0.05/0.10	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.05/0.10	% Command    : tptp2X_and_run_prover9 %d %s
0.10/0.31	% Computer   : n016.cluster.edu
0.10/0.31	% Model      : x86_64 x86_64
0.10/0.31	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.10/0.31	% Memory     : 8042.1875MB
0.10/0.31	% OS         : Linux 3.10.0-693.el7.x86_64
0.10/0.31	% CPULimit   : 1200
0.10/0.31	% DateTime   : Tue Jul 13 16:18:53 EDT 2021
0.10/0.31	% CPUTime    : 
0.80/1.09	============================== Prover9 ===============================
0.80/1.09	Prover9 (32) version 2009-11A, November 2009.
0.80/1.09	Process 26689 was started by sandbox2 on n016.cluster.edu,
0.80/1.09	Tue Jul 13 16:18:54 2021
0.80/1.09	The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1200 -f /tmp/Prover9_26536_n016.cluster.edu".
0.80/1.09	============================== end of head ===========================
0.80/1.09	
0.80/1.09	============================== INPUT =================================
0.80/1.09	
0.80/1.09	% Reading from file /tmp/Prover9_26536_n016.cluster.edu
0.80/1.09	
0.80/1.09	set(prolog_style_variables).
0.80/1.09	set(auto2).
0.80/1.09	    % set(auto2) -> set(auto).
0.80/1.09	    % set(auto) -> set(auto_inference).
0.80/1.09	    % set(auto) -> set(auto_setup).
0.80/1.09	    % set(auto_setup) -> set(predicate_elim).
0.80/1.09	    % set(auto_setup) -> assign(eq_defs, unfold).
0.80/1.09	    % set(auto) -> set(auto_limits).
0.80/1.09	    % set(auto_limits) -> assign(max_weight, "100.000").
0.80/1.09	    % set(auto_limits) -> assign(sos_limit, 20000).
0.80/1.09	    % set(auto) -> set(auto_denials).
0.80/1.09	    % set(auto) -> set(auto_process).
0.80/1.09	    % set(auto2) -> assign(new_constants, 1).
0.80/1.09	    % set(auto2) -> assign(fold_denial_max, 3).
0.80/1.09	    % set(auto2) -> assign(max_weight, "200.000").
0.80/1.09	    % set(auto2) -> assign(max_hours, 1).
0.80/1.09	    % assign(max_hours, 1) -> assign(max_seconds, 3600).
0.80/1.09	    % set(auto2) -> assign(max_seconds, 0).
0.80/1.09	    % set(auto2) -> assign(max_minutes, 5).
0.80/1.09	    % assign(max_minutes, 5) -> assign(max_seconds, 300).
0.80/1.09	    % set(auto2) -> set(sort_initial_sos).
0.80/1.09	    % set(auto2) -> assign(sos_limit, -1).
0.80/1.09	    % set(auto2) -> assign(lrs_ticks, 3000).
0.80/1.09	    % set(auto2) -> assign(max_megs, 400).
0.80/1.09	    % set(auto2) -> assign(stats, some).
0.80/1.09	    % set(auto2) -> clear(echo_input).
0.80/1.09	    % set(auto2) -> set(quiet).
0.80/1.09	    % set(auto2) -> clear(print_initial_clauses).
0.80/1.09	    % set(auto2) -> clear(print_given).
0.80/1.09	assign(lrs_ticks,-1).
0.80/1.09	assign(sos_limit,10000).
0.80/1.09	assign(order,kbo).
0.80/1.09	set(lex_order_vars).
0.80/1.09	clear(print_given).
0.80/1.09	
0.80/1.09	% formulas(sos).  % not echoed (77 formulas)
0.80/1.09	
0.80/1.09	============================== end of input ==========================
0.80/1.09	
0.80/1.09	% From the command line: assign(max_seconds, 1200).
