0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n026.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 : 180 0.12/0.33 % DateTime : Thu Aug 29 15:45:35 EDT 2019 0.12/0.33 % CPUTime : 0.80/1.10 ============================== Prover9 =============================== 0.80/1.10 Prover9 (32) version 2009-11A, November 2009. 0.80/1.10 Process 12421 was started by sandbox2 on n026.cluster.edu, 0.80/1.10 Thu Aug 29 15:45:36 2019 0.80/1.10 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 180 -f /tmp/Prover9_12268_n026.cluster.edu". 0.80/1.10 ============================== end of head =========================== 0.80/1.10 0.80/1.10 ============================== INPUT ================================= 0.80/1.10 0.80/1.10 % Reading from file /tmp/Prover9_12268_n026.cluster.edu 0.80/1.10 0.80/1.10 set(prolog_style_variables). 0.80/1.10 set(auto2). 0.80/1.10 % set(auto2) -> set(auto). 0.80/1.10 % set(auto) -> set(auto_inference). 0.80/1.10 % set(auto) -> set(auto_setup). 0.80/1.10 % set(auto_setup) -> set(predicate_elim). 0.80/1.10 % set(auto_setup) -> assign(eq_defs, unfold). 0.80/1.10 % set(auto) -> set(auto_limits). 0.80/1.10 % set(auto_limits) -> assign(max_weight, "100.000"). 0.80/1.10 % set(auto_limits) -> assign(sos_limit, 20000). 0.80/1.10 % set(auto) -> set(auto_denials). 0.80/1.10 % set(auto) -> set(auto_process). 0.80/1.10 % set(auto2) -> assign(new_constants, 1). 0.80/1.10 % set(auto2) -> assign(fold_denial_max, 3). 0.80/1.10 % set(auto2) -> assign(max_weight, "200.000"). 0.80/1.10 % set(auto2) -> assign(max_hours, 1). 0.80/1.10 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.80/1.10 % set(auto2) -> assign(max_seconds, 0). 0.80/1.10 % set(auto2) -> assign(max_minutes, 5). 0.80/1.10 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.80/1.10 % set(auto2) -> set(sort_initial_sos). 0.80/1.10 % set(auto2) -> assign(sos_limit, -1). 0.80/1.10 % set(auto2) -> assign(lrs_ticks, 3000). 0.80/1.10 % set(auto2) -> assign(max_megs, 400). 0.80/1.10 % set(auto2) -> assign(stats, some). 0.80/1.10 % set(auto2) -> clear(echo_input). 0.80/1.10 % set(auto2) -> set(quiet). 0.80/1.10 % set(auto2) -> clear(print_initial_clauses). 0.80/1.10 % set(auto2) -> clear(print_given). 0.80/1.10 assign(lrs_ticks,-1). 0.80/1.10 assign(sos_limit,10000). 0.80/1.10 assign(order,kbo). 0.80/1.10 set(lex_order_vars). 0.80/1.10 clear(print_given). 0.80/1.10 0.80/1.10 % formulas(sos). % not echoed (77 formulas) 0.80/1.10 0.80/1.10 ============================== end of input ========================== 0.80/1.10 0.80/1.10 % From the command line: assign(max_seconds, 180). 0.80/1.10 0.80/1.10 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.80/1.10 0.80/1.10 % Formulas that are not ordinary clauses: 0.80/1.10 1 (all X all Y is_a_theorem(implies(X,implies(Y,and(X,Y))))) <-> and_3 # label(and_3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 2 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]. 0.80/1.10 3 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]. 0.80/1.10 4 (all P all Q all R is_a_theorem(implies(implies(P,Q),implies(not(and(Q,R)),not(and(R,P)))))) <-> kn3 # label(kn3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 5 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]. 0.80/1.10 6 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]. 0.80/1.10 7 (all X all Y is_a_theorem(implies(equiv(X,Y),implies(X,Y)))) <-> equivalence_1 # label(equivalence_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 8 (all X all Y is_a_theorem(implies(implies(X,Y),implies(implies(Y,X),equiv(X,Y))))) <-> equivalence_3 # label(equivalence_3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 9 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]. 