0.06/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.12	% Command    : tptp2X_and_run_prover9 %d %s
0.11/0.32	% Computer   : n024.cluster.edu
0.11/0.32	% Model      : x86_64 x86_64
0.11/0.32	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.32	% Memory     : 8042.1875MB
0.11/0.32	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.32	% CPULimit   : 960
0.11/0.32	% DateTime   : Thu Jul  2 08:26:01 EDT 2020
0.11/0.32	% CPUTime    : 
0.76/1.10	============================== Prover9 ===============================
0.76/1.10	Prover9 (32) version 2009-11A, November 2009.
0.76/1.10	Process 643 was started by sandbox2 on n024.cluster.edu,
0.76/1.10	Thu Jul  2 08:26:02 2020
0.76/1.10	The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_457_n024.cluster.edu".
0.76/1.10	============================== end of head ===========================
0.76/1.10	
0.76/1.10	============================== INPUT =================================
0.76/1.10	
0.76/1.10	% Reading from file /tmp/Prover9_457_n024.cluster.edu
0.76/1.10	
0.76/1.10	set(prolog_style_variables).
0.76/1.10	set(auto2).
0.76/1.10	    % set(auto2) -> set(auto).
0.76/1.10	    % set(auto) -> set(auto_inference).
0.76/1.10	    % set(auto) -> set(auto_setup).
0.76/1.10	    % set(auto_setup) -> set(predicate_elim).
0.76/1.10	    % set(auto_setup) -> assign(eq_defs, unfold).
0.76/1.10	    % set(auto) -> set(auto_limits).
0.76/1.10	    % set(auto_limits) -> assign(max_weight, "100.000").
0.76/1.10	    % set(auto_limits) -> assign(sos_limit, 20000).
0.76/1.10	    % set(auto) -> set(auto_denials).
0.76/1.10	    % set(auto) -> set(auto_process).
0.76/1.10	    % set(auto2) -> assign(new_constants, 1).
0.76/1.10	    % set(auto2) -> assign(fold_denial_max, 3).
0.76/1.10	    % set(auto2) -> assign(max_weight, "200.000").
0.76/1.10	    % set(auto2) -> assign(max_hours, 1).
0.76/1.10	    % assign(max_hours, 1) -> assign(max_seconds, 3600).
0.76/1.10	    % set(auto2) -> assign(max_seconds, 0).
0.76/1.10	    % set(auto2) -> assign(max_minutes, 5).
0.76/1.10	    % assign(max_minutes, 5) -> assign(max_seconds, 300).
0.76/1.10	    % set(auto2) -> set(sort_initial_sos).
0.76/1.10	    % set(auto2) -> assign(sos_limit, -1).
0.76/1.10	    % set(auto2) -> assign(lrs_ticks, 3000).
0.76/1.10	    % set(auto2) -> assign(max_megs, 400).
0.76/1.10	    % set(auto2) -> assign(stats, some).
0.76/1.10	    % set(auto2) -> clear(echo_input).
0.76/1.10	    % set(auto2) -> set(quiet).
0.76/1.10	    % set(auto2) -> clear(print_initial_clauses).
0.76/1.10	    % set(auto2) -> clear(print_given).
0.76/1.10	assign(lrs_ticks,-1).
0.76/1.10	assign(sos_limit,10000).
0.76/1.10	assign(order,kbo).
0.76/1.10	set(lex_order_vars).
0.76/1.10	clear(print_given).
0.76/1.10	
0.76/1.10	% formulas(sos).  % not echoed (88 formulas)
0.76/1.10	
0.76/1.10	============================== end of input ==========================
0.76/1.10	
0.76/1.10	% From the command line: assign(max_seconds, 960).
0.76/1.10	
0.76/1.10	============================== PROCESS NON-CLAUSAL FORMULAS ==========
0.76/1.10	
0.76/1.10	% Formulas that are not ordinary clauses:
0.76/1.10	1 (all X all Y is_a_theorem(implies(equiv(X,Y),implies(X,Y)))) <-> equivalence_1 # label(equivalence_1) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	2 (all X all Y is_a_theorem(implies(Y,or(X,Y)))) <-> or_2 # label(or_2) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	3 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	4 (all X all Y is_a_theorem(implies(X,implies(Y,and(X,Y))))) <-> and_3 # label(and_3) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	5 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].
