TSTP Solution File: LCL523+1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : LCL523+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n010.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 13:47:21 EDT 2022
% Result : Theorem 37.93s 38.23s
% Output : Refutation 37.93s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : LCL523+1 : TPTP v8.1.0. Released v3.3.0.
% 0.06/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.12/0.31 % Computer : n010.cluster.edu
% 0.12/0.31 % Model : x86_64 x86_64
% 0.12/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.31 % Memory : 8042.1875MB
% 0.12/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.31 % CPULimit : 300
% 0.12/0.31 % WCLimit : 600
% 0.12/0.31 % DateTime : Sat Jul 2 10:57:18 EDT 2022
% 0.12/0.32 % CPUTime :
% 0.39/0.98 ============================== Prover9 ===============================
% 0.39/0.98 Prover9 (32) version 2009-11A, November 2009.
% 0.39/0.98 Process 16892 was started by sandbox on n010.cluster.edu,
% 0.39/0.98 Sat Jul 2 10:57:19 2022
% 0.39/0.98 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_16739_n010.cluster.edu".
% 0.39/0.98 ============================== end of head ===========================
% 0.39/0.98
% 0.39/0.98 ============================== INPUT =================================
% 0.39/0.98
% 0.39/0.98 % Reading from file /tmp/Prover9_16739_n010.cluster.edu
% 0.39/0.98
% 0.39/0.98 set(prolog_style_variables).
% 0.39/0.98 set(auto2).
% 0.39/0.98 % set(auto2) -> set(auto).
% 0.39/0.98 % set(auto) -> set(auto_inference).
% 0.39/0.98 % set(auto) -> set(auto_setup).
% 0.39/0.98 % set(auto_setup) -> set(predicate_elim).
% 0.39/0.98 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.39/0.98 % set(auto) -> set(auto_limits).
% 0.39/0.98 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.39/0.98 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.39/0.98 % set(auto) -> set(auto_denials).
% 0.39/0.98 % set(auto) -> set(auto_process).
% 0.39/0.98 % set(auto2) -> assign(new_constants, 1).
% 0.39/0.98 % set(auto2) -> assign(fold_denial_max, 3).
% 0.39/0.98 % set(auto2) -> assign(max_weight, "200.000").
% 0.39/0.98 % set(auto2) -> assign(max_hours, 1).
% 0.39/0.98 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.39/0.98 % set(auto2) -> assign(max_seconds, 0).
% 0.39/0.98 % set(auto2) -> assign(max_minutes, 5).
% 0.39/0.98 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.39/0.98 % set(auto2) -> set(sort_initial_sos).
% 0.39/0.98 % set(auto2) -> assign(sos_limit, -1).
% 0.39/0.98 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.39/0.98 % set(auto2) -> assign(max_megs, 400).
% 0.39/0.98 % set(auto2) -> assign(stats, some).
% 0.39/0.98 % set(auto2) -> clear(echo_input).
% 0.39/0.98 % set(auto2) -> set(quiet).
% 0.39/0.98 % set(auto2) -> clear(print_initial_clauses).
% 0.39/0.98 % set(auto2) -> clear(print_given).
% 0.39/0.98 assign(lrs_ticks,-1).
% 0.39/0.98 assign(sos_limit,10000).
% 0.39/0.98 assign(order,kbo).
% 0.39/0.98 set(lex_order_vars).
% 0.39/0.98 clear(print_given).
% 0.39/0.98
% 0.39/0.98 % formulas(sos). % not echoed (82 formulas)
% 0.39/0.98
% 0.39/0.98 ============================== end of input ==========================
% 0.39/0.98
% 0.39/0.98 % From the command line: assign(max_seconds, 300).
% 0.39/0.98
% 0.39/0.98 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.39/0.98
% 0.39/0.98 % Formulas that are not ordinary clauses:
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 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].
% 0.39/0.98 10 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 11 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 16 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 17 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 20 cn2 <-> (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) # label(cn2) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 21 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 22 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 23 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 24 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 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].
% 0.39/0.98 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 0.39/0.98 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].
% 0.71/1.01 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].