0.80/1.10 10 (all P is_a_theorem(implies(P,and(P,P)))) <-> kn1 # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 11 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 12 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 13 (all P is_a_theorem(implies(implies(not(P),P),P))) <-> cn3 # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 14 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 15 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 16 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 17 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]. 0.80/1.10 18 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 19 (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]. 0.80/1.10 20 (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]. 0.80/1.10 21 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]. 0.80/1.10 22 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 23 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 24 (all P all Q all R is_a_theorem(implies(or(P,or(Q,R)),or(Q,or(P,R))))) <-> r4 # label(r4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 25 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 26 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 27 op_and -> (all X all Y not(or(not(X),not(Y))) = and(X,Y)) # label(op_and) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 28 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 29 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]. 0.80/1.10 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]. 0.80/1.10 31 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]. 0.80/1.10 32 (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]. 0.80/1.10 33 (all P all Q is_a_theorem(strict_implies(possibly(and(P,Q)),P))) <-> axiom_m7 # label(axiom_m7) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 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]. 0.80/1.10 35 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 36 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 37 (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_s4 # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 38 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]. 0.80/1.10 39 (all X all Y (is_a_theorem(strict_equiv(X,Y)) -> X = Y)) <-> substitution_strict_equiv # label(substitution_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 40 axiom_m8 <-> (all P all Q is_a_theorem(strict_implies(strict_implies(P,Q),strict_implies(possibly(P),possibly(Q))))) # label(axiom_m8) # label(axiom) # label(non_clause). [assumption]. 0.80/1.10 41 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 42 (all X all Y is_a_theorem(implies(necessarily(implies(X,Y)),implies(necessarily(X),necessarily(Y))))) <-> axiom_K # label(axiom_K) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 43 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]. 0.84/1.13 44 (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_4 # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 45 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]. 0.84/1.13 46 (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) <-> axiom_m10 # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 47 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]. 0.84/1.13 48 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 49 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 50 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 51 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 52 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 53 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]. 0.84/1.13 54 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]. 0.84/1.13 55 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 56 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]. 0.84/1.13 57 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]. 