0.76/1.10	6 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	7 (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].
0.76/1.10	8 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	9 cn2 <-> (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) # label(cn2) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	10 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	11 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	12 (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].
0.76/1.10	13 (all X all Y is_a_theorem(implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y))))) <-> equivalence_3 # label(equivalence_3) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	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].
0.76/1.10	15 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	16 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].
0.76/1.10	17 (all X all Y is_a_theorem(implies(X,or(X,Y)))) <-> or_1 # label(or_1) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	18 (all P is_a_theorem(implies(implies(not(P),P),P))) <-> cn3 # label(cn3) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	19 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	20 (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].
0.76/1.10	21 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	22 (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].
0.76/1.10	23 (all P all Q all R is_a_theorem(implies(or(P,or(Q,R)),or(Q,or(P,R))))) <-> r4 # label(r4) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	24 (all X all Y (is_a_theorem(X) & is_a_theorem(implies(X,Y)) -> is_a_theorem(Y))) <-> modus_ponens # label(modus_ponens) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	25 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].
0.76/1.10	26 (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].
0.76/1.10	27 op_and -> (all X all Y and(X,Y) = not(or(not(X),not(Y)))) # label(op_and) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	28 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].
0.76/1.10	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].
0.76/1.10	30 op_implies_and -> (all X all Y not(and(X,not(Y))) = implies(X,Y)) # label(op_implies_and) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	31 op_implies_or -> (all X all Y or(not(X),Y) = implies(X,Y)) # label(op_implies_or) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	32 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].
0.76/1.10	33 (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_4 # label(axiom_4) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	34 (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].
0.76/1.10	35 (all X is_a_theorem(implies(necessarily(X),X))) <-> axiom_M # label(axiom_M) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	36 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].
0.76/1.10	37 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].
0.76/1.10	38 (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_s4 # label(axiom_s4) # label(axiom) # label(non_clause).  [assumption].
0.76/1.10	39 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].
0.76/1.10	40 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].
0.76/1.10	41 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	42 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	43 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	44 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].
0.76/1.12	45 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	46 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].
0.76/1.12	47 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].
0.76/1.12	48 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	49 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	50 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	51 (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].
0.76/1.12	52 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].
0.76/1.12	53 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	54 modus_ponens_strict_implies <-> (all X all Y (is_a_theorem(strict_implies(X,Y)) & is_a_theorem(X) -> is_a_theorem(Y))) # label(modus_ponens_strict_implies) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	55 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	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].
0.76/1.12	57 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause).  [assumption].
0.76/1.12	58 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].
0.76/1.12	
0.76/1.12	============================== end of process non-clausal formulas ===
0.76/1.12	
0.76/1.12	============================== PROCESS INITIAL CLAUSES ===============
0.76/1.12	
0.76/1.12	============================== PREDICATE ELIMINATION =================
0.76/1.12	
0.76/1.12	============================== end predicate elimination =============
0.76/1.12	
0.76/1.12	Auto_denials:  (non-Horn, no changes).
0.76/1.12	
0.76/1.12	Term ordering decisions:
0.76/1.12	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.
0.76/1.12	
0.76/1.12	============================== end of process initial clauses ========
0.76/1.12	
0.76/1.12	============================== CLAUSES FOR SEARCH ====================
0.76/1.12	
0.76/1.12	============================== end of clauses for search =============
6.27/6.58	
6.27/6.58	============================== SEARCH ================================
6.27/6.58	
6.27/6.58	% Starting search at 0.04 seconds.
