TSTP Solution File: LCL547+1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : LCL547+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n009.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 13:47:30 EDT 2022
% Result : Theorem 121.44s 121.71s
% Output : Refutation 121.44s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : LCL547+1 : TPTP v8.1.0. Released v3.3.0.
% 0.11/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.12/0.33 % Computer : n009.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 : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Sun Jul 3 07:49:22 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.41/1.00 ============================== Prover9 ===============================
% 0.41/1.00 Prover9 (32) version 2009-11A, November 2009.
% 0.41/1.00 Process 16103 was started by sandbox on n009.cluster.edu,
% 0.41/1.00 Sun Jul 3 07:49:23 2022
% 0.41/1.00 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_15950_n009.cluster.edu".
% 0.41/1.00 ============================== end of head ===========================
% 0.41/1.00
% 0.41/1.00 ============================== INPUT =================================
% 0.41/1.00
% 0.41/1.00 % Reading from file /tmp/Prover9_15950_n009.cluster.edu
% 0.41/1.00
% 0.41/1.00 set(prolog_style_variables).
% 0.41/1.00 set(auto2).
% 0.41/1.00 % set(auto2) -> set(auto).
% 0.41/1.00 % set(auto) -> set(auto_inference).
% 0.41/1.00 % set(auto) -> set(auto_setup).
% 0.41/1.00 % set(auto_setup) -> set(predicate_elim).
% 0.41/1.00 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.41/1.00 % set(auto) -> set(auto_limits).
% 0.41/1.00 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.41/1.00 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.41/1.00 % set(auto) -> set(auto_denials).
% 0.41/1.00 % set(auto) -> set(auto_process).
% 0.41/1.00 % set(auto2) -> assign(new_constants, 1).
% 0.41/1.00 % set(auto2) -> assign(fold_denial_max, 3).
% 0.41/1.00 % set(auto2) -> assign(max_weight, "200.000").
% 0.41/1.00 % set(auto2) -> assign(max_hours, 1).
% 0.41/1.00 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.41/1.00 % set(auto2) -> assign(max_seconds, 0).
% 0.41/1.00 % set(auto2) -> assign(max_minutes, 5).
% 0.41/1.00 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.41/1.00 % set(auto2) -> set(sort_initial_sos).
% 0.41/1.00 % set(auto2) -> assign(sos_limit, -1).
% 0.41/1.00 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.41/1.00 % set(auto2) -> assign(max_megs, 400).
% 0.41/1.00 % set(auto2) -> assign(stats, some).
% 0.41/1.00 % set(auto2) -> clear(echo_input).
% 0.41/1.00 % set(auto2) -> set(quiet).
% 0.41/1.00 % set(auto2) -> clear(print_initial_clauses).
% 0.41/1.00 % set(auto2) -> clear(print_given).
% 0.41/1.00 assign(lrs_ticks,-1).
% 0.41/1.00 assign(sos_limit,10000).
% 0.41/1.00 assign(order,kbo).
% 0.41/1.00 set(lex_order_vars).
% 0.41/1.00 clear(print_given).
% 0.41/1.00
% 0.41/1.00 % formulas(sos). % not echoed (89 formulas)
% 0.41/1.00
% 0.41/1.00 ============================== end of input ==========================
% 0.41/1.00
% 0.41/1.00 % From the command line: assign(max_seconds, 300).
% 0.41/1.00
% 0.41/1.00 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.41/1.00
% 0.41/1.00 % Formulas that are not ordinary clauses:
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 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].
% 0.41/1.00 10 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 11 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 16 kn1 <-> (all P is_a_theorem(implies(P,and(P,P)))) # label(kn1) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 17 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 20 cn2 <-> (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) # label(cn2) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 21 cn3 <-> (all P is_a_theorem(implies(implies(not(P),P),P))) # label(cn3) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 22 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 23 r2 <-> (all P all Q is_a_theorem(implies(Q,or(P,Q)))) # label(r2) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 24 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 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].
% 0.41/1.00 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 40 axiom_5 <-> (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) # label(axiom_5) # label(axiom) # label(non_clause). [assumption].
% 0.41/1.00 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].
% 0.74/1.03 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].
% 0.74/1.03 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].
% 0.74/1.03 44 axiom_s4 <-> (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_s4) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.03 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].
% 0.74/1.03 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].
% 0.74/1.03 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].
% 0.74/1.03 48 axiom_m4 <-> (all X is_a_theorem(strict_implies(X,and(X,X)))) # label(axiom_m4) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.03 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].
% 0.74/1.03 50 axiom_m6 <-> (all X is_a_theorem(strict_implies(X,possibly(X)))) # label(axiom_m6) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.03 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].
% 0.74/1.03 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].
% 0.74/1.03 53 axiom_m9 <-> (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) # label(axiom_m9) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.03 54 axiom_m10 <-> (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) # label(axiom_m10) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.03 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.03 56 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.03 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].
% 0.74/1.03 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].
% 0.74/1.03
% 0.74/1.03 ============================== end of process non-clausal formulas ===
% 0.74/1.03
% 0.74/1.03 ============================== PROCESS INITIAL CLAUSES ===============
% 0.74/1.03
% 0.74/1.03 ============================== PREDICATE ELIMINATION =================
% 0.74/1.03
% 0.74/1.03 ============================== end predicate elimination =============
% 0.74/1.03
% 0.74/1.03 Auto_denials: (non-Horn, no changes).
% 0.74/1.03
% 0.74/1.03 Term ordering decisions:
% 0.74/1.03 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.
% 0.74/1.03
% 0.74/1.03 ============================== end of process initial clauses ========
% 0.74/1.03
% 0.74/1.03 ============================== CLAUSES FOR SEARCH ====================
% 121.44/121.71
% 121.44/121.71 ============================== end of clauses for search =============
% 121.44/121.71
% 121.44/121.71 ============================== SEARCH ================================
% 121.44/121.71
% 121.44/121.71 % Starting search at 0.05 seconds.
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=22.000, iters=3405
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=20.000, iters=3364
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=19.000, iters=3480
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=18.000, iters=3375
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=17.000, iters=3345
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=16.000, iters=3343
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=15.000, iters=3346
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=14.000, iters=3333
% 121.44/121.71
% 121.44/121.71 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 68 (0.00 of 1.50 sec).
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=13.000, iters=3336
% 121.44/121.71
% 121.44/121.71 Low Water (displace): id=5257, wt=29.000
% 121.44/121.71
% 121.44/121.71 Low Water (displace): id=5248, wt=28.000
% 121.44/121.71
% 121.44/121.71 Low Water (displace): id=5531, wt=27.000
% 121.44/121.71
% 121.44/121.71 Low Water (displace): id=10931, wt=9.000
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=11.000, iters=3385
% 121.44/121.71
% 121.44/121.71 Low Water (keep): wt=10.000, iters=3335
% 121.44/121.71
% 121.44/121.71 ============================== PROOF =================================
% 121.44/121.71 % SZS status Theorem
% 121.44/121.71 % SZS output start Refutation
% 121.44/121.71
% 121.44/121.71 % Proof 1 at 114.64 (+ 6.08) seconds.
% 121.44/121.71 % Length of proof is 364.
% 121.44/121.71 % Level of proof is 42.
% 121.44/121.71 % Maximum clause weight is 17.000.
% 121.44/121.71 % Given clauses 31035.
% 121.44/121.71
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 4 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 7 and_1 <-> (all X all Y is_a_theorem(implies(and(X,Y),X))) # label(and_1) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 8 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 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].