0.84/1.13 58 op_possibly -> (all X not(necessarily(not(X))) = possibly(X)) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.84/1.13 0.84/1.13 ============================== end of process non-clausal formulas === 0.84/1.13 0.84/1.13 ============================== PROCESS INITIAL CLAUSES =============== 0.84/1.13 0.84/1.13 ============================== PREDICATE ELIMINATION ================= 0.84/1.13 0.84/1.13 ============================== end predicate elimination ============= 0.84/1.13 0.84/1.13 Auto_denials: (non-Horn, no changes). 0.84/1.13 0.84/1.13 Term ordering decisions: 0.84/1.13 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. 0.84/1.13 0.84/1.13 ============================== end of process initial clauses ======== 0.84/1.13 0.84/1.13 ============================== CLAUSES FOR SEARCH ==================== 154.46/154.76 154.46/154.76 ============================== end of clauses for search ============= 154.46/154.76 154.46/154.76 ============================== SEARCH ================================ 154.46/154.76 154.46/154.76 % Starting search at 0.04 seconds. 154.46/154.76 154.46/154.76 Low Water (keep): wt=55.000, iters=3355 154.46/154.76 154.46/154.76 Low Water (keep): wt=34.000, iters=3402 154.46/154.76 154.46/154.76 Low Water (keep): wt=33.000, iters=3389 154.46/154.76 154.46/154.76 Low Water (keep): wt=25.000, iters=3338 154.46/154.76 154.46/154.76 Low Water (keep): wt=23.000, iters=3425 154.46/154.76 154.46/154.76 Low Water (keep): wt=22.000, iters=3383 154.46/154.76 154.46/154.76 Low Water (keep): wt=21.000, iters=3402 154.46/154.76 154.46/154.76 Low Water (keep): wt=19.000, iters=3366 154.46/154.76 154.46/154.76 Low Water (keep): wt=18.000, iters=3398 154.46/154.76 154.46/154.76 Low Water (keep): wt=17.000, iters=3345 154.46/154.76 154.46/154.76 Low Water (keep): wt=16.000, iters=3333 154.46/154.76 154.46/154.76 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 416 (0.00 of 2.08 sec). 154.46/154.76 % back CAC tautology: 17597 equiv(not(A),B) = equiv(B,not(A)). [back_rewrite(280),rewrite([17041(4),279(4)])]. 154.46/154.76 % back CAC tautology: 17506 equiv(possibly(necessarily(not(A))),B) = equiv(B,possibly(necessarily(not(A)))). [back_rewrite(2708),rewrite([17041(8),2594(8)])]. 154.46/154.76 % back CAC tautology: 17504 equiv(implies(A,necessarily(not(B))),C) = equiv(C,implies(A,necessarily(not(B)))). [back_rewrite(2710),rewrite([17041(8),2597(8)])]. 154.46/154.76 % back CAC tautology: 17503 equiv(implies(A,and(B,not(C))),D) = equiv(D,implies(A,and(B,not(C)))). [back_rewrite(2711),rewrite([17041(8),2599(8)])]. 154.46/154.76 % back CAC tautology: 17176 equiv(necessarily(not(A)),possibly(A)) = equiv(possibly(A),necessarily(not(A))). [back_rewrite(13390),rewrite([17041(7),13389(7)]),flip(a)]. 154.46/154.76 154.46/154.76 Low Water (displace): id=21011, wt=48.000 154.46/154.76 154.46/154.76 Low Water (displace): id=21017, wt=41.000 154.46/154.76 154.46/154.76 Low Water (displace): id=21015, wt=40.000 154.46/154.76 154.46/154.76 Low Water (displace): id=17845, wt=37.000 154.46/154.76 154.46/154.76 Low Water (displace): id=21022, wt=36.000 154.46/154.76 154.46/154.76 Low Water (displace): id=21039, wt=33.000 154.46/154.76 154.46/154.76 Low Water (displace): id=26841, wt=17.000 154.46/154.76 154.46/154.76 Low Water (displace): id=26962, wt=16.000 154.46/154.76 154.46/154.76 Low Water (displace): id=27460, wt=15.000 154.46/154.76 154.46/154.76 Low Water (displace): id=27845, wt=14.000 154.46/154.76 154.46/154.76 Low Water (displace): id=31262, wt=10.000 154.46/154.76 154.46/154.76 Low Water (keep): wt=15.000, iters=3341 154.46/154.76 154.46/154.76 Low Water (displace): id=45601, wt=8.000 154.46/154.76 154.46/154.76 Low Water (keep): wt=14.000, iters=3333 154.46/154.76 154.46/154.76 Low Water (keep): wt=13.000, iters=3333 154.46/154.76 154.46/154.76 Low Water (keep): wt=12.000, iters=3824 154.46/154.76 154.46/154.76 ============================== PROOF ================================= 154.46/154.76 % SZS status Theorem 154.46/154.76 % SZS output start Refutation 154.46/154.76 154.46/154.76 % Proof 1 at 149.19 (+ 4.48) seconds. 