0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.12	% Command    : tptp2X_and_run_prover9 %d %s
0.12/0.33	% Computer   : n008.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   : 1200
0.12/0.33	% DateTime   : Tue Jul 13 16:18:59 EDT 2021
0.12/0.33	% CPUTime    : 
0.42/1.02	============================== Prover9 ===============================
0.42/1.02	Prover9 (32) version 2009-11A, November 2009.
0.42/1.02	Process 24299 was started by sandbox2 on n008.cluster.edu,
0.42/1.02	Tue Jul 13 16:19:00 2021
0.42/1.02	The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1200 -f /tmp/Prover9_24146_n008.cluster.edu".
0.42/1.02	============================== end of head ===========================
0.42/1.02	
0.42/1.02	============================== INPUT =================================
0.42/1.02	
0.42/1.02	% Reading from file /tmp/Prover9_24146_n008.cluster.edu
0.42/1.02	
0.42/1.02	set(prolog_style_variables).
0.42/1.02	set(auto2).
0.42/1.02	    % set(auto2) -> set(auto).
0.42/1.02	    % set(auto) -> set(auto_inference).
0.42/1.02	    % set(auto) -> set(auto_setup).
0.42/1.02	    % set(auto_setup) -> set(predicate_elim).
0.42/1.02	    % set(auto_setup) -> assign(eq_defs, unfold).
0.42/1.02	    % set(auto) -> set(auto_limits).
0.42/1.02	    % set(auto_limits) -> assign(max_weight, "100.000").
0.42/1.02	    % set(auto_limits) -> assign(sos_limit, 20000).
0.42/1.02	    % set(auto) -> set(auto_denials).
0.42/1.02	    % set(auto) -> set(auto_process).
0.42/1.02	    % set(auto2) -> assign(new_constants, 1).
0.42/1.02	    % set(auto2) -> assign(fold_denial_max, 3).
0.42/1.02	    % set(auto2) -> assign(max_weight, "200.000").
0.42/1.02	    % set(auto2) -> assign(max_hours, 1).
0.42/1.02	    % assign(max_hours, 1) -> assign(max_seconds, 3600).
0.42/1.02	    % set(auto2) -> assign(max_seconds, 0).
0.42/1.02	    % set(auto2) -> assign(max_minutes, 5).
0.42/1.02	    % assign(max_minutes, 5) -> assign(max_seconds, 300).
0.42/1.02	    % set(auto2) -> set(sort_initial_sos).
0.42/1.02	    % set(auto2) -> assign(sos_limit, -1).
0.42/1.02	    % set(auto2) -> assign(lrs_ticks, 3000).
0.42/1.02	    % set(auto2) -> assign(max_megs, 400).
0.42/1.02	    % set(auto2) -> assign(stats, some).
0.42/1.02	    % set(auto2) -> clear(echo_input).
0.42/1.02	    % set(auto2) -> set(quiet).
0.42/1.02	    % set(auto2) -> clear(print_initial_clauses).
0.42/1.02	    % set(auto2) -> clear(print_given).
0.42/1.02	assign(lrs_ticks,-1).
0.42/1.02	assign(sos_limit,10000).
0.42/1.02	assign(order,kbo).
0.42/1.02	set(lex_order_vars).
0.42/1.02	clear(print_given).
0.42/1.02	
0.42/1.02	% formulas(sos).  % not echoed (82 formulas)
0.42/1.02	
0.42/1.02	============================== end of input ==========================
0.42/1.02	
0.42/1.02	% From the command line: assign(max_seconds, 1200).
0.42/1.02	
0.42/1.02	============================== PROCESS NON-CLAUSAL FORMULAS ==========
0.42/1.03	
0.42/1.03	% Formulas that are not ordinary clauses:
0.42/1.03	1 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].
0.42/1.03	2 (all P all Q all R is_a_theorem(implies(implies(P,Q),implies(implies(Q,R),implies(P,R))))) <-> cn1 # label(cn1) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	4 (all X all Y is_a_theorem(implies(implies(X,implies(X,Y)),implies(X,Y)))) <-> implies_2 # label(implies_2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	5 (all X all Y (is_a_theorem(implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) <-> modus_ponens # label(modus_ponens) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	6 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	7 (all X all Y is_a_theorem(implies(equiv(X,Y),implies(Y,X)))) <-> equivalence_2 # label(equivalence_2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	8 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].