0.80/1.09	
0.80/1.09	============================== PROCESS NON-CLAUSAL FORMULAS ==========
0.80/1.09	
0.80/1.09	% Formulas that are not ordinary clauses:
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	3 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	6 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	10 (all P is_a_theorem(implies(or(P,P),P))) <-> r1 # label(r1) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	11 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	12 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	15 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	16 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	17 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	20 (all P all Q is_a_theorem(implies(and(P,Q),P))) <-> kn2 # label(kn2) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	23 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	24 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	25 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	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].
0.80/1.09	35 (all X is_a_theorem(implies(X,necessarily(possibly(X))))) <-> axiom_B # label(axiom_B) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	36 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.09	38 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	39 (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) <-> axiom_m10 # label(axiom_m10) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	40 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause).  [assumption].
0.80/1.09	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].
0.80/1.11	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].
0.80/1.11	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].
0.80/1.11	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].
0.80/1.11	45 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause).  [assumption].
0.80/1.11	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].
0.80/1.11	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].
0.80/1.11	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].
0.80/1.11	49 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause).  [assumption].
0.80/1.11	50 (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) <-> necessitation # label(necessitation) # label(axiom) # label(non_clause).  [assumption].
0.80/1.11	51 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause).  [assumption].
0.80/1.11	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].
0.80/1.11	53 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause).  [assumption].
0.80/1.11	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].
0.80/1.11	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].
0.80/1.11	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].
0.80/1.11	57 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause).  [assumption].
0.80/1.11	58 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause).  [assumption].
0.80/1.11	
0.80/1.11	============================== end of process non-clausal formulas ===
0.80/1.11	
0.80/1.11	============================== PROCESS INITIAL CLAUSES ===============
0.80/1.11	
0.80/1.11	============================== PREDICATE ELIMINATION =================
0.80/1.11	
0.80/1.11	============================== end predicate elimination =============
0.80/1.11	
0.80/1.11	Auto_denials:  (non-Horn, no changes).
0.80/1.11	
0.80/1.11	Term ordering decisions:
0.80/1.11	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.
0.80/1.11	
0.80/1.11	============================== end of process initial clauses ========
0.80/1.11	
0.80/1.11	============================== CLAUSES FOR SEARCH ====================
97.18/97.82	
97.18/97.82	============================== end of clauses for search =============
97.18/97.82	
97.18/97.82	============================== SEARCH ================================
97.18/97.82	
97.18/97.82	% Starting search at 0.04 seconds.
97.18/97.82	
97.18/97.82	Low Water (keep): wt=24.000, iters=3370
97.18/97.82	
97.18/97.82	Low Water (keep): wt=22.000, iters=3368
97.18/97.82	
97.18/97.82	Low Water (keep): wt=21.000, iters=3423
97.18/97.82	
97.18/97.82	Low Water (keep): wt=20.000, iters=3357
97.18/97.82	
97.18/97.82	Low Water (keep): wt=18.000, iters=3335
97.18/97.82	
97.18/97.82	Low Water (keep): wt=17.000, iters=3346
97.18/97.82	
97.18/97.82	NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 123 (0.00 of 4.26 sec).
97.18/97.82	
97.18/97.82	Low Water (keep): wt=16.000, iters=3335
97.18/97.82	
97.18/97.82	Low Water (keep): wt=15.000, iters=3378
97.18/97.82	
97.18/97.82	Low Water (displace): id=19366, wt=48.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=19370, wt=40.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=19364, wt=37.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=19419, wt=36.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=19359, wt=33.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=19381, wt=32.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=18068, wt=31.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=17973, wt=30.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=24542, wt=17.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=24591, wt=15.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=25486, wt=14.000
97.18/97.82	
97.18/97.82	Low Water (displace): id=30013, wt=12.000
97.18/97.82	% back CAC tautology: 29751 or(or(A,B),not(C)) = or(not(C),or(A,B)).  [para(29689(a,2),26332(a,1)),rewrite([26332(6)]),flip(a)].