% 0.71/1.01 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].
% 0.71/1.01 44 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption].
% 0.71/1.01 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].
% 0.71/1.01 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].
% 0.71/1.01 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].
% 0.71/1.01 48 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption].
% 0.71/1.01 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].
% 0.71/1.01 50 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption].
% 0.71/1.01 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].
% 0.71/1.01 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].
% 0.71/1.01 53 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption].
% 0.71/1.01 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption].
% 0.71/1.01 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 0.71/1.01 56 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption].
% 0.71/1.01 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].
% 0.71/1.01 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].
% 0.71/1.01
% 0.71/1.01 ============================== end of process non-clausal formulas ===
% 0.71/1.01
% 0.71/1.01 ============================== PROCESS INITIAL CLAUSES ===============
% 0.71/1.01
% 0.71/1.01 ============================== PREDICATE ELIMINATION =================
% 0.71/1.01
% 0.71/1.01 ============================== end predicate elimination =============
% 0.71/1.01
% 0.71/1.01 Auto_denials: (non-Horn, no changes).
% 0.71/1.01
% 0.71/1.01 Term ordering decisions:
% 0.71/1.01 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.
% 0.71/1.01
% 0.71/1.01 ============================== end of process initial clauses ========
% 0.71/1.01
% 0.71/1.01 ============================== CLAUSES FOR SEARCH ====================
% 37.93/38.23
% 37.93/38.23 ============================== end of clauses for search =============
% 37.93/38.23
% 37.93/38.23 ============================== SEARCH ================================
% 37.93/38.23
% 37.93/38.23 % Starting search at 0.05 seconds.
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=26.000, iters=3336
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=22.000, iters=3380
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=20.000, iters=3360
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=19.000, iters=3358
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=18.000, iters=3376
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=17.000, iters=3345
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=16.000, iters=3366
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=15.000, iters=3379
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=14.000, iters=3454
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=13.000, iters=3487
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=12.000, iters=3340
% 37.93/38.23
% 37.93/38.23 Low Water (displace): id=10136, wt=30.000
% 37.93/38.23
% 37.93/38.23 Low Water (displace): id=15713, wt=11.000
% 37.93/38.23
% 37.93/38.23 Low Water (displace): id=15724, wt=10.000
% 37.93/38.23
% 37.93/38.23 Low Water (displace): id=16516, wt=9.000
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=11.000, iters=3341
% 37.93/38.23
% 37.93/38.23 Low Water (displace): id=25209, wt=8.000
% 37.93/38.23
% 37.93/38.23 Low Water (keep): wt=10.000, iters=3335
% 37.93/38.23
% 37.93/38.23 ============================== PROOF =================================
% 37.93/38.23 % SZS status Theorem
% 37.93/38.23 % SZS output start Refutation
% 37.93/38.23
% 37.93/38.23 % Proof 1 at 35.44 (+ 1.82) seconds.
% 37.93/38.23 % Length of proof is 131.
% 37.93/38.23 % Level of proof is 21.
% 37.93/38.23 % Maximum clause weight is 15.000.
% 37.93/38.23 % Given clauses 11350.
% 37.93/38.23
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 37.93/38.23 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 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].
% 37.93/38.23 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 37.93/38.23 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 37.93/38.23 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 37.93/38.23 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 37.93/38.23 59 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(1)].
% 37.93/38.23 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(2)].
% 37.93/38.23 66 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(3)].
% 37.93/38.23 68 -implies_1 | is_a_theorem(implies(A,implies(B,A))) # label(implies_1) # label(axiom). [clausify(4)].
% 37.93/38.23 70 -implies_2 | is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) # label(implies_2) # label(axiom). [clausify(5)].
% 37.93/38.23 72 -implies_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) # label(implies_3) # label(axiom). [clausify(6)].
% 37.93/38.23 74 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(7)].
% 37.93/38.23 76 -and_2 | is_a_theorem(implies(and(A,B),B)) # label(and_2) # label(axiom). [clausify(8)].
% 37.93/38.23 78 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(9)].
% 37.93/38.23 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)].
% 37.93/38.23 90 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(15)].