% 121.44/121.71 10 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 11 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 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].
% 121.44/121.71 32 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 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].
% 121.44/121.71 37 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 38 axiom_4 <-> (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) # label(axiom_4) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 39 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 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].
% 121.44/121.71 55 op_possibly -> (all X possibly(X) = not(necessarily(not(X)))) # label(op_possibly) # label(axiom) # label(non_clause). [assumption].
% 121.44/121.71 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].
% 121.44/121.71 59 -modus_ponens | -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B) # label(modus_ponens) # label(axiom). [clausify(1)].
% 121.44/121.71 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(2)].
% 121.44/121.71 66 -modus_tollens | is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))) # label(modus_tollens) # label(axiom). [clausify(3)].
% 121.44/121.71 68 -implies_1 | is_a_theorem(implies(A,implies(B,A))) # label(implies_1) # label(axiom). [clausify(4)].
% 121.44/121.71 70 -implies_2 | is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))) # label(implies_2) # label(axiom). [clausify(5)].
% 121.44/121.71 72 -implies_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))) # label(implies_3) # label(axiom). [clausify(6)].
% 121.44/121.71 74 -and_1 | is_a_theorem(implies(and(A,B),A)) # label(and_1) # label(axiom). [clausify(7)].
% 121.44/121.71 76 -and_2 | is_a_theorem(implies(and(A,B),B)) # label(and_2) # label(axiom). [clausify(8)].
% 121.44/121.71 78 -and_3 | is_a_theorem(implies(A,implies(B,and(A,B)))) # label(and_3) # label(axiom). [clausify(9)].
% 121.44/121.71 80 -or_1 | is_a_theorem(implies(A,or(A,B))) # label(or_1) # label(axiom). [clausify(10)].
% 121.44/121.71 82 -or_2 | is_a_theorem(implies(A,or(B,A))) # label(or_2) # label(axiom). [clausify(11)].
% 121.44/121.71 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)].
% 121.44/121.71 88 -equivalence_2 | is_a_theorem(implies(equiv(A,B),implies(B,A))) # label(equivalence_2) # label(axiom). [clausify(14)].
% 121.44/121.71 90 -equivalence_3 | is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))) # label(equivalence_3) # label(axiom). [clausify(15)].
% 121.44/121.71 114 -op_or | or(A,B) = not(and(not(A),not(B))) # label(op_or) # label(axiom). [clausify(27)].
% 121.44/121.71 115 -op_or | not(and(not(A),not(B))) = or(A,B). [copy(114),flip(b)].
% 121.44/121.71 118 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(29)].
% 121.44/121.71 120 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B) # label(op_equiv) # label(axiom). [clausify(31)].
% 121.44/121.71 121 op_or # label(hilbert_op_or) # label(axiom). [assumption].
% 121.44/121.71 122 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption].
% 121.44/121.71 123 op_equiv # label(hilbert_op_equiv) # label(axiom). [assumption].
% 121.44/121.71 124 modus_ponens # label(hilbert_modus_ponens) # label(axiom). [assumption].
% 121.44/121.71 125 modus_tollens # label(hilbert_modus_tollens) # label(axiom). [assumption].
% 121.44/121.71 126 implies_1 # label(hilbert_implies_1) # label(axiom). [assumption].
% 121.44/121.71 127 implies_2 # label(hilbert_implies_2) # label(axiom). [assumption].
% 121.44/121.71 128 implies_3 # label(hilbert_implies_3) # label(axiom). [assumption].
% 121.44/121.71 129 and_1 # label(hilbert_and_1) # label(axiom). [assumption].
% 121.44/121.71 130 and_2 # label(hilbert_and_2) # label(axiom). [assumption].
% 121.44/121.71 131 and_3 # label(hilbert_and_3) # label(axiom). [assumption].
% 121.44/121.71 132 or_1 # label(hilbert_or_1) # label(axiom). [assumption].
% 121.44/121.71 133 or_2 # label(hilbert_or_2) # label(axiom). [assumption].
% 121.44/121.71 134 or_3 # label(hilbert_or_3) # label(axiom). [assumption].
% 121.44/121.71 136 equivalence_2 # label(hilbert_equivalence_2) # label(axiom). [assumption].
% 121.44/121.71 137 equivalence_3 # label(hilbert_equivalence_3) # label(axiom). [assumption].
% 121.44/121.71 138 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption].
% 121.44/121.71 139 -necessitation | -is_a_theorem(A) | is_a_theorem(necessarily(A)) # label(necessitation) # label(axiom). [clausify(32)].
% 121.44/121.71 153 -axiom_K | is_a_theorem(implies(necessarily(implies(A,B)),implies(necessarily(A),necessarily(B)))) # label(axiom_K) # label(axiom). [clausify(36)].
% 121.44/121.71 155 -axiom_M | is_a_theorem(implies(necessarily(A),A)) # label(axiom_M) # label(axiom). [clausify(37)].
% 121.44/121.71 157 -axiom_4 | is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))) # label(axiom_4) # label(axiom). [clausify(38)].
% 121.44/121.71 159 -axiom_B | is_a_theorem(implies(A,necessarily(possibly(A)))) # label(axiom_B) # label(axiom). [clausify(39)].
% 121.44/121.71 168 axiom_s3 | -is_a_theorem(strict_implies(strict_implies(c74,c75),strict_implies(not(possibly(c75)),not(possibly(c74))))) # label(axiom_s3) # label(axiom). [clausify(43)].
% 121.44/121.71 191 -op_possibly | possibly(A) = not(necessarily(not(A))) # label(op_possibly) # label(axiom). [clausify(55)].
% 121.44/121.71 192 -op_possibly | not(necessarily(not(A))) = possibly(A). [copy(191),flip(b)].
% 121.44/121.71 195 -op_strict_implies | strict_implies(A,B) = necessarily(implies(A,B)) # label(op_strict_implies) # label(axiom). [clausify(57)].
% 121.44/121.71 196 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B). [copy(195),flip(b)].
% 121.44/121.71 199 op_possibly # label(km4b_op_possibly) # label(axiom). [assumption].
% 121.44/121.71 200 necessitation # label(km4b_necessitation) # label(axiom). [assumption].
% 121.44/121.71 201 axiom_K # label(km4b_axiom_K) # label(axiom). [assumption].
% 121.44/121.71 202 axiom_M # label(km4b_axiom_M) # label(axiom). [assumption].
% 121.44/121.71 203 axiom_4 # label(km4b_axiom_4) # label(axiom). [assumption].
% 121.44/121.71 204 axiom_B # label(km4b_axiom_B) # label(axiom). [assumption].
% 121.44/121.71 206 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption].
% 121.44/121.71 208 -axiom_s3 # label(s1_0_m6s3m9b_axiom_s3) # label(negated_conjecture). [assumption].
% 121.44/121.71 209 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,121)].
% 121.44/121.71 210 not(and(A,not(B))) = implies(A,B). [back_unit_del(118),unit_del(a,122)].
% 121.44/121.71 211 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(120),unit_del(a,123)].
% 121.44/121.71 212 -is_a_theorem(A) | -is_a_theorem(implies(A,B)) | is_a_theorem(B). [back_unit_del(59),unit_del(a,124)].
% 121.44/121.71 213 is_a_theorem(implies(implies(not(A),not(B)),implies(B,A))). [back_unit_del(66),unit_del(a,125)].
% 121.44/121.71 214 is_a_theorem(implies(A,implies(B,A))). [back_unit_del(68),unit_del(a,126)].