0.42/1.03	9 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].
0.42/1.03	10 (all P is_a_theorem(implies(or(P,P),P))) <-> r1 # label(r1) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	11 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	12 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	13 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].
0.42/1.03	14 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].
0.42/1.03	15 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	17 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	18 (all P all Q all R is_a_theorem(implies(implies(Q,R),implies(or(P,Q),or(P,R))))) <-> r5 # label(r5) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	19 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].
0.42/1.03	20 (all P all Q is_a_theorem(implies(and(P,Q),P))) <-> kn2 # label(kn2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	21 (all X all Y is_a_theorem(implies(implies(not(Y),not(X)),implies(X,Y)))) <-> modus_tollens # label(modus_tollens) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	22 (all X all Y all Z is_a_theorem(implies(implies(X,Z),implies(implies(Y,Z),implies(or(X,Y),Z))))) <-> or_3 # label(or_3) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	23 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	24 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	25 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	26 (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) <-> substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	27 op_implies_or -> (all X all Y implies(X,Y) = or(not(X),Y)) # label(op_implies_or) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	28 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].
0.42/1.03	29 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	30 op_equiv -> (all X all Y and(implies(X,Y),implies(Y,X)) = equiv(X,Y)) # label(op_equiv) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	31 op_and -> (all X all Y not(or(not(X),not(Y))) = and(X,Y)) # label(op_and) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	32 (all P all Q is_a_theorem(strict_implies(possibly(and(P,Q)),and(possibly(P),possibly(Q))))) <-> axiom_s2 # label(axiom_s2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	33 (all X all Y all Z is_a_theorem(strict_implies(and(and(X,Y),Z),and(X,and(Y,Z))))) <-> axiom_m3 # label(axiom_m3) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	34 (all X all Y (is_a_theorem(X) & is_a_theorem(strict_implies(X,Y)) -> is_a_theorem(Y))) <-> modus_ponens_strict_implies # label(modus_ponens_strict_implies) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	35 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	36 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	37 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].
0.42/1.03	38 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	39 (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) <-> axiom_m10 # label(axiom_m10) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	40 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause).  [assumption].
0.42/1.03	41 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].
0.42/1.04	42 adjunction <-> (all X all Y (is_a_theorem(Y) & is_a_theorem(X) -> is_a_theorem(and(X,Y)))) # label(adjunction) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	43 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	44 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].
0.42/1.04	45 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	46 substitution_strict_equiv <-> (all X all Y (is_a_theorem(strict_equiv(X,Y)) -> Y = X)) # label(substitution_strict_equiv) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	47 (all X all Y all Z is_a_theorem(implies(and(necessarily(implies(X,Y)),necessarily(implies(Y,Z))),necessarily(implies(X,Z))))) <-> axiom_s1 # label(axiom_s1) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	48 (all X all Y all Z is_a_theorem(strict_implies(and(strict_implies(X,Y),strict_implies(Y,Z)),strict_implies(X,Z)))) <-> axiom_m5 # label(axiom_m5) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	49 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	50 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	51 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	52 axiom_m8 <-> (all P all Q is_a_theorem(strict_implies(strict_implies(P,Q),strict_implies(possibly(P),possibly(Q))))) # label(axiom_m8) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	53 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	54 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].
0.42/1.04	55 op_strict_equiv -> (all X all Y and(strict_implies(X,Y),strict_implies(Y,X)) = strict_equiv(X,Y)) # label(op_strict_equiv) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	56 op_strict_implies -> (all X all Y necessarily(implies(X,Y)) = strict_implies(X,Y)) # label(op_strict_implies) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	57 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	58 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause).  [assumption].
0.42/1.04	
0.42/1.04	============================== end of process non-clausal formulas ===
0.42/1.04	
0.42/1.04	============================== PROCESS INITIAL CLAUSES ===============
0.42/1.04	
0.42/1.04	============================== PREDICATE ELIMINATION =================
0.42/1.04	
0.42/1.04	============================== end predicate elimination =============
0.42/1.04	
0.42/1.04	Auto_denials:  (non-Horn, no changes).