97.18/97.82	
97.18/97.82	Low Water (displace): id=34578, wt=10.000
97.18/97.82	
97.18/97.82	Low Water (keep): wt=14.000, iters=3399
97.18/97.82	
97.18/97.82	Low Water (displace): id=49703, wt=9.000
97.18/97.82	
97.18/97.82	============================== PROOF =================================
97.18/97.82	% SZS status Theorem
97.18/97.82	% SZS output start Refutation
97.18/97.82	
97.18/97.82	% Proof 1 at 93.94 (+ 2.48) seconds.
97.18/97.82	% Length of proof is 140.
97.18/97.82	% Level of proof is 30.
97.18/97.82	% Maximum clause weight is 16.000.
97.18/97.82	% Given clauses 8706.
97.18/97.82	
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	51 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause).  [assumption].
97.18/97.82	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].
97.18/97.82	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].
97.18/97.82	67 is_a_theorem(implies(c11,c12)) | modus_ponens # label(modus_ponens) # label(axiom).  [clausify(5)].
97.18/97.82	68 is_a_theorem(c11) | modus_ponens # label(modus_ponens) # label(axiom).  [clausify(5)].
97.18/97.82	69 -is_a_theorem(c12) | modus_ponens # label(modus_ponens) # label(axiom).  [clausify(5)].
97.18/97.82	113 -is_a_theorem(equiv(A,B)) | B = A | -substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom).  [clausify(26)].
97.18/97.82	115 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom).  [clausify(28)].
97.18/97.82	116 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom).  [clausify(29)].
97.18/97.82	117 -op_or | not(and(not(A),not(B))) = or(A,B).  [copy(116),flip(b)].
97.18/97.82	118 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom).  [clausify(30)].
97.18/97.82	119 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B).  [copy(118),flip(b)].
97.18/97.82	128 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(B) | -modus_ponens_strict_implies # label(modus_ponens_strict_implies) # label(axiom).  [clausify(34)].
97.18/97.82	141 -axiom_m1 | is_a_theorem(strict_implies(and(A,B),and(B,A))) # label(axiom_m1) # label(axiom).  [clausify(41)].
97.18/97.82	143 -adjunction | -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(B,A)) # label(adjunction) # label(axiom).  [clausify(42)].
97.18/97.82	147 -axiom_m2 | is_a_theorem(strict_implies(and(A,B),A)) # label(axiom_m2) # label(axiom).  [clausify(43)].
97.18/97.82	153 -substitution_strict_equiv | -is_a_theorem(strict_equiv(A,B)) | B = A # label(substitution_strict_equiv) # label(axiom).  [clausify(46)].
97.18/97.82	159 is_a_theorem(strict_implies(and(strict_implies(A,B),strict_implies(B,C)),strict_implies(A,C))) | -axiom_m5 # label(axiom_m5) # label(axiom).  [clausify(48)].
97.18/97.82	166 is_a_theorem(strict_implies(A,and(A,A))) | -axiom_m4 # label(axiom_m4) # label(axiom).  [clausify(51)].
97.18/97.82	173 -op_strict_equiv | strict_equiv(A,B) = and(strict_implies(A,B),strict_implies(B,A)) # label(op_strict_equiv) # label(axiom).  [clausify(55)].
97.18/97.82	174 -op_strict_equiv | and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B).  [copy(173),flip(b)].
97.18/97.82	175 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B) # label(op_strict_implies) # label(axiom).  [clausify(56)].
97.18/97.82	180 substitution_strict_equiv # label(s1_0_substitution_strict_equiv) # label(axiom).  [assumption].
97.18/97.82	181 adjunction # label(s1_0_adjunction) # label(axiom).  [assumption].
97.18/97.82	182 axiom_m2 # label(s1_0_axiom_m2) # label(axiom).  [assumption].
97.18/97.82	186 axiom_m1 # label(s1_0_axiom_m1) # label(axiom).  [assumption].
97.18/97.82	187 op_equiv # label(s1_0_op_equiv) # label(axiom).  [assumption].
97.18/97.82	188 op_strict_equiv # label(s1_0_op_strict_equiv) # label(axiom).  [assumption].
97.18/97.82	189 op_or # label(s1_0_op_or) # label(axiom).  [assumption].