6.27/6.58	
6.27/6.58	Low Water (keep): wt=26.000, iters=3368
6.27/6.58	
6.27/6.58	Low Water (keep): wt=25.000, iters=3365
6.27/6.58	
6.27/6.58	Low Water (keep): wt=24.000, iters=3352
6.27/6.58	
6.27/6.58	Low Water (keep): wt=22.000, iters=3343
6.27/6.58	
6.27/6.58	Low Water (keep): wt=20.000, iters=3348
6.27/6.58	
6.27/6.58	Low Water (keep): wt=19.000, iters=3390
6.27/6.58	
6.27/6.58	Low Water (keep): wt=18.000, iters=3342
6.27/6.58	
6.27/6.58	Low Water (keep): wt=17.000, iters=3351
6.27/6.58	
6.27/6.58	Low Water (keep): wt=16.000, iters=3336
6.27/6.58	
6.27/6.58	Low Water (keep): wt=15.000, iters=3354
6.27/6.58	
6.27/6.58	Low Water (keep): wt=14.000, iters=3391
6.27/6.58	
6.27/6.58	Low Water (keep): wt=13.000, iters=3391
6.27/6.58	
6.27/6.58	Low Water (keep): wt=12.000, iters=3335
6.27/6.58	
6.27/6.58	Low Water (displace): id=15105, wt=32.000
6.27/6.58	
6.27/6.58	Low Water (displace): id=20486, wt=11.000
6.27/6.58	
6.27/6.58	Low Water (displace): id=20487, wt=10.000
6.27/6.58	
6.27/6.58	Low Water (displace): id=20636, wt=9.000
6.27/6.58	
6.27/6.58	============================== PROOF =================================
6.27/6.58	% SZS status Theorem
6.27/6.58	% SZS output start Refutation
6.27/6.58	
6.27/6.58	% Proof 1 at 5.44 (+ 0.05) seconds.
6.27/6.58	% Length of proof is 88.
6.27/6.58	% Level of proof is 13.
6.27/6.58	% Maximum clause weight is 14.000.
6.27/6.58	% Given clauses 1320.
6.27/6.58	
6.27/6.58	4 (all X all Y is_a_theorem(implies(X,implies(Y,and(X,Y))))) <-> and_3 # label(and_3) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	7 (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].
6.27/6.58	12 (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].
6.27/6.58	13 (all X all Y is_a_theorem(implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y))))) <-> equivalence_3 # label(equivalence_3) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	17 (all X all Y is_a_theorem(implies(X,or(X,Y)))) <-> or_1 # label(or_1) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	21 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	24 (all X all Y (is_a_theorem(X) & is_a_theorem(implies(X,Y)) -> is_a_theorem(Y))) <-> modus_ponens # label(modus_ponens) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	25 implies_2 <-> (all X all Y is_a_theorem(implies(implies(X,implies(X,Y)),implies(X,Y)))) # label(implies_2) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	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].
6.27/6.58	30 op_implies_and -> (all X all Y not(and(X,not(Y))) = implies(X,Y)) # label(op_implies_and) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	35 (all X is_a_theorem(implies(necessarily(X),X))) <-> axiom_M # label(axiom_M) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	43 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	45 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	53 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	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].
6.27/6.58	57 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause).  [assumption].
6.27/6.58	66 is_a_theorem(implies(A,implies(B,and(A,B)))) | -and_3 # label(and_3) # label(axiom).  [clausify(4)].
6.27/6.58	72 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) | -modus_tollens # label(modus_tollens) # label(axiom).  [clausify(7)].
6.27/6.58	83 -is_a_theorem(equiv(A,B)) | B = A | -substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom).  [clausify(12)].
6.27/6.58	85 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) | -equivalence_3 # label(equivalence_3) # label(axiom).  [clausify(13)].
6.27/6.58	93 is_a_theorem(implies(A,or(A,B))) | -or_1 # label(or_1) # label(axiom).  [clausify(17)].
6.27/6.58	101 is_a_theorem(implies(and(A,B),A)) | -and_1 # label(and_1) # label(axiom).  [clausify(21)].
6.27/6.58	109 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) | -modus_ponens # label(modus_ponens) # label(axiom).  [clausify(24)].
6.27/6.58	110 -implies_2 | is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) # label(implies_2) # label(axiom).  [clausify(25)].
6.27/6.58	116 -op_or | not(and(not(A),not(B))) = or(A,B) # label(op_or) # label(axiom).  [clausify(29)].
6.27/6.58	117 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom).  [clausify(30)].
6.27/6.58	119 op_implies_and # label(hilbert_op_implies_and) # label(axiom).  [assumption].
6.27/6.58	120 equivalence_3 # label(hilbert_equivalence_3) # label(axiom).  [assumption].
6.27/6.58	121 modus_tollens # label(hilbert_modus_tollens) # label(axiom).  [assumption].