% 121.44/121.71 215 is_a_theorem(implies(implies(A,implies(A,B)),implies(A,B))). [back_unit_del(70),unit_del(a,127)].
% 121.44/121.71 216 is_a_theorem(implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [back_unit_del(72),unit_del(a,128)].
% 121.44/121.71 217 is_a_theorem(implies(and(A,B),A)). [back_unit_del(74),unit_del(a,129)].
% 121.44/121.71 218 is_a_theorem(implies(and(A,B),B)). [back_unit_del(76),unit_del(a,130)].
% 121.44/121.71 219 is_a_theorem(implies(A,implies(B,and(A,B)))). [back_unit_del(78),unit_del(a,131)].
% 121.44/121.71 220 is_a_theorem(implies(A,or(A,B))). [back_unit_del(80),unit_del(a,132)].
% 121.44/121.71 221 is_a_theorem(implies(A,or(B,A))). [back_unit_del(82),unit_del(a,133)].
% 121.44/121.71 222 is_a_theorem(implies(implies(A,B),implies(implies(C,B),implies(or(A,C),B)))). [back_unit_del(84),unit_del(a,134)].
% 121.44/121.71 224 is_a_theorem(implies(equiv(A,B),implies(B,A))). [back_unit_del(88),unit_del(a,136)].
% 121.44/121.71 225 is_a_theorem(implies(implies(A,B),implies(implies(B,A),equiv(A,B)))). [back_unit_del(90),unit_del(a,137)].
% 121.44/121.71 226 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,138)].
% 121.44/121.71 228 not(necessarily(not(A))) = possibly(A). [back_unit_del(192),unit_del(a,199)].
% 121.44/121.71 229 -is_a_theorem(A) | is_a_theorem(necessarily(A)). [back_unit_del(139),unit_del(a,200)].
% 121.44/121.71 230 is_a_theorem(implies(necessarily(implies(A,B)),implies(necessarily(A),necessarily(B)))). [back_unit_del(153),unit_del(a,201)].
% 121.44/121.71 231 is_a_theorem(implies(necessarily(A),A)). [back_unit_del(155),unit_del(a,202)].
% 121.44/121.71 232 is_a_theorem(implies(necessarily(A),necessarily(necessarily(A)))). [back_unit_del(157),unit_del(a,203)].
% 121.44/121.71 233 is_a_theorem(implies(A,necessarily(possibly(A)))). [back_unit_del(159),unit_del(a,204)].
% 121.44/121.71 234 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(196),unit_del(a,206)].
% 121.44/121.71 236 -is_a_theorem(strict_implies(strict_implies(c74,c75),strict_implies(not(possibly(c75)),not(possibly(c74))))). [back_unit_del(168),unit_del(a,208)].
% 121.44/121.71 237 implies(not(A),B) = or(A,B). [back_rewrite(209),rewrite([210(4)])].
% 121.44/121.71 241 is_a_theorem(implies(strict_implies(A,B),implies(necessarily(A),necessarily(B)))). [back_rewrite(230),rewrite([234(2)])].
% 121.44/121.71 244 is_a_theorem(implies(or(A,not(B)),implies(B,A))). [back_rewrite(213),rewrite([237(3)])].
% 121.44/121.71 256 not(and(A,implies(B,C))) = implies(A,and(B,not(C))). [para(210(a,1),210(a,1,1,2))].
% 121.44/121.71 257 -is_a_theorem(A) | is_a_theorem(implies(B,A)). [resolve(214,a,212,b)].
% 121.44/121.71 260 -is_a_theorem(implies(A,implies(A,B))) | is_a_theorem(implies(A,B)). [resolve(215,a,212,b)].
% 121.44/121.71 263 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,C),implies(A,C))). [resolve(216,a,212,b)].
% 121.44/121.71 281 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(C,B),implies(or(A,C),B))). [resolve(222,a,212,b)].
% 121.44/121.71 290 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(implies(B,A),equiv(A,B))). [resolve(225,a,212,b)].
% 121.44/121.71 308 not(and(A,possibly(B))) = implies(A,necessarily(not(B))). [para(228(a,1),210(a,1,1,2))].
% 121.44/121.71 309 possibly(and(A,not(B))) = not(strict_implies(A,B)). [para(210(a,1),228(a,1,1,1)),rewrite([234(2)]),flip(a)].
% 121.44/121.71 310 not(necessarily(possibly(A))) = possibly(necessarily(not(A))). [para(228(a,1),228(a,1,1,1))].
% 121.44/121.71 320 is_a_theorem(strict_implies(implies(A,B),implies(implies(B,C),implies(A,C)))). [resolve(229,a,216,a),rewrite([234(6)])].
% 121.44/121.71 322 is_a_theorem(strict_implies(A,implies(B,A))). [resolve(229,a,214,a),rewrite([234(3)])].
% 121.44/121.71 331 -is_a_theorem(necessarily(A)) | is_a_theorem(A). [resolve(231,a,212,b)].
% 121.44/121.71 349 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(210(a,1),237(a,1,1)),flip(a)].
% 121.44/121.71 350 and(or(A,B),implies(B,not(A))) = equiv(not(A),B). [para(237(a,1),211(a,1,1))].
% 121.44/121.71 355 is_a_theorem(implies(or(A,B),implies(implies(B,C),or(A,C)))). [para(237(a,1),216(a,1,1)),rewrite([237(4)])].
% 121.44/121.71 367 or(necessarily(not(A)),B) = implies(possibly(A),B). [para(228(a,1),237(a,1,1)),flip(a)].
% 121.44/121.71 369 necessarily(or(A,B)) = strict_implies(not(A),B). [para(237(a,1),234(a,1,1))].
% 121.44/121.71 370 is_a_theorem(strict_implies(strict_implies(A,B),implies(necessarily(A),necessarily(B)))). [resolve(241,a,229,a),rewrite([234(6)])].
% 121.44/121.71 372 -is_a_theorem(strict_implies(A,B)) | is_a_theorem(implies(necessarily(A),necessarily(B))). [resolve(241,a,212,b)].
% 121.44/121.71 379 -is_a_theorem(or(A,not(B))) | is_a_theorem(implies(B,A)). [resolve(244,a,212,b)].
% 121.44/121.71 384 is_a_theorem(implies(or(A,not(not(B))),or(B,A))). [para(237(a,1),244(a,1,2))].
% 121.44/121.71 438 -is_a_theorem(strict_implies(A,B)) | is_a_theorem(implies(A,B)). [para(234(a,1),331(a,1))].
% 121.44/121.71 479 is_a_theorem(implies(A,implies(necessarily(B),B))). [resolve(257,a,231,a)].
% 121.44/121.71 646 implies(implies(A,B),and(B,not(A))) = not(equiv(A,B)). [para(211(a,1),256(a,1,1)),flip(a)].
% 121.44/121.71 648 implies(implies(A,and(B,not(C))),D) = or(and(A,implies(B,C)),D). [para(256(a,1),237(a,1,1))].
% 121.44/121.71 874 is_a_theorem(implies(implies(A,B),implies(or(A,A),B))). [resolve(260,a,222,a)].
% 121.44/121.71 875 is_a_theorem(implies(A,and(A,A))). [resolve(260,a,219,a)].
% 121.44/121.71 877 is_a_theorem(implies(A,A)). [resolve(260,a,214,a)].
% 121.44/121.71 884 is_a_theorem(implies(A,implies(B,B))). [resolve(877,a,257,a)].
% 121.44/121.71 891 is_a_theorem(strict_implies(A,A)). [resolve(877,a,229,a),rewrite([234(2)])].