97.18/97.82	190 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom).  [assumption].
97.18/97.82	191 axiom_m5 # label(s1_0_axiom_m5) # label(axiom).  [assumption].
97.18/97.82	192 axiom_m4 # label(s1_0_axiom_m4) # label(axiom).  [assumption].
97.18/97.82	193 modus_ponens_strict_implies # label(s1_0_modus_ponens_strict_implies) # label(axiom).  [assumption].
97.18/97.82	194 op_implies_and # label(hilbert_op_implies_and) # label(axiom).  [assumption].
97.18/97.82	195 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom).  [assumption].
97.18/97.82	196 -modus_ponens # label(hilbert_modus_ponens) # label(negated_conjecture).  [assumption].
97.18/97.82	198 -is_a_theorem(strict_equiv(A,B)) | B = A.  [back_unit_del(153),unit_del(a,180)].
97.18/97.82	199 -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(B,A)).  [back_unit_del(143),unit_del(a,181)].
97.18/97.82	200 is_a_theorem(strict_implies(and(A,B),A)).  [back_unit_del(147),unit_del(a,182)].
97.18/97.82	203 is_a_theorem(strict_implies(and(A,B),and(B,A))).  [back_unit_del(141),unit_del(a,186)].
97.18/97.82	204 and(implies(A,B),implies(B,A)) = equiv(A,B).  [back_unit_del(119),unit_del(a,187)].
97.18/97.82	205 and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B).  [back_unit_del(174),unit_del(a,188)].
97.18/97.82	206 not(and(not(A),not(B))) = or(A,B).  [back_unit_del(117),unit_del(a,189)].
97.18/97.82	207 necessarily(implies(A,B)) = strict_implies(A,B).  [back_unit_del(175),unit_del(a,190)].
97.18/97.82	208 is_a_theorem(strict_implies(and(strict_implies(A,B),strict_implies(B,C)),strict_implies(A,C))).  [back_unit_del(159),unit_del(b,191)].
97.18/97.82	209 is_a_theorem(strict_implies(A,and(A,A))).  [back_unit_del(166),unit_del(b,192)].
97.18/97.82	210 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(B).  [back_unit_del(128),unit_del(d,193)].
97.18/97.82	211 not(and(A,not(B))) = implies(A,B).  [back_unit_del(115),unit_del(a,194)].
97.18/97.82	212 -is_a_theorem(equiv(A,B)) | B = A.  [back_unit_del(113),unit_del(c,195)].
97.18/97.82	213 -is_a_theorem(c12).  [back_unit_del(69),unit_del(b,196)].
97.18/97.82	214 is_a_theorem(c11).  [back_unit_del(68),unit_del(b,196)].
97.18/97.82	215 is_a_theorem(implies(c11,c12)).  [back_unit_del(67),unit_del(b,196)].
97.18/97.82	220 implies(not(A),B) = or(A,B).  [back_rewrite(206),rewrite([211(4)])].
97.18/97.82	232 -is_a_theorem(A) | is_a_theorem(and(strict_implies(and(B,C),B),A)).  [resolve(200,a,199,b)].
97.18/97.82	233 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(and(B,C),B))).  [resolve(200,a,199,a)].
97.18/97.82	241 -is_a_theorem(A) | is_a_theorem(and(strict_implies(and(B,C),and(C,B)),A)).  [resolve(203,a,199,b)].
97.18/97.82	258 -is_a_theorem(A) | is_a_theorem(and(strict_implies(B,and(B,B)),A)).  [resolve(209,a,199,b)].
97.18/97.82	259 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(B,and(B,B)))).  [resolve(209,a,199,a)].
97.18/97.82	264 -is_a_theorem(and(strict_implies(A,B),strict_implies(B,C))) | is_a_theorem(strict_implies(A,C)).  [resolve(210,b,208,a)].
97.18/97.82	270 not(and(A,implies(B,C))) = implies(A,and(B,not(C))).  [para(211(a,1),211(a,1,1,2))].