154.46/154.76 % Length of proof is 255. 154.46/154.76 % Level of proof is 54. 154.46/154.76 % Maximum clause weight is 17.000. 154.46/154.76 % Given clauses 24841. 154.46/154.76 154.46/154.76 3 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]. 154.46/154.76 11 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 154.46/154.76 28 op_or -> (all X all Y or(X,Y) = not(and(not(X),not(Y)))) # label(op_or) # label(axiom) # label(non_clause). [assumption]. 154.46/154.76 29 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]. 154.46/154.76 31 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]. 154.46/154.76 32 (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]. 154.46/154.76 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]. 154.46/154.76 38 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]. 154.46/154.76 39 (all X all Y (is_a_theorem(strict_equiv(X,Y)) -> X = Y)) <-> substitution_strict_equiv # label(substitution_strict_equiv) # label(axiom) # label(non_clause). [assumption]. 154.46/154.76 47 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]. 154.46/154.76 49 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 154.46/154.76 52 axiom_m2 <-> (all X all Y is_a_theorem(strict_implies(and(X,Y),X))) # label(axiom_m2) # label(axiom) # label(non_clause). [assumption]. 154.46/154.76 54 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]. 154.46/154.76 56 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]. 154.46/154.76 57 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]. 154.46/154.76 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(3)]. 154.46/154.76 82 -is_a_theorem(implies(and(c24,c25),c24)) | and_1 # label(and_1) # label(axiom). [clausify(11)]. 154.46/154.76 115 -op_or | not(and(not(A),not(B))) = or(A,B) # label(op_or) # label(axiom). [clausify(28)]. 154.46/154.76 116 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(29)]. 154.46/154.76 117 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(116),flip(b)]. 154.46/154.76 119 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(31)]. 154.46/154.76 121 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(32)]. 154.46/154.76 124 -adjunction | -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(A,B)) # label(adjunction) # label(axiom). [clausify(34)]. 154.46/154.76 134 -axiom_m1 | is_a_theorem(strict_implies(and(A,B),and(B,A))) # label(axiom_m1) # label(axiom). [clausify(38)]. 154.46/154.76 138 -is_a_theorem(strict_equiv(A,B)) | B = A | -substitution_strict_equiv # label(substitution_strict_equiv) # label(axiom). [clausify(39)]. 154.46/154.76 154 -modus_ponens_strict_implies | -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(B) # label(modus_ponens_strict_implies) # label(axiom). [clausify(47)]. 154.46/154.76 161 is_a_theorem(strict_implies(A,and(A,A))) | -axiom_m4 # label(axiom_m4) # label(axiom). [clausify(49)]. 154.46/154.76 166 -axiom_m2 | is_a_theorem(strict_implies(and(A,B),A)) # label(axiom_m2) # label(axiom). [clausify(52)]. 154.46/154.76 170 -axiom_m3 | is_a_theorem(strict_implies(and(and(A,B),C),and(A,and(B,C)))) # label(axiom_m3) # label(axiom). [clausify(54)]. 154.46/154.76 174 -op_strict_equiv | strict_equiv(A,B) = and(strict_implies(A,B),strict_implies(B,A)) # label(op_strict_equiv) # label(axiom). [clausify(56)]. 154.46/154.76 175 -op_strict_equiv | and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B). [copy(174),flip(b)]. 154.46/154.76 176 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B) # label(op_strict_implies) # label(axiom). [clausify(57)]. 