0.42/1.04	
0.42/1.04	Term ordering decisions:
0.42/1.04	Function symbol KB weights:  c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. c16=1. c17=1. c18=1. c19=1. c20=1. c21=1. c22=1. c23=1. c24=1. c25=1. c26=1. c27=1. c28=1. c29=1. c30=1. c31=1. c32=1. c33=1. c34=1. c35=1. c36=1. c37=1. c38=1. c39=1. c40=1. c41=1. c42=1. c43=1. c44=1. c45=1. c46=1. c47=1. c48=1. c49=1. c50=1. c51=1. c52=1. c53=1. c54=1. c55=1. c56=1. c57=1. c58=1. c59=1. c60=1. c61=1. c62=1. c63=1. c64=1. c65=1. c66=1. c67=1. c68=1. c69=1. c70=1. c71=1. c72=1. c73=1. c74=1. c75=1. c76=1. c77=1. c78=1. c79=1. c80=1. c81=1. c82=1. c83=1. c84=1. c85=1. c86=1. c87=1. c88=1. c89=1. c90=1. c91=1. c92=1. c93=1. c94=1. implies=1. and=1. strict_implies=1. or=1. equiv=1. strict_equiv=1. necessarily=1. possibly=1. not=1.
0.42/1.04	
0.42/1.04	============================== end of process initial clauses ========
0.42/1.04	
0.42/1.04	============================== CLAUSES FOR SEARCH ====================
24.76/25.05	
24.76/25.05	============================== end of clauses for search =============
24.76/25.05	
24.76/25.05	============================== SEARCH ================================
24.76/25.05	
24.76/25.05	% Starting search at 0.03 seconds.
24.76/25.05	
24.76/25.05	Low Water (keep): wt=28.000, iters=3357
24.76/25.05	
24.76/25.05	Low Water (keep): wt=26.000, iters=3375
24.76/25.05	
24.76/25.05	Low Water (keep): wt=22.000, iters=3436
24.76/25.05	
24.76/25.05	Low Water (keep): wt=20.000, iters=3383
24.76/25.05	
24.76/25.05	Low Water (keep): wt=19.000, iters=3392
24.76/25.05	
24.76/25.05	Low Water (keep): wt=18.000, iters=3336
24.76/25.05	
24.76/25.05	Low Water (keep): wt=17.000, iters=3343
24.76/25.05	
24.76/25.05	Low Water (keep): wt=16.000, iters=3345
24.76/25.05	
24.76/25.05	Low Water (keep): wt=15.000, iters=3352
24.76/25.05	
24.76/25.05	Low Water (keep): wt=13.000, iters=3349
24.76/25.05	
24.76/25.05	Low Water (keep): wt=12.000, iters=3393
24.76/25.05	
24.76/25.05	Low Water (keep): wt=11.000, iters=3337
24.76/25.05	
24.76/25.05	Low Water (displace): id=8256, wt=32.000
24.76/25.05	
24.76/25.05	Low Water (displace): id=6286, wt=30.000
24.76/25.05	
24.76/25.05	Low Water (displace): id=14279, wt=10.000
24.76/25.05	
24.76/25.05	Low Water (displace): id=19963, wt=9.000
24.76/25.05	
24.76/25.05	============================== PROOF =================================
24.76/25.05	% SZS status Theorem
24.76/25.05	% SZS output start Refutation
24.76/25.05	
24.76/25.05	% Proof 1 at 22.83 (+ 1.20) seconds.
24.76/25.05	% Length of proof is 80.
24.76/25.05	% Level of proof is 10.
24.76/25.05	% Maximum clause weight is 15.000.
24.76/25.05	% Given clauses 9161.
24.76/25.05	
24.76/25.05	3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	4 (all X all Y is_a_theorem(implies(implies(X,implies(X,Y)),implies(X,Y)))) <-> implies_2 # label(implies_2) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	5 (all X all Y (is_a_theorem(implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) <-> modus_ponens # label(modus_ponens) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	8 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].
24.76/25.05	13 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].
24.76/25.05	21 (all X all Y is_a_theorem(implies(implies(not(Y),not(X)),implies(X,Y)))) <-> modus_tollens # label(modus_tollens) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	22 (all X all Y all Z is_a_theorem(implies(implies(X,Z),implies(implies(Y,Z),implies(or(X,Y),Z))))) <-> or_3 # label(or_3) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	26 (all X all Y (is_a_theorem(equiv(X,Y)) -> X = Y)) <-> substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	28 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].
24.76/25.05	29 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	35 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	38 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	53 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	58 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause).  [assumption].
24.76/25.05	64 is_a_theorem(implies(and(A,B),A)) | -and_1 # label(and_1) # label(axiom).  [clausify(3)].