97.18/97.82	276 -is_a_theorem(A) | is_a_theorem(and(A,implies(c11,c12))).  [resolve(215,a,199,a)].
97.18/97.82	282 necessarily(or(A,B)) = strict_implies(not(A),B).  [para(220(a,1),207(a,1,1))].
97.18/97.82	283 or(and(A,not(B)),C) = implies(implies(A,B),C).  [para(211(a,1),220(a,1,1)),flip(a)].
97.18/97.82	373 is_a_theorem(and(strict_implies(and(A,B),A),strict_implies(C,and(C,C)))).  [resolve(232,a,209,a)].
97.18/97.82	423 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(and(B,C),B))).  [resolve(233,a,209,a)].
97.18/97.82	794 is_a_theorem(and(strict_implies(and(A,B),and(B,A)),strict_implies(and(C,D),and(D,C)))).  [resolve(241,a,203,a)].
97.18/97.82	1762 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(B,and(B,B)))).  [resolve(258,a,209,a)].
97.18/97.82	2524 implies(A,and(not(B),not(C))) = not(and(A,or(B,C))).  [para(220(a,1),270(a,1,1,2)),flip(a)].
97.18/97.82	7160 is_a_theorem(strict_equiv(and(A,A),A)).  [para(205(a,1),373(a,1))].
97.18/97.82	7192 and(A,A) = A.  [resolve(7160,a,198,a),flip(a)].
97.18/97.82	7527 is_a_theorem(and(strict_implies(A,A),strict_implies(B,B))).  [back_rewrite(1762),rewrite([7192(1),7192(2)])].
97.18/97.82	7608 is_a_theorem(and(strict_implies(A,A),strict_implies(and(B,C),B))).  [back_rewrite(423),rewrite([7192(1)])].
97.18/97.82	7611 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(B,B))).  [back_rewrite(259),rewrite([7192(2)])].
97.18/97.82	7613 is_a_theorem(strict_implies(A,A)).  [back_rewrite(209),rewrite([7192(1)])].
97.18/97.82	7620 strict_equiv(A,A) = strict_implies(A,A).  [para(7192(a,1),205(a,1)),flip(a)].
97.18/97.82	7621 or(A,A) = not(not(A)).  [para(7192(a,1),211(a,1,1)),rewrite([220(4)]),flip(a)].
97.18/97.82	7634 implies(or(A,A),B) = or(not(A),B).  [para(7192(a,1),283(a,1,1)),rewrite([220(4)]),flip(a)].
97.18/97.82	7764 strict_implies(or(A,A),B) = strict_implies(not(not(A)),B).  [para(7621(a,2),282(a,2,1)),rewrite([282(3)]),flip(a)].
97.18/97.82	7779 not(or(A,A)) = not(not(not(A))).  [para(7621(a,2),7621(a,2,1)),rewrite([7621(3)]),flip(a)].
97.18/97.82	8939 implies(A,or(B,B)) = implies(A,not(not(B))).  [para(7779(a,1),211(a,1,1,2)),rewrite([211(5)]),flip(a)].
97.18/97.82	15073 is_a_theorem(strict_equiv(and(A,B),and(B,A))).  [para(205(a,1),794(a,1))].
97.18/97.82	15100 and(A,B) = and(B,A).  [resolve(15073,a,198,a)].
97.18/97.82	15771 equiv(A,B) = equiv(B,A).  [para(15100(a,1),204(a,1)),rewrite([204(3)])].
97.18/97.82	15772 strict_equiv(A,B) = strict_equiv(B,A).  [para(15100(a,1),205(a,1)),rewrite([205(3)])].
97.18/97.82	15773 -is_a_theorem(and(strict_implies(A,B),strict_implies(C,A))) | is_a_theorem(strict_implies(C,B)).  [para(15100(a,1),264(a,1))].
97.18/97.82	15839 -is_a_theorem(equiv(A,B)) | A = B.  [para(15771(a,1),212(a,1))].