154.46/154.76 179 axiom_m5 # label(s1_0_axiom_m5) # label(axiom). [assumption]. 154.46/154.76 180 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption]. 154.46/154.76 181 axiom_m3 # label(s1_0_axiom_m3) # label(axiom). [assumption]. 154.46/154.76 183 op_strict_equiv # label(s1_0_op_strict_equiv) # label(axiom). [assumption]. 154.46/154.76 184 axiom_m2 # label(s1_0_axiom_m2) # label(axiom). [assumption]. 154.46/154.76 185 axiom_m4 # label(s1_0_axiom_m4) # label(axiom). [assumption]. 154.46/154.76 186 axiom_m1 # label(s1_0_axiom_m1) # label(axiom). [assumption]. 154.46/154.76 188 op_equiv # label(s1_0_op_equiv) # label(axiom). [assumption]. 154.46/154.76 189 adjunction # label(s1_0_adjunction) # label(axiom). [assumption]. 154.46/154.76 190 modus_ponens_strict_implies # label(s1_0_modus_ponens_strict_implies) # label(axiom). [assumption]. 154.46/154.76 191 op_or # label(s1_0_op_or) # label(axiom). [assumption]. 154.46/154.76 192 substitution_strict_equiv # label(s1_0_substitution_strict_equiv) # label(axiom). [assumption]. 154.46/154.76 193 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 154.46/154.76 194 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 154.46/154.76 195 -and_1 # label(hilbert_and_1) # label(negated_conjecture). [assumption]. 154.46/154.76 197 is_a_theorem(strict_implies(and(strict_implies(A,B),strict_implies(B,C)),strict_implies(A,C))). [back_unit_del(121),unit_del(b,179)]. 154.46/154.76 198 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(176),unit_del(a,180)]. 154.46/154.76 199 is_a_theorem(strict_implies(and(and(A,B),C),and(A,and(B,C)))). [back_unit_del(170),unit_del(a,181)]. 154.46/154.76 201 and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B). [back_unit_del(175),unit_del(a,183)]. 154.46/154.76 202 is_a_theorem(strict_implies(and(A,B),A)). [back_unit_del(166),unit_del(a,184)]. 154.46/154.76 203 is_a_theorem(strict_implies(A,and(A,A))). [back_unit_del(161),unit_del(b,185)]. 154.46/154.76 204 is_a_theorem(strict_implies(and(A,B),and(B,A))). [back_unit_del(134),unit_del(a,186)]. 154.46/154.76 205 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(117),unit_del(a,188)]. 154.46/154.76 206 -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(A,B)). [back_unit_del(124),unit_del(a,189)]. 154.46/154.76 207 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(B). [back_unit_del(154),unit_del(a,190)]. 154.46/154.76 208 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,191)]. 154.46/154.76 209 -is_a_theorem(strict_equiv(A,B)) | B = A. [back_unit_del(138),unit_del(c,192)]. 154.46/154.76 210 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,193)]. 154.46/154.76 211 not(and(A,not(B))) = implies(A,B). [back_unit_del(119),unit_del(a,194)]. 154.46/154.76 212 -is_a_theorem(implies(and(c24,c25),c24)). [back_unit_del(82),unit_del(b,195)]. 154.46/154.76 217 implies(not(A),B) = or(A,B). [back_rewrite(208),rewrite([211(4)])]. 154.46/154.76 253 -is_a_theorem(A) | is_a_theorem(and(strict_implies(and(B,C),and(C,B)),A)). [resolve(206,a,204,a)]. 154.46/154.76 254 -is_a_theorem(A) | is_a_theorem(and(strict_implies(B,and(B,B)),A)). [resolve(206,a,203,a)]. 154.46/154.76 263 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(and(B,C),B))). [resolve(206,b,202,a)]. 154.46/154.76 272 -is_a_theorem(and(strict_implies(A,B),strict_implies(B,C))) | is_a_theorem(strict_implies(A,C)). [resolve(207,b,197,a)]. 154.46/154.76 275 not(and(A,implies(B,C))) = implies(A,and(B,not(C))). [para(211(a,1),211(a,1,1,2))]. 154.46/154.76 277 necessarily(or(A,B)) = strict_implies(not(A),B). [para(217(a,1),198(a,1,1))]. 154.46/154.76 279 and(or(A,B),implies(B,not(A))) = equiv(not(A),B). [para(217(a,1),205(a,1,1))]. 154.46/154.76 280 and(implies(A,not(B)),or(B,A)) = equiv(A,not(B)). [para(217(a,1),205(a,1,2))]. 154.46/154.76 281 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(211(a,1),217(a,1,1)),flip(a)]. 