6.27/6.58	122 implies_2 # label(hilbert_implies_2) # label(axiom).  [assumption].
6.27/6.58	124 and_1 # label(hilbert_and_1) # label(axiom).  [assumption].
6.27/6.58	127 or_1 # label(hilbert_or_1) # label(axiom).  [assumption].
6.27/6.58	129 and_3 # label(hilbert_and_3) # label(axiom).  [assumption].
6.27/6.58	132 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom).  [assumption].
6.27/6.58	133 modus_ponens # label(hilbert_modus_ponens) # label(axiom).  [assumption].
6.27/6.58	134 op_or # label(hilbert_op_or) # label(axiom).  [assumption].
6.27/6.58	144 is_a_theorem(implies(necessarily(A),A)) | -axiom_M # label(axiom_M) # label(axiom).  [clausify(35)].
6.27/6.58	162 -necessitation | -is_a_theorem(A) | is_a_theorem(necessarily(A)) # label(necessitation) # label(axiom).  [clausify(43)].
6.27/6.58	167 -is_a_theorem(strict_implies(possibly(possibly(c77)),possibly(c77))) | axiom_m9 # label(axiom_m9) # label(axiom).  [clausify(45)].
6.27/6.58	184 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))) | -axiom_5 # label(axiom_5) # label(axiom).  [clausify(53)].
6.27/6.58	191 -op_strict_implies | strict_implies(A,B) = necessarily(implies(A,B)) # label(op_strict_implies) # label(axiom).  [clausify(56)].
6.27/6.58	192 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B).  [copy(191),flip(b)].
6.27/6.58	193 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom).  [clausify(57)].
6.27/6.58	194 -op_possibly | not(necessarily(not(A))) = possibly(A).  [copy(193),flip(b)].
6.27/6.58	197 axiom_M # label(km5_axiom_M) # label(axiom).  [assumption].
6.27/6.58	198 necessitation # label(km5_necessitation) # label(axiom).  [assumption].
6.27/6.58	200 op_possibly # label(km5_op_possibly) # label(axiom).  [assumption].
6.27/6.58	201 axiom_5 # label(km5_axiom_5) # label(axiom).  [assumption].
6.27/6.58	204 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom).  [assumption].
6.27/6.58	205 -axiom_m9 # label(s1_0_m6s3m9b_axiom_m9) # label(negated_conjecture).  [assumption].
6.27/6.58	206 not(and(A,not(B))) = implies(A,B).  [back_unit_del(117),unit_del(a,119)].
6.27/6.58	207 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))).  [back_unit_del(85),unit_del(b,120)].
6.27/6.58	208 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))).  [back_unit_del(72),unit_del(b,121)].
6.27/6.58	209 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))).  [back_unit_del(110),unit_del(a,122)].
6.27/6.58	211 is_a_theorem(implies(and(A,B),A)).  [back_unit_del(101),unit_del(b,124)].
6.27/6.58	214 is_a_theorem(implies(A,or(A,B))).  [back_unit_del(93),unit_del(b,127)].
6.27/6.58	216 is_a_theorem(implies(A,implies(B,and(A,B)))).  [back_unit_del(66),unit_del(b,129)].
6.27/6.58	219 -is_a_theorem(equiv(A,B)) | B = A.  [back_unit_del(83),unit_del(c,132)].
6.27/6.58	220 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B).  [back_unit_del(109),unit_del(d,133)].
6.27/6.58	221 implies(not(A),B) = or(A,B).  [back_unit_del(116),rewrite([206(5)]),unit_del(a,134)].
6.27/6.58	225 is_a_theorem(implies(necessarily(A),A)).  [back_unit_del(144),unit_del(b,197)].
6.27/6.58	226 -is_a_theorem(A) | is_a_theorem(necessarily(A)).  [back_unit_del(162),unit_del(a,198)].
6.27/6.58	228 not(necessarily(not(A))) = possibly(A).  [back_unit_del(194),unit_del(a,200)].
6.27/6.58	229 is_a_theorem(implies(possibly(A),necessarily(possibly(A)))).  [back_unit_del(184),unit_del(b,201)].
6.27/6.58	231 necessarily(implies(A,B)) = strict_implies(A,B).  [back_unit_del(192),unit_del(a,204)].