97.18/97.82	16869 strict_implies(A,or(B,B)) = strict_implies(A,not(not(B))).  [para(8939(a,1),207(a,1,1)),rewrite([207(4)]),flip(a)].
97.18/97.82	17712 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,or(B,B))) | is_a_theorem(not(not(B))).  [para(16869(a,2),210(b,1))].
97.18/97.82	21935 not(and(not(A),or(B,C))) = or(A,and(not(B),not(C))).  [para(2524(a,1),220(a,1))].
97.18/97.82	21947 not(and(A,or(B,C))) = not(and(A,or(C,B))).  [para(15100(a,1),2524(a,1,2)),rewrite([2524(4)])].
97.18/97.82	25903 not(and(or(A,B),or(B,A))) = not(or(A,B)).  [para(7192(a,1),21947(a,1,1)),flip(a)].
97.18/97.82	26326 not(or(A,B)) = not(or(B,A)).  [para(15100(a,1),25903(a,1,1)),rewrite([25903(4)])].
97.18/97.82	26331 implies(A,or(B,C)) = implies(A,or(C,B)).  [para(26326(a,1),211(a,1,1,2)),rewrite([211(4)])].
97.18/97.82	26350 strict_implies(A,or(B,C)) = strict_implies(A,or(C,B)).  [para(26331(a,1),207(a,1,1)),rewrite([207(3)])].
97.18/97.82	29679 or(A,and(not(B),not(C))) = implies(or(B,C),A).  [para(15100(a,1),21935(a,1,1)),rewrite([211(4)]),flip(a)].
97.18/97.82	29689 or(not(A),B) = or(B,not(A)).  [para(7192(a,1),29679(a,1,2)),rewrite([7634(4)]),flip(a)].
97.18/97.82	29693 implies(or(A,B),C) = implies(or(B,A),C).  [para(15100(a,1),29679(a,1,2)),rewrite([29679(4)])].
97.18/97.82	29740 strict_implies(not(not(A)),B) = strict_implies(not(B),not(A)).  [para(29689(a,1),282(a,1,1)),rewrite([282(3)]),flip(a)].
97.18/97.82	29833 strict_implies(or(A,A),B) = strict_implies(not(B),not(A)).  [back_rewrite(7764),rewrite([29740(5)])].
97.18/97.82	29881 strict_implies(or(A,B),C) = strict_implies(or(B,A),C).  [para(29693(a,1),207(a,1,1)),rewrite([207(3)])].
97.18/97.82	29996 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(B,B))).  [para(29833(a,2),7527(a,1,1))].
97.18/97.82	30011 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(and(B,C),B))).  [para(29833(a,2),7608(a,1,1))].
97.18/97.82	30807 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(or(B,B),B))).  [para(29833(a,2),29996(a,1,2))].
97.18/97.82	31025 strict_equiv(A,or(B,C)) = strict_equiv(A,or(C,B)).  [para(29881(a,1),205(a,1,1)),rewrite([26350(4),15100(5),205(5),15772(4)])].
97.18/97.82	31079 -is_a_theorem(strict_equiv(A,or(B,C))) | or(C,B) = A.  [para(31025(a,1),198(a,1))].
97.18/97.82	31081 strict_equiv(A,or(B,B)) = strict_equiv(A,not(not(B))).  [para(7621(a,1),31025(a,1,2)),flip(a)].
97.18/97.82	31085 or(A,B) = or(B,A).  [para(7620(a,1),31079(a,1)),unit_del(a,7613)].
97.18/97.82	31114 strict_implies(not(A),B) = strict_implies(not(B),A).  [para(31085(a,1),282(a,1,1)),rewrite([282(2)])].
97.18/97.82	31133 and(strict_implies(A,not(B)),strict_implies(not(A),B)) = strict_equiv(A,not(B)).  [para(31114(a,1),205(a,1,1)),rewrite([15100(5),15772(7)])].
97.18/97.82	31213 strict_equiv(A,not(or(B,B))) = strict_equiv(A,not(not(not(B)))).  [para(7621(a,2),31081(a,2,2,1)),rewrite([7621(3)]),flip(a)].