% 37.93/38.23 114 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(27)].
% 37.93/38.23 115 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(114),flip(b)].
% 37.93/38.23 118 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(29)].
% 37.93/38.23 120 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B) # label(op_equiv) # label(axiom). [clausify(31)].
% 37.93/38.23 121 op_or # label(hilbert_op_or) # label(axiom). [assumption].
% 37.93/38.23 122 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption].
% 37.93/38.23 123 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption].
% 37.93/38.23 124 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption].
% 37.93/38.23 125 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption].
% 37.93/38.23 126 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption].
% 37.93/38.23 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption].
% 37.93/38.23 128 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption].
% 37.93/38.23 129 and_1 # label(hilbert_and_1) # label(axiom). [assumption].
% 37.93/38.23 130 and_2 # label(hilbert_and_2) # label(axiom). [assumption].
% 37.93/38.23 131 and_3 # label(hilbert_and_3) # label(axiom). [assumption].
% 37.93/38.23 134 or_3 # label(hilbert_or_3) # label(axiom). [assumption].
% 37.93/38.23 137 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption].
% 37.93/38.23 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption].
% 37.93/38.23 155 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(37)].
% 37.93/38.23 158 axiom_4 | -is_a_theorem(implies(necessarily(c66),necessarily(necessarily(c66)))) # label(axiom_4) # label(axiom). [clausify(38)].
% 37.93/38.23 161 -axiom_5 | is_a_theorem(implies(possibly(A),necessarily(possibly(A)))) # label(axiom_5) # label(axiom). [clausify(40)].
% 37.93/38.23 191 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(55)].
% 37.93/38.23 192 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(191),flip(b)].
% 37.93/38.23 199 op_possibly # label(km5_op_possibly) # label(axiom). [assumption].
% 37.93/38.23 202 axiom_M # label(km5_axiom_M) # label(axiom). [assumption].
% 37.93/38.23 203 axiom_5 # label(km5_axiom_5) # label(axiom). [assumption].
% 37.93/38.23 204 -axiom_4 # label(km4b_axiom_4) # label(negated_conjecture). [assumption].
% 37.93/38.23 205 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,121)].
% 37.93/38.23 206 not(and(A,not(B))) = implies(A,B). [back_unit_del(118),unit_del(a,122)].
% 37.93/38.23 207 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(120),unit_del(a,123)].
% 37.93/38.23 208 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(59),unit_del(a,124)].
% 37.93/38.23 209 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(66),unit_del(a,125)].
% 37.93/38.23 210 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(68),unit_del(a,126)].
% 37.93/38.23 211 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(70),unit_del(a,127)].
% 37.93/38.23 212 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(72),unit_del(a,128)].
% 37.93/38.23 213 is_a_theorem(implies(and(A,B),A)). [back_unit_del(74),unit_del(a,129)].
% 37.93/38.23 214 is_a_theorem(implies(and(A,B),B)). [back_unit_del(76),unit_del(a,130)].
% 37.93/38.23 215 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(78),unit_del(a,131)].
% 37.93/38.23 218 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(84),unit_del(a,134)].
% 37.93/38.23 221 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(90),unit_del(a,137)].
% 37.93/38.23 222 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,138)].
% 37.93/38.23 224 not(necessarily(not(A))) = possibly(A). [back_unit_del(192),unit_del(a,199)].
% 37.93/38.23 227 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(155),unit_del(a,202)].
% 37.93/38.23 228 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))). [back_unit_del(161),unit_del(a,203)].
% 37.93/38.23 229 -is_a_theorem(implies(necessarily(c66),necessarily(necessarily(c66)))). [back_unit_del(158),unit_del(a,204)].
% 37.93/38.23 230 implies(not(A),B) = or(A,B). [back_rewrite(205),rewrite([206(4)])].
% 37.93/38.23 234 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(209),rewrite([230(3)])].
% 37.93/38.23 247 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(210,a,208,b)].
% 37.93/38.23 250 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(211,a,208,b)].
% 37.93/38.23 253 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(212,a,208,b)].