6.27/6.59	232 -is_a_theorem(strict_implies(possibly(possibly(c77)),possibly(c77))).  [back_unit_del(167),unit_del(b,205)].
6.27/6.59	235 is_a_theorem(implies(or(A,not(B)),implies(B,A))).  [back_rewrite(208),rewrite([221(3)])].
6.27/6.59	283 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)).  [resolve(220,b,209,a)].
6.27/6.59	284 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))).  [resolve(220,b,207,a)].
6.27/6.59	291 is_a_theorem(or(A,or(not(A),B))).  [para(221(a,1),214(a,1))].
6.27/6.59	347 not(necessarily(possibly(A))) = possibly(necessarily(not(A))).  [para(228(a,1),228(a,1,1,1))].
6.27/6.59	356 necessarily(or(A,B)) = strict_implies(not(A),B).  [para(221(a,1),231(a,1,1))].
6.27/6.59	360 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)).  [resolve(235,a,220,b)].
6.27/6.59	3355 is_a_theorem(implies(A,and(A,A))).  [resolve(283,a,216,a)].
6.27/6.59	3462 is_a_theorem(implies(implies(necessarily(possibly(A)),possibly(A)),equiv(possibly(A),necessarily(possibly(A))))).  [resolve(284,a,229,a)].
6.27/6.59	3549 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))).  [resolve(3355,a,284,a)].
6.27/6.59	20935 is_a_theorem(equiv(possibly(A),necessarily(possibly(A)))).  [resolve(3462,a,220,b),unit_del(a,225)].
6.27/6.59	20951 necessarily(possibly(A)) = possibly(A).  [resolve(20935,a,219,a)].
6.27/6.59	20967 possibly(necessarily(not(A))) = not(possibly(A)).  [back_rewrite(347),rewrite([20951(2)]),flip(a)].
6.27/6.59	21003 not(not(possibly(A))) = possibly(possibly(A)).  [para(228(a,1),20967(a,1,1,1)),rewrite([20951(2),20967(5)]),flip(a)].
6.27/6.59	21229 is_a_theorem(equiv(A,and(A,A))).  [resolve(3549,a,220,b),unit_del(a,211)].
6.27/6.59	21260 and(A,A) = A.  [resolve(21229,a,219,a)].
6.27/6.59	21263 not(not(A)) = or(A,A).  [para(21260(a,1),206(a,1,1)),rewrite([221(4)])].
6.27/6.59	21310 is_a_theorem(or(A,not(not(not(A))))).  [para(21263(a,2),291(a,1,2))].
6.27/6.59	22275 is_a_theorem(or(not(A),A)).  [resolve(21310,a,360,a),rewrite([221(3)])].
6.27/6.59	22304 is_a_theorem(strict_implies(not(not(A)),A)).  [resolve(22275,a,226,a),rewrite([356(3)])].
6.27/6.59	22346 is_a_theorem(strict_implies(possibly(possibly(A)),possibly(A))).  [para(21003(a,1),22304(a,1,1))].
6.27/6.59	22347 $F.  [resolve(22346,a,232,a)].
6.27/6.59	
6.27/6.59	% SZS output end Refutation
6.27/6.59	============================== end of proof ==========================
6.27/6.59	
6.27/6.59	============================== STATISTICS ============================
6.27/6.59	
6.27/6.59	Given=1320. Generated=58428. Kept=22284. proofs=1.
6.27/6.59	Usable=851. Sos=7217. Demods=95. Limbo=28, Disabled=14333. Hints=0.
6.27/6.59	Megabytes=19.36.
6.27/6.59	User_CPU=5.44, System_CPU=0.05, Wall_clock=6.
6.27/6.59	
6.27/6.59	============================== end of statistics =====================
6.27/6.59	
6.27/6.59	============================== end of search =========================
6.27/6.59	
6.27/6.59	THEOREM PROVED
6.27/6.59	% SZS status Theorem
6.27/6.59	
6.27/6.59	Exiting with 1 proof.
6.27/6.59	
6.27/6.59	Process 643 exit (max_proofs) Thu Jul  2 08:26:08 2020
6.27/6.59	Prover9 interrupted
6.27/6.59	EOF