154.46/154.76 1024 is_a_theorem(and(strict_implies(and(A,B),and(B,A)),strict_implies(and(C,D),and(D,C)))). [resolve(253,a,204,a)]. 154.46/154.76 1136 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(and(B,C),B))). [resolve(254,a,202,a)]. 154.46/154.76 2588 implies(A,and(not(B),not(C))) = not(and(A,or(B,C))). [para(217(a,1),275(a,1,1,2)),flip(a)]. 154.46/154.76 2593 and(or(A,not(B)),or(B,not(A))) = equiv(not(A),not(B)). [para(217(a,1),279(a,1,2))]. 154.46/154.76 9787 is_a_theorem(strict_implies(A,A)). [resolve(1136,a,272,a)]. 154.46/154.76 9793 is_a_theorem(strict_equiv(A,and(A,A))). [para(201(a,1),1136(a,1))]. 154.46/154.76 9869 is_a_theorem(and(strict_implies(A,A),strict_implies(and(B,C),B))). [resolve(9787,a,263,a)]. 154.46/154.76 9876 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(B,B))). [resolve(9787,a,254,a)]. 154.46/154.76 9878 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(B,B))). [resolve(9787,a,206,b)]. 154.46/154.76 10049 and(A,A) = A. [resolve(9793,a,209,a)]. 154.46/154.76 10055 is_a_theorem(and(strict_implies(A,A),strict_implies(B,B))). [back_rewrite(9876),rewrite([10049(1)])]. 154.46/154.76 10567 is_a_theorem(strict_implies(and(A,B),and(A,and(A,B)))). [para(10049(a,1),199(a,1,1,1))]. 154.46/154.76 10570 strict_equiv(A,A) = strict_implies(A,A). [para(10049(a,1),201(a,1)),flip(a)]. 154.46/154.76 10572 not(not(A)) = or(A,A). [para(10049(a,1),211(a,1,1)),rewrite([217(4)])]. 154.46/154.76 10585 implies(or(A,A),B) = or(not(A),B). [para(10049(a,1),281(a,1,1)),rewrite([217(4)]),flip(a)]. 154.46/154.76 10985 strict_implies(not(not(A)),B) = strict_implies(or(A,A),B). [para(10572(a,1),277(a,2,1)),rewrite([277(3)])]. 154.46/154.76 10996 equiv(not(not(A)),B) = equiv(or(A,A),B). [para(10572(a,1),279(a,2,1)),rewrite([279(6)])]. 154.46/154.76 10999 equiv(A,not(not(B))) = equiv(A,or(B,B)). [para(10572(a,1),280(a,2,2)),rewrite([280(6)])]. 154.46/154.76 11003 not(not(not(A))) = not(or(A,A)). [para(10572(a,1),10572(a,1,1)),rewrite([10572(5,R)]),flip(a)]. 154.46/154.76 11214 is_a_theorem(and(and(strict_implies(A,A),strict_implies(B,B)),strict_implies(C,C))). [resolve(10055,a,9878,a)]. 154.46/154.76 11267 -is_a_theorem(A) | is_a_theorem(and(A,and(strict_implies(B,B),strict_implies(C,C)))). [resolve(10055,a,206,b)]. 154.46/154.76 12186 implies(A,not(not(B))) = implies(A,or(B,B)). [para(11003(a,1),211(a,1,1,2)),rewrite([211(4)]),flip(a)]. 154.46/154.76 12187 or(not(not(A)),B) = or(or(A,A),B). [para(11003(a,1),217(a,1,1)),rewrite([217(3)]),flip(a)]. 154.46/154.76 12217 not(not(not(not(not(A))))) = not(not(not(or(A,A)))). [para(11003(a,1),11003(a,1,1,1)),rewrite([10572(9,R)]),flip(a)]. 154.46/154.76 15904 -is_a_theorem(or(A,A)) | -is_a_theorem(strict_implies(not(not(A)),B)) | is_a_theorem(B). [para(10985(a,2),207(b,1))]. 154.46/154.76 16281 equiv(not(not(A)),or(B,B)) = equiv(or(A,A),not(not(B))). [para(10999(a,1),10996(a,1))]. 154.46/154.76 16302 strict_implies(A,not(not(B))) = strict_implies(A,or(B,B)). [para(12186(a,1),198(a,1,1)),rewrite([198(3)]),flip(a)]. 154.46/154.76 16303 or(A,not(not(B))) = or(A,or(B,B)). [para(12186(a,1),217(a,1)),rewrite([217(3)]),flip(a)]. 154.46/154.76 16396 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,or(B,B))) | is_a_theorem(not(not(B))). [para(16302(a,1),207(b,1))]. 154.46/154.76 16466 or(not(not(A)),or(B,B)) = or(or(A,A),not(not(B))). [para(16303(a,1),12187(a,1))]. 154.46/154.76 16467 or(not(not(A)),not(not(B))) = or(or(A,A),or(B,B)). [para(16303(a,1),12187(a,2))]. 154.46/154.76 17014 is_a_theorem(strict_equiv(and(A,B),and(B,A))). [para(201(a,1),1024(a,1))]. 154.46/154.76 17041 and(A,B) = and(B,A). [resolve(17014,a,209,a)]. 154.46/154.76 17597 equiv(not(A),B) = equiv(B,not(A)). [back_rewrite(280),rewrite([17041(4),279(4)])]. 