24.76/25.05	66 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) | -implies_2 # label(implies_2) # label(axiom).  [clausify(4)].
24.76/25.05	70 -is_a_theorem(implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B) | -modus_ponens # label(modus_ponens) # label(axiom).  [clausify(5)].
24.76/25.05	75 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom).  [clausify(8)].
24.76/25.05	85 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom).  [clausify(13)].
24.76/25.05	102 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) | -modus_tollens # label(modus_tollens) # label(axiom).  [clausify(21)].
24.76/25.05	104 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))) | -or_3 # label(or_3) # label(axiom).  [clausify(22)].
24.76/25.05	113 -is_a_theorem(equiv(A,B)) | B = A | -substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom).  [clausify(26)].
24.76/25.05	115 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom).  [clausify(28)].
24.76/25.05	116 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom).  [clausify(29)].
24.76/25.05	117 -op_or | not(and(not(A),not(B))) = or(A,B).  [copy(116),flip(b)].
24.76/25.05	122 and_1 # label(hilbert_and_1) # label(axiom).  [assumption].
24.76/25.05	123 equivalence_3 # label(hilbert_equivalence_3) # label(axiom).  [assumption].
24.76/25.05	125 op_implies_and # label(hilbert_op_implies_and) # label(axiom).  [assumption].
24.76/25.05	127 implies_2 # label(hilbert_implies_2) # label(axiom).  [assumption].
24.76/25.05	128 or_3 # label(hilbert_or_3) # label(axiom).  [assumption].
24.76/25.05	129 op_or # label(hilbert_op_or) # label(axiom).  [assumption].
24.76/25.05	130 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom).  [assumption].
24.76/25.05	133 and_3 # label(hilbert_and_3) # label(axiom).  [assumption].
24.76/25.05	134 modus_ponens # label(hilbert_modus_ponens) # label(axiom).  [assumption].
24.76/25.05	135 modus_tollens # label(hilbert_modus_tollens) # label(axiom).  [assumption].
24.76/25.05	147 -is_a_theorem(implies(c63,necessarily(possibly(c63)))) | axiom_B # label(axiom_B) # label(axiom).  [clausify(35)].
24.76/25.05	154 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))) | -axiom_5 # label(axiom_5) # label(axiom).  [clausify(38)].
24.76/25.05	187 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom).  [clausify(53)].
24.76/25.05	196 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom).  [clausify(58)].
24.76/25.05	197 -op_possibly | not(necessarily(not(A))) = possibly(A).  [copy(196),flip(b)].
24.76/25.05	198 axiom_M # label(km5_axiom_M) # label(axiom).  [assumption].
24.76/25.05	199 axiom_5 # label(km5_axiom_5) # label(axiom).  [assumption].
24.76/25.05	200 op_possibly # label(km5_op_possibly) # label(axiom).  [assumption].
24.76/25.05	203 -axiom_B # label(km4b_axiom_B) # label(negated_conjecture).  [assumption].
24.76/25.05	205 is_a_theorem(implies(and(A,B),A)).  [back_unit_del(64),unit_del(b,122)].
24.76/25.05	206 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))).  [back_unit_del(75),unit_del(a,123)].
24.76/25.05	208 not(and(A,not(B))) = implies(A,B).  [back_unit_del(115),unit_del(a,125)].
24.76/25.05	210 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))).  [back_unit_del(66),unit_del(b,127)].
24.76/25.05	211 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))).  [back_unit_del(104),unit_del(b,128)].
24.76/25.05	212 implies(not(A),B) = or(A,B).  [back_unit_del(117),rewrite([208(5)]),unit_del(a,129)].
24.76/25.05	213 -is_a_theorem(equiv(A,B)) | B = A.  [back_unit_del(113),unit_del(c,130)].
24.76/25.05	216 is_a_theorem(implies(A,implies(B,and(A,B)))).  [back_unit_del(85),unit_del(a,133)].
24.76/25.05	217 -is_a_theorem(implies(A,B)) | -is_a_theorem(A) | is_a_theorem(B).  [back_unit_del(70),unit_del(d,134)].
24.76/25.05	218 is_a_theorem(implies(or(A,not(B)),implies(B,A))).  [back_unit_del(102),rewrite([212(3)]),unit_del(b,135)].
24.76/25.05	223 is_a_theorem(implies(necessarily(A),A)).  [back_unit_del(187),unit_del(a,198)].
