ============================== STATISTICS ============================
% 73.63/73.91
% 73.63/73.91 Given=21724. Generated=9185267. Kept=65037. proofs=1.
% 73.63/73.91 Usable=4706. Sos=6315. Demods=1019. Limbo=297, Disabled=53866. Hints=0.
% 73.63/73.91 Megabytes=48.59.
% 73.63/73.91 User_CPU=68.56, System_CPU=4.43, Wall_clock=73.
% 73.63/73.91
% 73.63/73.91 ============================== end of statistics =====================
% 73.63/73.91
% 73.63/73.91 ============================== end of search =========================
% 73.63/73.91
% 73.63/73.91 THEOREM PROVED
% 73.63/73.91 % SZS status Theorem
% 73.63/73.91
% 73.63/73.91 Exiting with 1 proof.
% 73.63/73.91
% 73.63/73.91 Process 25249 exit (max_proofs) Sat Jul 2 17:01:22 2022
% 73.63/73.91 Prover9 interrupted
%------------------------------------------------------------------------------