154.46/154.76 17607 is_a_theorem(strict_implies(and(A,and(B,C)),and(B,and(C,A)))). [back_rewrite(199),rewrite([17041(2)])]. 154.46/154.76 17743 and(or(A,B),implies(B,not(A))) = equiv(B,not(A)). [back_rewrite(279),rewrite([17597(6)])]. 154.46/154.76 17745 strict_equiv(A,B) = strict_equiv(B,A). [para(17041(a,1),201(a,1)),rewrite([201(3)])]. 154.46/154.76 17746 is_a_theorem(strict_implies(and(A,B),B)). [para(17041(a,1),202(a,1,1))]. 154.46/154.76 17747 equiv(A,B) = equiv(B,A). [para(17041(a,1),205(a,1)),rewrite([205(3)])]. 154.46/154.76 17751 -is_a_theorem(and(strict_implies(A,B),strict_implies(C,A))) | is_a_theorem(strict_implies(C,B)). [para(17041(a,1),272(a,1))]. 154.46/154.76 17762 is_a_theorem(strict_implies(and(A,B),and(B,and(A,B)))). [para(17041(a,1),10567(a,1,1)),rewrite([17041(2)])]. 154.46/154.76 17984 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(and(B,C),C))). [resolve(17746,a,206,b)]. 154.46/154.76 18010 -is_a_theorem(strict_equiv(A,B)) | A = B. [para(17745(a,1),209(a,1))]. 154.46/154.76 18018 -is_a_theorem(equiv(A,B)) | A = B. [para(17747(a,1),210(a,1))]. 154.46/154.76 18670 is_a_theorem(and(strict_implies(and(A,B),and(B,and(A,B))),strict_implies(and(C,D),D))). [resolve(17984,a,17762,a)]. 154.46/154.76 21162 -is_a_theorem(A) | is_a_theorem(and(A,and(and(strict_implies(B,B),strict_implies(C,C)),strict_implies(D,D)))). [resolve(11214,a,206,b)]. 154.46/154.76 21249 is_a_theorem(and(and(strict_implies(A,A),strict_implies(B,B)),and(strict_implies(C,C),strict_implies(D,D)))). [resolve(11267,a,10055,a)]. 154.46/154.76 22802 is_a_theorem(strict_implies(and(A,and(B,C)),and(C,and(A,B)))). [para(17041(a,1),17607(a,1,1,2)),rewrite([17041(3)])]. 154.46/154.76 24238 not(and(not(A),or(B,C))) = or(A,and(not(B),not(C))). [para(2588(a,1),217(a,1))]. 154.46/154.76 24250 not(and(A,or(B,C))) = not(and(A,or(C,B))). [para(17041(a,1),2588(a,1,2)),rewrite([2588(4)])]. 154.46/154.76 25899 not(and(or(A,B),or(B,A))) = not(or(A,B)). [para(10049(a,1),24250(a,1,1)),flip(a)]. 154.46/154.76 26478 not(or(A,B)) = not(or(B,A)). [para(17041(a,1),25899(a,1,1)),rewrite([25899(4)])]. 154.46/154.76 26483 implies(A,or(B,C)) = implies(A,or(C,B)). [para(26478(a,1),211(a,1,1,2)),rewrite([211(4)])]. 154.46/154.76 26512 strict_implies(A,or(B,C)) = strict_implies(A,or(C,B)). [para(26483(a,1),198(a,1,1)),rewrite([198(3)])]. 154.46/154.76 26758 and(strict_implies(A,or(B,C)),strict_implies(or(C,B),A)) = strict_equiv(A,or(C,B)). [para(26512(a,1),201(a,1,1))]. 154.46/154.76 30126 or(A,and(not(B),not(C))) = implies(or(B,C),A). [para(17041(a,1),24238(a,1,1)),rewrite([211(4)]),flip(a)]. 154.46/154.76 30137 or(not(A),B) = or(B,not(A)). [para(10049(a,1),30126(a,1,2)),rewrite([10585(4)]),flip(a)]. 154.46/154.76 30141 implies(or(A,B),C) = implies(or(B,A),C). [para(17041(a,1),30126(a,1,2)),rewrite([30126(4)])]. 154.46/154.76 30187 implies(or(A,A),B) = or(B,not(A)). [back_rewrite(10585),rewrite([30137(4)])]. 154.46/154.76 30193 strict_implies(not(not(A)),B) = strict_implies(not(B),not(A)). [para(30137(a,1),277(a,1,1)),rewrite([277(3)]),flip(a)]. 154.46/154.76 30196 or(not(A),not(not(B))) = or(or(B,B),not(A)). [para(30137(a,1),16303(a,2))]. 154.46/154.76 30227 or(not(not(A)),not(not(B))) = or(or(B,B),or(A,A)). [para(30137(a,1),16467(a,1))]. 154.46/154.76 30282 -is_a_theorem(or(A,A)) | -is_a_theorem(strict_implies(not(B),not(A))) | is_a_theorem(B). [back_rewrite(15904),rewrite([30193(5)])]. 154.46/154.76 30287 strict_implies(not(A),not(B)) = strict_implies(or(B,B),A). [back_rewrite(10985),rewrite([30193(3)])]. 154.46/154.76 30341 strict_implies(or(A,B),C) = strict_implies(or(B,A),C). [para(30141(a,1),198(a,1,1)),rewrite([198(3)])]. 154.46/154.76 30343 strict_equiv(A,or(B,C)) = strict_equiv(A,or(C,B)). [back_rewrite(26758),rewrite([30341(4),201(5)])]. 154.46/154.76 30444 is_a_theorem(strict_implies(or(A,A),A)). [para(30287(a,1),9787(a,1))]. 