24.76/25.05	224 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))).  [back_unit_del(154),unit_del(b,199)].
24.76/25.05	225 not(necessarily(not(A))) = possibly(A).  [back_unit_del(197),unit_del(a,200)].
24.76/25.05	228 -is_a_theorem(implies(c63,necessarily(possibly(c63)))).  [back_unit_del(147),unit_del(b,203)].
24.76/25.05	261 or(and(A,not(B)),C) = implies(implies(A,B),C).  [para(208(a,1),212(a,1,1)),flip(a)].
24.76/25.05	281 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)).  [resolve(217,a,210,a)].
24.76/25.05	284 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))).  [resolve(217,a,206,a)].
24.76/25.05	288 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)).  [resolve(218,a,217,a)].
24.76/25.05	328 or(necessarily(not(A)),B) = implies(possibly(A),B).  [para(225(a,1),212(a,1,1)),flip(a)].
24.76/25.05	2041 implies(implies(A,necessarily(not(B))),C) = or(and(A,possibly(B)),C).  [para(225(a,1),261(a,1,1,2)),flip(a)].
24.76/25.05	5338 is_a_theorem(implies(A,and(A,A))).  [resolve(281,a,216,a)].
24.76/25.05	5339 is_a_theorem(implies(implies(A,B),implies(or(A,A),B))).  [resolve(281,a,211,a)].
24.76/25.05	5434 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))).  [resolve(284,a,5338,a)].
24.76/25.05	5456 is_a_theorem(implies(implies(A,necessarily(A)),equiv(necessarily(A),A))).  [resolve(284,a,223,a)].
24.76/25.05	15232 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(or(A,A),B)).  [resolve(5339,a,217,a)].
24.76/25.05	15255 -is_a_theorem(implies(A,necessarily(A))) | is_a_theorem(equiv(necessarily(A),A)).  [resolve(5456,a,217,a)].
24.76/25.05	17974 is_a_theorem(implies(or(necessarily(A),necessarily(A)),A)).  [resolve(15232,a,223,a)].
24.76/25.05	17993 is_a_theorem(or(and(possibly(A),possibly(A)),not(A))).  [para(328(a,1),17974(a,1,1)),rewrite([2041(6)])].
24.76/25.05	19268 is_a_theorem(implies(A,and(possibly(A),possibly(A)))).  [resolve(17993,a,288,a)].
24.76/25.05	22300 is_a_theorem(equiv(necessarily(possibly(A)),possibly(A))).  [resolve(15255,a,224,a)].
24.76/25.05	22308 necessarily(possibly(A)) = possibly(A).  [resolve(22300,a,213,a),flip(a)].
24.76/25.05	22321 -is_a_theorem(implies(c63,possibly(c63))).  [back_rewrite(228),rewrite([22308(4)])].
24.76/25.05	22960 is_a_theorem(equiv(A,and(A,A))).  [resolve(5434,a,217,a),unit_del(a,205)].
24.76/25.05	22989 and(A,A) = A.  [resolve(22960,a,213,a)].
24.76/25.05	23007 is_a_theorem(implies(A,possibly(A))).  [back_rewrite(19268),rewrite([22989(3)])].
24.76/25.05	23008 $F.  [resolve(23007,a,22321,a)].
24.76/25.05	
24.76/25.05	% SZS output end Refutation
24.76/25.05	============================== end of proof ==========================
24.76/25.05	
24.76/25.05	============================== STATISTICS ============================
24.76/25.05	
24.76/25.05	Given=9161. Generated=2254828. Kept=22944. proofs=1.
24.76/25.05	Usable=8311. Sos=8440. Demods=32. Limbo=18, Disabled=6314. Hints=0.
24.76/25.05	Megabytes=27.68.
24.76/25.05	User_CPU=22.83, System_CPU=1.20, Wall_clock=24.
24.76/25.05	
24.76/25.05	============================== end of statistics =====================
24.76/25.05	
24.76/25.05	============================== end of search =========================
24.76/25.05	
24.76/25.05	THEOREM PROVED
24.76/25.05	% SZS status Theorem
24.76/25.05	
24.76/25.05	Exiting with 1 proof.
24.76/25.05	
24.76/25.05	Process 24299 exit (max_proofs) Tue Jul 13 16:19:24 2021
24.76/25.05	Prover9 interrupted
24.76/25.05	EOF