97.18/97.82	31334 is_a_theorem(strict_implies(not(A),not(and(A,B)))).  [resolve(30011,a,264,a),rewrite([7621(3),31114(4)])].
97.18/97.82	31429 -is_a_theorem(not(A)) | is_a_theorem(not(and(A,B))).  [resolve(31334,a,210,b)].
97.18/97.82	31701 -is_a_theorem(or(A,A)) | is_a_theorem(implies(B,A)).  [para(7621(a,2),31429(a,1)),rewrite([15100(4),211(5)])].
97.18/97.82	33177 is_a_theorem(strict_implies(not(A),not(or(A,A)))).  [resolve(30807,a,15773,a),rewrite([7621(3),31114(4)])].
97.18/97.82	33203 is_a_theorem(strict_implies(not(A),not(not(not(A))))).  [para(7621(a,1),33177(a,1,2,1))].
97.18/97.82	33270 is_a_theorem(and(strict_implies(not(A),not(not(not(A)))),strict_implies(B,B))).  [resolve(33203,a,7611,a)].
97.18/97.82	40274 -is_a_theorem(strict_equiv(A,not(not(not(B))))) | not(or(B,B)) = A.  [para(31213(a,1),198(a,1))].
97.18/97.82	43285 is_a_theorem(strict_equiv(not(A),not(not(not(A))))).  [para(31133(a,1),33270(a,1))].
97.18/97.82	43546 not(or(A,A)) = not(A).  [resolve(43285,a,40274,a)].
97.18/97.82	44011 implies(A,or(B,B)) = implies(A,B).  [para(43546(a,1),211(a,1,1,2)),rewrite([211(3)]),flip(a)].
97.18/97.82	44666 strict_implies(A,or(B,B)) = strict_implies(A,B).  [para(44011(a,1),207(a,1,1)),rewrite([207(2)]),flip(a)].
97.18/97.82	44748 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(not(not(B))).  [back_rewrite(17712),rewrite([44666(3)])].
97.18/97.82	45952 -is_a_theorem(A) | is_a_theorem(not(not(A))).  [resolve(44748,b,7613,a)].
97.18/97.82	47140 is_a_theorem(or(c11,c11)).  [resolve(45952,a,214,a),rewrite([7621(3,R)])].
97.18/97.82	47181 is_a_theorem(implies(A,c11)).  [resolve(47140,a,31701,a)].
97.18/97.82	47382 is_a_theorem(and(implies(A,c11),implies(c11,c12))).  [resolve(47181,a,276,a)].
97.18/97.82	50037 is_a_theorem(equiv(c11,c12)).  [para(204(a,1),47382(a,1)),rewrite([15771(3)])].
97.18/97.82	50087 c12 = c11.  [resolve(50037,a,15839,a),flip(a)].
97.18/97.82	53975 $F.  [back_rewrite(213),rewrite([50087(1)]),unit_del(a,214)].
97.18/97.82	
97.18/97.82	% SZS output end Refutation
97.18/97.82	============================== end of proof ==========================
97.18/97.82	
97.18/97.82	============================== STATISTICS ============================
97.18/97.82	
97.18/97.82	Given=8706. Generated=2346740. Kept=53911. proofs=1.
97.18/97.82	Usable=2573. Sos=5561. Demods=197. Limbo=3888, Disabled=42024. Hints=0.
97.18/97.82	Megabytes=47.35.
97.18/97.82	User_CPU=93.94, System_CPU=2.48, Wall_clock=96.
97.18/97.82	
97.18/97.82	============================== end of statistics =====================
97.18/97.82	
97.18/97.82	============================== end of search =========================
97.18/97.82	
97.18/97.82	THEOREM PROVED
97.18/97.82	% SZS status Theorem
97.18/97.82	
97.18/97.82	Exiting with 1 proof.
97.18/97.82	
97.18/97.82	Process 26689 exit (max_proofs) Tue Jul 13 16:20:30 2021
97.18/97.82	Prover9 interrupted
97.18/97.83	EOF