% 121.44/121.71 895 is_a_theorem(or(A,not(A))). [para(237(a,1),877(a,1))].
% 121.44/121.71 900 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(891,a,257,a)].
% 121.44/121.71 959 is_a_theorem(or(and(A,implies(B,C)),implies(A,and(B,not(C))))). [para(256(a,1),895(a,1,2))].
% 121.44/121.71 1088 is_a_theorem(implies(implies(necessarily(possibly(A)),B),implies(A,B))). [resolve(263,a,233,a)].
% 121.44/121.71 1095 is_a_theorem(implies(implies(or(A,B),C),implies(B,C))). [resolve(263,a,221,a)].
% 121.44/121.71 1102 is_a_theorem(implies(implies(implies(A,B),C),implies(B,C))). [resolve(263,a,214,a)].
% 121.44/121.71 1103 -is_a_theorem(or(A,B)) | is_a_theorem(implies(implies(B,C),or(A,C))). [para(237(a,1),263(a,1)),rewrite([237(5)])].
% 121.44/121.71 1662 is_a_theorem(implies(implies(implies(A,A),B),implies(C,B))). [resolve(884,a,263,a)].
% 121.44/121.71 1817 is_a_theorem(implies(implies(strict_implies(A,A),B),implies(C,B))). [resolve(900,a,263,a)].
% 121.44/121.71 2534 is_a_theorem(implies(implies(A,B),implies(or(B,A),B))). [resolve(281,a,877,a)].
% 121.44/121.71 2549 is_a_theorem(implies(implies(A,B),implies(or(and(B,C),A),B))). [resolve(281,a,217,a)].
% 121.44/121.71 4056 is_a_theorem(implies(implies(strict_implies(A,A),B),equiv(B,strict_implies(A,A)))). [resolve(290,a,900,a)].
% 121.44/121.71 4057 is_a_theorem(implies(implies(implies(A,A),B),equiv(B,implies(A,A)))). [resolve(290,a,884,a)].
% 121.44/121.71 4058 is_a_theorem(implies(implies(and(A,A),A),equiv(A,and(A,A)))). [resolve(290,a,875,a)].
% 121.44/121.71 4067 is_a_theorem(implies(implies(necessarily(possibly(A)),A),equiv(A,necessarily(possibly(A))))). [resolve(290,a,233,a)].
% 121.44/121.71 4068 is_a_theorem(implies(implies(necessarily(necessarily(A)),necessarily(A)),equiv(necessarily(A),necessarily(necessarily(A))))). [resolve(290,a,232,a)].
% 121.44/121.71 4071 is_a_theorem(implies(implies(implies(A,B),equiv(B,A)),equiv(equiv(B,A),implies(A,B)))). [resolve(290,a,224,a)].
% 121.44/121.71 4075 is_a_theorem(implies(implies(or(A,B),A),equiv(A,or(A,B)))). [resolve(290,a,220,a)].
% 121.44/121.71 4076 is_a_theorem(implies(implies(implies(A,and(B,A)),B),equiv(B,implies(A,and(B,A))))). [resolve(290,a,219,a)].
% 121.44/121.71 4081 is_a_theorem(implies(implies(implies(A,B),B),equiv(B,implies(A,B)))). [resolve(290,a,214,a)].
% 121.44/121.71 4607 is_a_theorem(implies(implies(implies(necessarily(A),A),B),implies(C,B))). [resolve(479,a,263,a)].
% 121.44/121.71 4623 not(strict_implies(A,necessarily(not(B)))) = possibly(and(A,possibly(B))). [para(308(a,1),228(a,1,1,1)),rewrite([234(4)])].
% 121.44/121.71 4802 implies(possibly(necessarily(not(A))),B) = or(necessarily(possibly(A)),B). [para(310(a,1),237(a,1,1))].
% 121.44/121.71 7974 is_a_theorem(implies(or(A,not(B)),implies(or(B,C),or(A,C)))). [para(237(a,1),355(a,1,2,1))].
% 121.44/121.71 8800 is_a_theorem(implies(necessarily(strict_implies(A,B)),strict_implies(necessarily(A),necessarily(B)))). [resolve(372,a,370,a),rewrite([234(6)])].
% 121.44/121.71 8802 is_a_theorem(implies(necessarily(A),strict_implies(B,A))). [resolve(372,a,322,a),rewrite([234(3)])].
% 121.44/121.71 8804 is_a_theorem(implies(strict_implies(A,B),strict_implies(implies(B,C),implies(A,C)))). [resolve(372,a,320,a),rewrite([234(2),234(5)])].
% 121.44/121.71 15807 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(or(A,A),B)). [resolve(874,a,212,b)].
% 121.44/121.71 15985 -is_a_theorem(implies(necessarily(possibly(A)),B)) | is_a_theorem(implies(A,B)). [resolve(1088,a,212,b)].
% 121.44/121.71 15990 -is_a_theorem(implies(or(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(1095,a,212,b)].
% 121.44/121.71 16163 -is_a_theorem(implies(implies(A,B),C)) | is_a_theorem(implies(B,C)). [resolve(1102,a,212,b)].
% 121.44/121.71 16287 is_a_theorem(implies(implies(implies(A,A),B),B)). [resolve(1662,a,260,a)].
% 121.44/121.71 16892 is_a_theorem(implies(implies(strict_implies(A,A),B),B)). [resolve(1817,a,260,a)].
% 121.44/121.71 18255 -is_a_theorem(implies(implies(A,A),B)) | is_a_theorem(B). [resolve(16287,a,212,b)].
% 121.44/121.71 18266 -is_a_theorem(implies(strict_implies(A,A),B)) | is_a_theorem(B). [resolve(16892,a,212,b)].
% 121.44/121.71 18299 is_a_theorem(implies(or(and(A,B),A),A)). [resolve(18255,a,2549,a)].
% 121.44/121.71 18413 is_a_theorem(equiv(A,and(A,A))). [resolve(4058,a,212,b),unit_del(a,217)].
% 121.44/121.71 18430 and(A,A) = A. [resolve(18413,a,226,a)].
% 121.44/121.71 18433 or(A,A) = not(not(A)). [para(18430(a,1),210(a,1,1)),rewrite([237(4)]),flip(a)].
% 121.44/121.71 18434 equiv(A,A) = implies(A,A). [para(18430(a,1),211(a,1)),flip(a)].
% 121.44/121.71 18436 implies(possibly(A),necessarily(not(A))) = not(possibly(A)). [para(18430(a,1),308(a,1,1)),flip(a)].
% 121.44/121.71 18437 not(strict_implies(not(A),A)) = possibly(not(A)). [para(18430(a,1),309(a,1,1)),flip(a)].
% 121.44/121.71 18438 implies(or(A,A),B) = or(not(A),B). [para(18430(a,1),349(a,1,1)),rewrite([237(4)]),flip(a)].
% 121.44/121.71 18440 is_a_theorem(or(not(A),A)). [para(18430(a,1),18299(a,1,1,1)),rewrite([18438(2)])].
% 121.44/121.71 18465 -is_a_theorem(implies(A,B)) | is_a_theorem(or(not(A),B)). [back_rewrite(15807),rewrite([18438(4)])].
% 121.44/121.71 18474 is_a_theorem(implies(A,not(not(A)))). [resolve(18440,a,379,a)].
% 121.44/121.71 19136 strict_implies(not(A),A) = necessarily(not(not(A))). [para(18433(a,1),369(a,1,1)),flip(a)].