% 37.93/38.23 271 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(C,B),implies(or(A,C),B))). [resolve(218,a,208,b)].
% 37.93/38.23 280 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(221,a,208,b)].
% 37.93/38.23 298 not(and(A,possibly(B))) = implies(A,necessarily(not(B))). [para(224(a,1),206(a,1,1,2))].
% 37.93/38.23 300 not(necessarily(possibly(A))) = possibly(necessarily(not(A))). [para(224(a,1),224(a,1,1,1))].
% 37.93/38.23 332 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(206(a,1),230(a,1,1)),flip(a)].
% 37.93/38.23 350 or(necessarily(not(A)),B) = implies(possibly(A),B). [para(224(a,1),230(a,1,1)),flip(a)].
% 37.93/38.23 354 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(234,a,208,b)].
% 37.93/38.23 799 is_a_theorem(implies(A,and(A,A))). [resolve(250,a,215,a)].
% 37.93/38.23 801 is_a_theorem(implies(A,A)). [resolve(250,a,210,a)].
% 37.93/38.23 808 is_a_theorem(implies(A,implies(B,B))). [resolve(801,a,247,a)].
% 37.93/38.23 996 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(253,a,210,a)].
% 37.93/38.23 1113 is_a_theorem(implies(implies(implies(A,A),B),implies(C,B))). [resolve(808,a,253,a)].
% 37.93/38.23 2648 is_a_theorem(implies(implies(A,B),implies(or(and(C,B),A),B))). [resolve(271,a,214,a)].
% 37.93/38.23 3708 is_a_theorem(implies(implies(implies(A,A),B),equiv(B,implies(A,A)))). [resolve(280,a,808,a)].
% 37.93/38.23 3709 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(280,a,799,a)].
% 37.93/38.23 3732 is_a_theorem(implies(implies(necessarily(possibly(A)),possibly(A)),equiv(possibly(A),necessarily(possibly(A))))). [resolve(280,a,228,a)].
% 37.93/38.23 14996 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(996,a,208,b)].
% 37.93/38.23 15080 is_a_theorem(implies(implies(implies(A,A),B),B)). [resolve(1113,a,250,a)].
% 37.93/38.23 15089 -is_a_theorem(implies(implies(A,A),B)) | is_a_theorem(B). [resolve(15080,a,208,b)].
% 37.93/38.23 18827 is_a_theorem(implies(or(and(A,B),B),B)). [resolve(2648,a,15089,a)].
% 37.93/38.23 20006 is_a_theorem(equiv(A,and(A,A))). [resolve(3709,a,208,b),unit_del(a,213)].
% 37.93/38.23 20045 and(A,A) = A. [resolve(20006,a,222,a)].
% 37.93/38.23 20051 or(A,A) = not(not(A)). [para(20045(a,1),206(a,1,1)),rewrite([230(4)]),flip(a)].
% 37.93/38.23 20052 equiv(A,A) = implies(A,A). [para(20045(a,1),207(a,1)),flip(a)].
% 37.93/38.23 20054 implies(possibly(A),necessarily(not(A))) = not(possibly(A)). [para(20045(a,1),298(a,1,1)),flip(a)].
% 37.93/38.23 20056 implies(or(A,A),B) = or(not(A),B). [para(20045(a,1),332(a,1,1)),rewrite([230(4)]),flip(a)].
% 37.93/38.23 20060 is_a_theorem(or(not(A),A)). [para(20045(a,1),18827(a,1,1,1)),rewrite([20056(2)])].
% 37.93/38.23 20165 is_a_theorem(implies(A,not(not(A)))). [resolve(20060,a,354,a)].
% 37.93/38.23 21160 possibly(necessarily(or(A,A))) = possibly(necessarily(not(not(A)))). [para(20051(a,2),300(a,2,1,1)),rewrite([300(4)]),flip(a)].
% 37.93/38.23 21703 is_a_theorem(equiv(possibly(A),necessarily(possibly(A)))). [resolve(3732,a,208,b),unit_del(a,227)].
% 37.93/38.23 21718 necessarily(possibly(A)) = possibly(A). [resolve(21703,a,222,a)].
% 37.93/38.23 21757 possibly(necessarily(not(A))) = not(possibly(A)). [back_rewrite(300),rewrite([21718(2)]),flip(a)].
% 37.93/38.23 21763 possibly(necessarily(or(A,A))) = not(possibly(not(A))). [back_rewrite(21160),rewrite([21757(7)])].
% 37.93/38.23 37507 is_a_theorem(implies(A,equiv(A,implies(B,B)))). [resolve(14996,a,3708,a)].
% 37.93/38.23 37627 is_a_theorem(equiv(implies(A,A),implies(B,B))). [resolve(37507,a,15089,a)].
% 37.93/38.23 37633 -is_a_theorem(A) | is_a_theorem(equiv(A,implies(B,B))). [resolve(37507,a,208,b)].
% 37.93/38.23 37758 implies(A,A) = implies(B,B). [resolve(37627,a,222,a)].
% 37.93/38.23 37760 implies(A,A) = c_0. [new_symbol(37758)].
% 37.93/38.23 37777 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(37633),rewrite([37760(2)])].
% 37.93/38.23 37783 is_a_theorem(c_0). [back_rewrite(37627),rewrite([37760(1),37760(2),20052(3),37760(3)])].
% 37.93/38.23 41638 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(37777,a,20165,a)].
% 37.93/38.23 41644 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(37777,a,20060,a)].
% 37.93/38.23 43914 or(not(A),A) = c_0. [resolve(41644,a,222,a),flip(a)].
% 37.93/38.23 44748 implies(A,not(not(A))) = c_0. [resolve(41638,a,222,a),flip(a)].
% 37.93/38.23 44749 equiv(A,not(not(A))) = c_0. [para(44748(a,1),207(a,1,1)),rewrite([230(4),43914(3),20045(3)]),flip(a)].
% 37.93/38.23 44754 not(not(A)) = A. [para(44749(a,1),222(a,1)),unit_del(a,37783)].
% 37.93/38.23 44791 or(A,A) = A. [back_rewrite(20051),rewrite([44754(3)])].
% 37.93/38.23 44838 not(possibly(not(A))) = possibly(necessarily(A)). [back_rewrite(21763),rewrite([44791(1)]),flip(a)].
% 37.93/38.23 45133 not(possibly(A)) = necessarily(not(A)). [para(44791(a,1),350(a,1)),rewrite([20054(6)]),flip(a)].
% 37.93/38.23 45140 possibly(necessarily(A)) = necessarily(A). [back_rewrite(44838),rewrite([45133(3),44754(2)]),flip(a)].
% 37.93/38.23 45211 necessarily(necessarily(A)) = necessarily(A). [para(45140(a,1),21718(a,1,1)),rewrite([45140(4)])].
% 37.93/38.23 45273 $F. [back_rewrite(229),rewrite([45211(5),37760(5)]),unit_del(a,37783)].
% 37.93/38.23
% 37.93/38.23 % SZS output end Refutation
% 37.93/38.23 ============================== end of proof ==========================
% 37.93/38.23
% 37.93/38.23 ============================== STATISTICS ============================
% 37.93/38.23
% 37.93/38.23 Given=11350. Generated=3697959. Kept=45208. proofs=1.
% 37.93/38.23 Usable=4662. Sos=4505. Demods=54. Limbo=62, Disabled=36119. Hints=0.
% 37.93/38.23 Megabytes=38.49.
% 37.93/38.23 User_CPU=35.44, System_CPU=1.82, Wall_clock=37.
% 37.93/38.23
% 37.93/38.23 ============================== end of statistics =====================
% 37.93/38.23
% 37.93/38.23 ============================== end of search =========================
% 37.93/38.23
% 37.93/38.23 THEOREM PROVED
% 37.93/38.23 % SZS status Theorem
% 37.93/38.23
% 37.93/38.23 Exiting with 1 proof.
% 37.93/38.23
% 37.93/38.23 Process 16892 exit (max_proofs) Sat Jul 2 10:57:56 2022
% 37.93/38.23 Prover9 interrupted
%------------------------------------------------------------------------------