% 121.44/121.71 19290 is_a_theorem(equiv(necessarily(A),necessarily(necessarily(A)))). [resolve(4068,a,212,b),unit_del(a,231)].
% 121.44/121.71 19409 necessarily(necessarily(A)) = necessarily(A). [resolve(19290,a,226,a)].
% 121.44/121.71 19424 necessarily(strict_implies(A,B)) = strict_implies(A,B). [para(234(a,1),19409(a,1,1)),rewrite([234(4)])].
% 121.44/121.71 19443 is_a_theorem(implies(strict_implies(A,B),strict_implies(necessarily(A),necessarily(B)))). [back_rewrite(8800),rewrite([19424(2)])].
% 121.44/121.71 20020 is_a_theorem(implies(implies(implies(necessarily(A),A),B),B)). [resolve(4607,a,260,a)].
% 121.44/121.71 24056 is_a_theorem(or(not(necessarily(A)),A)). [resolve(18465,a,231,a)].
% 121.44/121.71 24063 is_a_theorem(or(not(and(A,B)),A)). [resolve(18465,a,217,a)].
% 121.44/121.71 24128 is_a_theorem(implies(A,possibly(A))). [resolve(24056,a,379,a),rewrite([228(3)])].
% 121.44/121.71 24136 is_a_theorem(or(possibly(A),not(A))). [para(228(a,1),24056(a,1,1))].
% 121.44/121.71 24765 is_a_theorem(implies(or(A,B),or(possibly(A),B))). [resolve(24136,a,1103,a),rewrite([237(2)])].
% 121.44/121.71 24903 is_a_theorem(or(implies(A,B),A)). [para(210(a,1),24063(a,1,1))].
% 121.44/121.71 25964 -is_a_theorem(implies(implies(necessarily(A),A),B)) | is_a_theorem(B). [resolve(20020,a,212,b)].
% 121.44/121.71 31405 is_a_theorem(implies(A,not(possibly(necessarily(not(A)))))). [resolve(15985,a,18474,a),rewrite([310(3)])].
% 121.44/121.71 31408 is_a_theorem(implies(A,strict_implies(B,possibly(A)))). [resolve(15985,a,8802,a)].
% 121.44/121.71 31506 is_a_theorem(or(A,not(possibly(necessarily(not(not(A))))))). [para(237(a,1),31405(a,1))].
% 121.44/121.71 31753 is_a_theorem(or(necessarily(possibly(not(A))),A)). [resolve(31506,a,379,a),rewrite([4802(5)])].
% 121.44/121.71 31763 is_a_theorem(strict_implies(possibly(necessarily(not(not(A)))),A)). [resolve(31753,a,229,a),rewrite([369(5),310(4)])].
% 121.44/121.71 31774 is_a_theorem(strict_implies(possibly(strict_implies(not(A),A)),A)). [para(18433(a,2),31763(a,1,1,1,1)),rewrite([369(2)])].
% 121.44/121.71 31780 is_a_theorem(implies(possibly(strict_implies(not(A),A)),A)). [resolve(31774,a,438,a)].
% 121.44/121.71 31836 is_a_theorem(or(A,implies(A,B))). [resolve(15990,a,244,a),rewrite([237(3)])].
% 121.44/121.71 33607 is_a_theorem(implies(A,equiv(A,implies(B,A)))). [resolve(16163,a,4081,a)].
% 121.44/121.71 33608 is_a_theorem(implies(A,equiv(A,or(A,B)))). [resolve(16163,a,4075,a)].
% 121.44/121.71 33612 is_a_theorem(implies(A,equiv(A,implies(B,B)))). [resolve(16163,a,4057,a)].
% 121.44/121.71 33613 is_a_theorem(implies(A,equiv(A,strict_implies(B,B)))). [resolve(16163,a,4056,a)].
% 121.44/121.71 33624 -is_a_theorem(A) | is_a_theorem(equiv(A,implies(B,A))). [resolve(33607,a,212,b)].
% 121.44/121.71 33631 -is_a_theorem(A) | is_a_theorem(equiv(A,or(A,B))). [resolve(33608,a,212,b)].
% 121.44/121.71 33663 is_a_theorem(equiv(strict_implies(A,A),implies(B,B))). [resolve(33612,a,18266,a)].
% 121.44/121.71 33673 is_a_theorem(equiv(implies(necessarily(A),A),strict_implies(B,B))). [resolve(33613,a,25964,a)].
% 121.44/121.71 33681 is_a_theorem(equiv(implies(A,A),strict_implies(B,B))). [resolve(33613,a,18255,a)].
% 121.44/121.71 33684 -is_a_theorem(A) | is_a_theorem(equiv(A,strict_implies(B,B))). [resolve(33613,a,212,b)].
% 121.44/121.71 33745 strict_implies(A,A) = implies(B,B). [resolve(33663,a,226,a),flip(a)].
% 121.44/121.71 33747 strict_implies(A,A) = c_0. [new_symbol(33745)].
% 121.44/121.71 33749 implies(A,A) = c_0. [back_rewrite(33745),rewrite([33747(1)]),flip(a)].
% 121.44/121.71 33753 -is_a_theorem(A) | is_a_theorem(equiv(A,c_0)). [back_rewrite(33684),rewrite([33747(2)])].
% 121.44/121.71 33756 is_a_theorem(c_0). [back_rewrite(33681),rewrite([33749(1),33747(2),18434(3),33749(3)])].
% 121.44/121.71 33763 is_a_theorem(equiv(implies(necessarily(A),A),c_0)). [back_rewrite(33673),rewrite([33747(3)])].
% 121.44/121.71 33767 is_a_theorem(implies(A,equiv(A,c_0))). [back_rewrite(33613),rewrite([33747(1)])].
% 121.44/121.71 34653 is_a_theorem(implies(implies(c_0,A),A)). [back_rewrite(16892),rewrite([33747(1)])].
% 121.44/121.71 35159 is_a_theorem(equiv(c_0,or(c_0,A))). [resolve(33756,a,33631,a)].
% 121.44/121.71 35160 is_a_theorem(equiv(c_0,implies(A,c_0))). [resolve(33756,a,33624,a)].
% 121.44/121.71 36730 is_a_theorem(equiv(or(A,implies(A,B)),c_0)). [resolve(33753,a,31836,a)].
% 121.44/121.71 36860 is_a_theorem(equiv(or(implies(A,B),A),c_0)). [resolve(33753,a,24903,a)].
% 121.44/121.71 36895 is_a_theorem(equiv(implies(A,possibly(A)),c_0)). [resolve(33753,a,24128,a)].
% 121.44/121.71 36988 is_a_theorem(equiv(implies(A,not(not(A))),c_0)). [resolve(33753,a,18474,a)].
% 121.44/121.71 36995 is_a_theorem(equiv(or(not(A),A),c_0)). [resolve(33753,a,18440,a)].
% 121.44/121.71 37084 is_a_theorem(equiv(implies(A,or(B,A)),c_0)). [resolve(33753,a,221,a)].
% 121.44/121.71 37086 is_a_theorem(equiv(implies(and(A,B),B),c_0)). [resolve(33753,a,218,a)].
% 121.44/121.71 37087 is_a_theorem(equiv(implies(and(A,B),A),c_0)). [resolve(33753,a,217,a)].
% 121.44/121.71 37088 is_a_theorem(equiv(implies(A,implies(B,A)),c_0)). [resolve(33753,a,214,a)].
% 121.44/121.71 37115 is_a_theorem(equiv(implies(A,equiv(A,c_0)),c_0)). [resolve(33767,a,33753,a)].
% 121.44/121.71 37584 is_a_theorem(equiv(implies(implies(c_0,A),A),c_0)). [resolve(34653,a,33753,a)].
% 121.44/121.71 38583 or(c_0,A) = c_0. [resolve(35159,a,226,a)].
% 121.44/121.71 38608 implies(A,c_0) = c_0. [resolve(35160,a,226,a)].
% 121.44/121.71 38609 and(c_0,implies(c_0,A)) = equiv(A,c_0). [para(38608(a,1),211(a,1,1))].
% 121.44/121.71 38610 and(implies(c_0,A),c_0) = equiv(c_0,A). [para(38608(a,1),211(a,1,2))].
% 121.44/121.71 38860 implies(necessarily(A),A) = c_0. [resolve(33763,a,226,a),flip(a)].
% 121.44/121.71 39560 implies(A,possibly(A)) = c_0. [resolve(36895,a,226,a),flip(a)].
% 121.44/121.71 39610 or(not(A),A) = c_0. [resolve(36995,a,226,a),flip(a)].
% 121.44/121.71 40220 or(A,implies(A,B)) = c_0. [resolve(36730,a,226,a),flip(a)].
% 121.44/121.71 40221 or(not(A),or(A,B)) = c_0. [para(237(a,1),40220(a,1,2))].
% 121.44/121.71 40273 or(implies(A,B),A) = c_0. [resolve(36860,a,226,a),flip(a)].
% 121.44/121.71 40274 or(or(A,B),not(A)) = c_0. [para(237(a,1),40273(a,1,1))].
% 121.44/121.71 40357 implies(A,not(not(A))) = c_0. [resolve(36988,a,226,a),flip(a)].
% 121.44/121.71 40358 equiv(A,not(not(A))) = c_0. [para(40357(a,1),211(a,1,1)),rewrite([237(4),39610(3),18430(3)]),flip(a)].
% 121.44/121.71 40364 not(not(A)) = A. [para(40358(a,1),226(a,1)),unit_del(a,33756)].
% 121.44/121.71 40384 strict_implies(not(A),A) = necessarily(A). [back_rewrite(19136),rewrite([40364(4)])].
% 121.44/121.71 40386 or(A,A) = A. [back_rewrite(18433),rewrite([40364(3)])].
% 121.44/121.71 40393 is_a_theorem(implies(or(A,B),or(B,A))). [back_rewrite(384),rewrite([40364(2)])].
% 121.44/121.71 40394 is_a_theorem(implies(possibly(necessarily(A)),A)). [back_rewrite(31780),rewrite([40384(2)])].
% 121.44/121.71 40401 not(necessarily(A)) = possibly(not(A)). [back_rewrite(18437),rewrite([40384(2)])].
% 121.44/121.71 40422 or(not(A),B) = implies(A,B). [back_rewrite(18438),rewrite([40386(1)]),flip(a)].
% 121.44/121.71 40533 implies(A,or(A,B)) = c_0. [back_rewrite(40221),rewrite([40422(3)])].
% 121.44/121.71 40551 not(and(A,B)) = implies(A,not(B)). [para(40364(a,1),210(a,1,1,2))].
% 121.44/121.71 40552 not(implies(A,B)) = and(A,not(B)). [para(210(a,1),40364(a,1,1))].
% 121.44/121.71 40986 not(possibly(A)) = necessarily(not(A)). [para(40386(a,1),367(a,1)),rewrite([18436(6)]),flip(a)].
% 121.44/121.71 40999 is_a_theorem(implies(or(A,not(B)),implies(or(B,A),A))). [para(40386(a,1),7974(a,1,2,2))].
% 121.44/121.71 41006 is_a_theorem(implies(or(A,possibly(A)),possibly(A))). [para(40386(a,1),24765(a,1,2))].
% 121.44/121.71 41081 -is_a_theorem(strict_implies(strict_implies(c74,c75),strict_implies(necessarily(not(c75)),necessarily(not(c74))))). [back_rewrite(236),rewrite([40986(6),40986(9)])].
% 121.44/121.71 41305 is_a_theorem(implies(possibly(necessarily(A)),necessarily(A))). [para(19409(a,1),40394(a,1,1,1))].
% 121.44/121.71 42135 possibly(possibly(A)) = possibly(A). [para(40384(a,1),4623(a,1,1)),rewrite([19409(3),40401(3),40364(2),40401(4),40364(3),18430(4)]),flip(a)].
% 121.44/121.71 42216 is_a_theorem(implies(c_0,equiv(possibly(A),necessarily(possibly(A))))). [para(42135(a,1),4067(a,1,1,1,1)),rewrite([38860(4),42135(4)])].
% 121.44/121.71 42341 is_a_theorem(implies(possibly(A),strict_implies(B,possibly(A)))). [para(42135(a,1),31408(a,1,2,2))].
% 121.44/121.71 43695 is_a_theorem(strict_implies(implies(A,B),implies(c_0,implies(A,or(B,C))))). [para(40533(a,1),320(a,1,2,1))].
% 121.44/121.71 44458 is_a_theorem(implies(c_0,equiv(possibly(necessarily(A)),necessarily(A)))). [resolve(41305,a,290,a),rewrite([39560(4)])].
% 121.44/121.71 45075 implies(A,or(B,A)) = c_0. [resolve(37084,a,226,a),flip(a)].
% 121.44/121.71 45104 implies(and(A,B),B) = c_0. [resolve(37086,a,226,a),flip(a)].
% 121.44/121.71 45107 implies(equiv(A,B),implies(B,A)) = c_0. [para(211(a,1),45104(a,1,1))].
% 121.44/121.71 45112 is_a_theorem(implies(c_0,implies(or(A,and(B,A)),A))). [para(45104(a,1),2534(a,1,1))].
% 121.44/121.71 45124 implies(and(A,B),A) = c_0. [resolve(37087,a,226,a),flip(a)].
% 121.44/121.71 45144 implies(A,implies(B,A)) = c_0. [resolve(37088,a,226,a),flip(a)].
% 121.44/121.71 45225 implies(A,equiv(A,c_0)) = c_0. [resolve(37115,a,226,a),flip(a)].
% 121.44/121.71 45744 implies(implies(c_0,A),A) = c_0. [resolve(37584,a,226,a),flip(a)].
% 121.44/121.71 45745 equiv(implies(c_0,A),A) = c_0. [para(45744(a,1),211(a,1,1)),rewrite([45144(4),18430(3)]),flip(a)].
% 121.44/121.71 45766 implies(c_0,A) = A. [para(45745(a,1),226(a,1)),flip(b),unit_del(a,33756)].
% 121.44/121.71 45777 is_a_theorem(implies(or(A,and(B,A)),A)). [back_rewrite(45112),rewrite([45766(5)])].
% 121.44/121.71 45779 is_a_theorem(equiv(possibly(necessarily(A)),necessarily(A))). [back_rewrite(44458),rewrite([45766(6)])].
% 121.44/121.71 45789 is_a_theorem(strict_implies(implies(A,B),implies(A,or(B,C)))). [back_rewrite(43695),rewrite([45766(5)])].
% 121.44/121.71 45791 is_a_theorem(equiv(possibly(A),necessarily(possibly(A)))). [back_rewrite(42216),rewrite([45766(6)])].
% 121.44/121.71 45793 equiv(c_0,A) = and(A,c_0). [back_rewrite(38610),rewrite([45766(2)]),flip(a)].
% 121.44/121.71 45794 equiv(A,c_0) = and(c_0,A). [back_rewrite(38609),rewrite([45766(3)]),flip(a)].
% 121.44/121.71 45914 implies(A,and(c_0,A)) = c_0. [back_rewrite(45225),rewrite([45794(2)])].
% 121.44/121.71 45978 strict_implies(c_0,A) = necessarily(A). [para(45766(a,1),234(a,1,1)),flip(a)].
% 121.44/121.71 45991 implies(A,and(A,not(c_0))) = implies(A,not(c_0)). [para(45766(a,1),646(a,1,1)),rewrite([45793(6),40551(7)])].
% 121.44/121.71 45995 is_a_theorem(or(and(A,B),implies(A,and(c_0,not(B))))). [para(45766(a,1),959(a,1,1,2))].
% 121.44/121.71 46005 is_a_theorem(equiv(and(c_0,A),A)). [para(45766(a,1),4071(a,1,1,1)),rewrite([45794(2),45914(3),45794(3),45766(5),45766(5)])].
% 121.44/121.71 46006 is_a_theorem(equiv(A,and(A,c_0))). [para(45766(a,1),4076(a,1,1,1)),rewrite([45124(3),45766(5),45766(5)])].
% 121.44/121.71 46454 and(c_0,A) = A. [resolve(46005,a,226,a),flip(a)].
% 121.44/121.71 46456 is_a_theorem(or(and(A,B),implies(A,not(B)))). [back_rewrite(45995),rewrite([46454(4)])].
% 121.44/121.71 46457 equiv(A,c_0) = A. [back_rewrite(45794),rewrite([46454(4)])].
% 121.44/121.71 46479 -is_a_theorem(A) | c_0 = A. [para(46457(a,1),226(a,1))].
% 121.44/121.71 49497 implies(strict_implies(A,B),strict_implies(implies(B,C),implies(A,C))) = c_0. [resolve(46479,a,8804,a),flip(a)].
% 121.44/121.71 49644 implies(implies(implies(A,and(B,A)),B),equiv(B,implies(A,and(B,A)))) = c_0. [resolve(46479,a,4076,a),flip(a)].
% 121.44/121.71 49684 implies(implies(or(A,B),C),implies(B,C)) = c_0. [resolve(46479,a,1095,a),flip(a)].
% 121.44/121.71 49754 and(A,c_0) = A. [resolve(46006,a,226,a)].
% 121.44/121.71 49756 equiv(c_0,A) = A. [back_rewrite(45793),rewrite([49754(4)])].
% 121.44/121.71 49757 implies(A,not(c_0)) = not(A). [para(49756(a,1),646(a,2,1)),rewrite([45766(2),45991(4)])].
% 121.44/121.71 49846 possibly(necessarily(A)) = necessarily(A). [resolve(45779,a,226,a),flip(a)].
% 121.44/121.71 49929 possibly(strict_implies(A,B)) = strict_implies(A,B). [para(234(a,1),49846(a,1,1)),rewrite([234(4)])].
% 121.44/121.71 49956 necessarily(possibly(A)) = possibly(A). [resolve(45791,a,226,a)].
% 121.44/121.71 50188 equiv(A,not(c_0)) = not(A). [para(49757(a,1),211(a,1,1)),rewrite([237(4),38583(3),49754(3)]),flip(a)].
% 121.44/121.71 52683 implies(or(A,B),or(B,A)) = c_0. [resolve(40393,a,46479,a),flip(a)].
% 121.44/121.71 52686 is_a_theorem(equiv(or(A,B),or(B,A))). [resolve(40393,a,290,a),rewrite([52683(3),45766(5)])].
% 121.44/121.71 53571 is_a_theorem(equiv(or(A,possibly(A)),possibly(A))). [resolve(41006,a,290,a),rewrite([45075(4),45766(6)])].
% 121.44/121.71 53761 implies(possibly(A),strict_implies(B,possibly(A))) = c_0. [resolve(42341,a,46479,a),flip(a)].
% 121.44/121.71 53918 is_a_theorem(or(and(A,implies(B,A)),not(A))). [para(40422(a,1),45777(a,1,1)),rewrite([648(5)])].
% 121.44/121.71 54093 -is_a_theorem(strict_implies(A,B)) | is_a_theorem(strict_implies(necessarily(A),necessarily(B))). [resolve(19443,a,212,b)].
% 121.44/121.71 54094 is_a_theorem(implies(strict_implies(possibly(A),B),strict_implies(possibly(A),necessarily(B)))). [para(49956(a,1),19443(a,1,2,1))].
% 121.44/121.71 54952 or(A,B) = or(B,A). [resolve(52686,a,226,a)].
% 121.44/121.71 54982 is_a_theorem(implies(A,and(A,implies(B,A)))). [back_rewrite(53918),rewrite([54952(4),40422(4)])].
% 121.44/121.71 55188 is_a_theorem(implies(or(A,not(B)),implies(or(A,B),A))). [back_rewrite(40999),rewrite([54952(3)])].
% 121.44/121.71 55198 or(A,not(B)) = implies(B,A). [back_rewrite(40422),rewrite([54952(2)])].
% 121.44/121.71 55199 or(not(A),or(A,B)) = c_0. [back_rewrite(40274),rewrite([54952(3)])].
% 121.44/121.71 55257 is_a_theorem(implies(implies(A,B),implies(or(B,A),B))). [back_rewrite(55188),rewrite([55198(2)])].
% 121.44/121.71 56355 or(A,possibly(A)) = possibly(A). [resolve(53571,a,226,a),flip(a)].
% 121.44/121.71 56600 is_a_theorem(equiv(A,and(A,implies(B,A)))). [resolve(54982,a,290,a),rewrite([45124(3),45766(5)])].
% 121.44/121.71 56670 equiv(not(A),not(B)) = equiv(B,A). [para(55198(a,1),350(a,1,1)),rewrite([237(4),55198(3),211(3)]),flip(a)].
% 121.44/121.71 56672 strict_implies(not(A),not(B)) = strict_implies(B,A). [para(55198(a,1),369(a,1,1)),rewrite([234(2)]),flip(a)].
% 121.44/121.71 56675 or(not(A),implies(B,A)) = c_0. [para(45144(a,1),55198(a,2)),rewrite([54952(3)])].
% 121.44/121.71 56685 equiv(A,or(A,B)) = implies(or(A,B),A). [para(55199(a,1),350(a,1,1)),rewrite([40364(4),46454(4),40364(4)]),flip(a)].
% 121.44/121.71 57008 and(A,implies(B,A)) = A. [resolve(56600,a,226,a)].
% 121.44/121.71 57010 and(A,or(A,B)) = A. [para(237(a,1),57008(a,1,2)),rewrite([54952(1)])].
% 121.44/121.71 57023 equiv(A,implies(B,A)) = implies(implies(B,A),A). [para(56675(a,1),350(a,1,1)),rewrite([40364(4),46454(4),40364(4)]),flip(a)].
% 121.44/121.71 57393 implies(A,not(or(A,B))) = not(A). [para(57010(a,1),40551(a,1,1)),flip(a)].
% 121.44/121.71 57889 is_a_theorem(or(implies(A,B),and(not(B),A))). [para(237(a,1),46456(a,1,2)),rewrite([55198(4),54952(4)])].
% 121.44/121.71 59936 -is_a_theorem(equiv(A,B)) | not(A) = not(B). [para(56670(a,1),226(a,1))].
% 121.44/121.71 59945 strict_implies(A,not(B)) = strict_implies(B,not(A)). [para(40364(a,1),56672(a,1,1))].
% 121.44/121.71 60193 implies(or(A,B),not(A)) = not(A). [para(57393(a,1),55198(a,2)),rewrite([54952(4),55198(4)])].
% 121.44/121.71 60263 is_a_theorem(implies(equiv(A,B),equiv(B,A))). [para(646(a,1),57889(a,1,1)),rewrite([40551(5),40364(4),211(5),54952(4),55198(4)])].
% 121.44/121.71 60266 implies(equiv(A,B),equiv(B,A)) = c_0. [resolve(60263,a,46479,a),flip(a)].
% 121.44/121.71 60270 is_a_theorem(equiv(equiv(A,B),equiv(B,A))). [resolve(60263,a,290,a),rewrite([60266(3),45766(5)])].
% 121.44/121.71 60273 equiv(A,B) = equiv(B,A). [resolve(60270,a,226,a)].
% 121.44/121.71 60366 -is_a_theorem(equiv(A,B)) | A = B. [para(60273(a,1),226(a,1))].
% 121.44/121.71 60778 -is_a_theorem(not(A)) | not(A) = c_0. [para(50188(a,1),59936(a,1)),rewrite([40364(6)])].
% 121.44/121.71 60780 -is_a_theorem(possibly(not(A))) | possibly(not(A)) = c_0. [para(40401(a,1),60778(a,1)),rewrite([40401(5)])].
% 121.44/121.71 60981 implies(implies(A,B),A) = A. [para(40364(a,1),60193(a,1,2)),rewrite([54952(2),55198(2),40364(4)])].
% 121.44/121.71 60985 and(or(A,B),A) = A. [para(60193(a,1),40552(a,1,1)),rewrite([40364(2),40364(3)]),flip(a)].
% 121.44/121.71 60990 implies(equiv(A,implies(A,B)),A) = c_0. [para(60981(a,1),45107(a,1,2))].
% 121.44/121.71 60992 and(possibly(A),A) = A. [para(56355(a,1),60985(a,1,1))].
% 121.44/121.71 61211 -is_a_theorem(possibly(A)) | possibly(A) = c_0. [para(40364(a,1),60780(a,1,1)),rewrite([40364(4)])].
% 121.44/121.71 61213 -is_a_theorem(strict_implies(A,B)) | strict_implies(A,B) = c_0. [para(49929(a,1),61211(a,1)),rewrite([49929(4)])].
% 121.44/121.71 61370 implies(equiv(c_0,possibly(A)),possibly(A)) = c_0. [para(53761(a,1),60990(a,1,1,2)),rewrite([60273(3)])].
% 121.44/121.71 62640 is_a_theorem(strict_implies(implies(A,B),implies(A,possibly(B)))). [para(56355(a,1),45789(a,1,2,2))].
% 121.44/121.71 62681 strict_implies(implies(A,B),implies(A,possibly(B))) = c_0. [resolve(62640,a,61213,a)].
% 121.44/121.71 63667 implies(strict_implies(A,necessarily(B)),strict_implies(A,B)) = c_0. [para(38860(a,1),49497(a,1,2,1)),rewrite([45978(5),234(4)])].
% 121.44/121.71 63780 implies(possibly(A),equiv(c_0,possibly(A))) = c_0. [para(60992(a,1),49644(a,1,1,1,2)),rewrite([33749(1),45766(3),60992(4),33749(3),60273(4)])].
% 121.44/121.71 63781 equiv(possibly(A),equiv(c_0,possibly(A))) = c_0. [para(63780(a,1),211(a,1,1)),rewrite([61370(6),18430(3)]),flip(a)].
% 121.44/121.71 64150 equiv(c_0,possibly(A)) = possibly(A). [para(63781(a,1),226(a,1)),unit_del(a,33756)].
% 121.44/121.71 64155 equiv(c_0,strict_implies(A,B)) = strict_implies(A,B). [para(49929(a,1),64150(a,1,2)),rewrite([49929(5)])].
% 121.44/121.71 65319 is_a_theorem(equiv(implies(A,B),implies(or(B,A),B))). [resolve(55257,a,290,a),rewrite([49684(4),45766(6)])].
% 121.44/121.71 68799 is_a_theorem(strict_implies(strict_implies(A,B),strict_implies(A,possibly(B)))). [para(62681(a,1),54093(a,1)),rewrite([234(4),234(6)]),unit_del(a,33756)].
% 121.44/121.71 68800 strict_implies(strict_implies(A,B),strict_implies(A,possibly(B))) = c_0. [resolve(68799,a,61213,a)].
% 121.44/121.71 70626 implies(or(A,B),B) = implies(A,B). [resolve(65319,a,60366,a),rewrite([54952(2)]),flip(a)].
% 121.44/121.71 70627 implies(or(A,B),A) = implies(B,A). [resolve(65319,a,226,a)].
% 121.44/121.71 70629 equiv(A,or(A,B)) = implies(B,A). [back_rewrite(56685),rewrite([70627(4)])].
% 121.44/121.71 70643 implies(implies(A,B),B) = or(A,B). [para(237(a,1),70626(a,2)),rewrite([54952(2),55198(2)])].
% 121.44/121.71 70707 equiv(A,implies(B,A)) = or(A,B). [back_rewrite(57023),rewrite([70643(4),54952(3)])].
% 121.44/121.71 71241 -is_a_theorem(implies(A,B)) | or(A,B) = B. [para(70629(a,1),226(a,1)),rewrite([54952(3)])].
% 121.44/121.71 71682 or(strict_implies(A,B),strict_implies(A,necessarily(B))) = strict_implies(A,B). [para(63667(a,1),70707(a,1,2)),rewrite([60273(3),64155(3)]),flip(a)].
% 121.44/121.71 72587 strict_implies(possibly(A),necessarily(B)) = strict_implies(possibly(A),B). [resolve(71241,a,54094,a),rewrite([71682(6)]),flip(a)].
% 121.44/121.71 76673 strict_implies(necessarily(A),necessarily(B)) = strict_implies(necessarily(A),B). [para(49846(a,1),72587(a,1,1)),rewrite([49846(5)])].
% 121.44/121.71 76674 $F. [back_rewrite(41081),rewrite([76673(10),59945(9),40401(8),40364(7),68800(8)]),unit_del(a,33756)].
% 121.44/121.71
% 121.44/121.71 % SZS output end Refutation
% 121.44/121.71 ============================== end of proof ==========================
% 121.44/121.71
% 121.44/121.71 ============================== STATISTICS ============================
% 121.44/121.71
% 121.44/121.71 Given=31035. Generated=12182154. Kept=76609. proofs=1.
% 121.44/121.71 Usable=11322. Sos=5655. Demods=2158. Limbo=1, Disabled=59778. Hints=0.
% 121.44/121.71 Megabytes=59.16.
% 121.44/121.71 User_CPU=114.65, System_CPU=6.08, Wall_clock=120.
% 121.44/121.71
% 121.44/121.71 ============================== end of statistics =====================
% 121.44/121.71
% 121.44/121.71 ============================== end of search =========================
% 121.44/121.71
% 121.44/121.71 THEOREM PROVED
% 121.44/121.71 % SZS status Theorem
% 121.44/121.71
% 121.44/121.71 Exiting with 1 proof.
% 121.44/121.71
% 121.44/121.71 Process 16103 exit (max_proofs) Sun Jul 3 07:51:23 2022
% 121.44/121.71 Prover9 interrupted
%------------------------------------------------------------------------------