154.46/154.76 30448 strict_implies(or(A,A),not(B)) = strict_implies(or(B,B),not(A)). [para(10572(a,1),30287(a,1,1))]. 154.46/154.76 30449 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(B,B))). [para(30287(a,1),10055(a,1,1))]. 154.46/154.76 30456 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(and(B,C),B))). [para(30287(a,1),9869(a,1,1))]. 154.46/154.76 30859 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(or(B,B),B))). [para(30287(a,1),30449(a,1,2))]. 154.46/154.76 30964 -is_a_theorem(strict_equiv(A,or(B,C))) | or(C,B) = A. [para(30343(a,1),209(a,1))]. 154.46/154.76 30966 strict_equiv(A,not(not(B))) = strict_equiv(A,or(B,B)). [para(10572(a,2),30343(a,1,2))]. 154.46/154.76 30993 or(A,B) = or(B,A). [para(10570(a,1),30964(a,1)),unit_del(a,9787)]. 154.46/154.76 30994 -is_a_theorem(strict_equiv(A,not(not(B)))) | or(B,B) = A. [para(10572(a,2),30964(a,1,2))]. 154.46/154.76 31018 strict_implies(not(A),B) = strict_implies(not(B),A). [para(30993(a,1),277(a,1,1)),rewrite([277(2)])]. 154.46/154.76 31019 and(or(A,B),implies(A,not(B))) = equiv(A,not(B)). [para(30993(a,1),17743(a,1,1))]. 154.46/154.76 31040 and(strict_implies(A,not(B)),strict_implies(not(A),B)) = strict_equiv(A,not(B)). [para(31018(a,1),201(a,1,1)),rewrite([17041(5),17745(7)])]. 154.46/154.76 31042 -is_a_theorem(and(strict_implies(not(A),B),strict_implies(A,C))) | is_a_theorem(strict_implies(not(B),C)). [para(31018(a,1),272(a,1,1))]. 154.46/154.76 31100 strict_equiv(A,not(not(not(B)))) = strict_equiv(A,not(or(B,B))). [para(10572(a,1),30966(a,1,2,1)),rewrite([10572(6,R)]),flip(a)]. 154.46/154.76 31138 is_a_theorem(strict_implies(not(A),not(and(A,B)))). [resolve(30456,a,272,a),rewrite([10572(3,R),31018(4)])]. 154.46/154.76 31218 -is_a_theorem(not(A)) | is_a_theorem(not(and(A,B))). [resolve(31138,a,207,b)]. 154.46/154.76 31225 is_a_theorem(strict_implies(not(A),implies(A,B))). [para(211(a,1),31138(a,1,2))]. 154.46/154.76 31305 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(not(B),implies(B,C)))). [resolve(31225,a,206,b)]. 154.46/154.76 31488 -is_a_theorem(not(not(not(not(not(A)))))) | is_a_theorem(implies(B,not(or(A,A)))). [para(12217(a,2),31218(a,1)),rewrite([17041(10),211(11)])]. 154.46/154.76 32367 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(not(B),implies(B,C)))). [resolve(31305,a,30444,a)]. 154.46/154.76 32375 is_a_theorem(and(strict_implies(and(A,B),B),strict_implies(not(C),implies(C,D)))). [resolve(31305,a,17746,a)]. 154.46/154.76 32875 is_a_theorem(strict_implies(not(A),not(or(A,A)))). [resolve(30859,a,17751,a),rewrite([10572(3,R),31018(4)])]. 154.46/154.76 32916 is_a_theorem(strict_implies(not(A),not(not(not(A))))). [para(10572(a,2),32875(a,1,2,1))]. 154.46/154.76 33062 is_a_theorem(and(strict_implies(not(A),not(not(not(A)))),strict_implies(B,B))). [resolve(32916,a,9878,a)]. 154.46/154.76 36299 -is_a_theorem(strict_equiv(A,not(not(not(B))))) | not(or(B,B)) = A. [para(31100(a,2),209(a,1))]. 154.46/154.76 36420 is_a_theorem(strict_implies(not(not(not(A))),implies(A,B))). [resolve(32367,a,272,a),rewrite([10572(3,R)])]. 154.46/154.76 36431 -is_a_theorem(not(not(not(A)))) | is_a_theorem(implies(A,B)). [resolve(36420,a,207,b)]. 154.46/154.76 36462 -is_a_theorem(not(not(not(or(A,A))))) | is_a_theorem(or(B,not(A))). [para(11003(a,1),36431(a,1,1,1)),rewrite([217(8),30993(7)])]. 154.46/154.76 36766 is_a_theorem(and(strict_implies(not(A),implies(A,B)),strict_implies(and(C,D),D))). [para(17041(a,1),32375(a,1))]. 154.46/154.76 40248 is_a_theorem(strict_equiv(not(A),not(not(not(A))))). [para(31040(a,1),33062(a,1))]. 154.46/154.76 40621 not(or(A,A)) = not(A). [resolve(40248,a,36299,a)]. 154.46/154.76 40627 not(not(not(A))) = not(A). [resolve(40248,a,30994,a),rewrite([10572(3,R)])]. 154.46/154.76 40725 -is_a_theorem(not(A)) | is_a_theorem(or(B,not(A))). [back_rewrite(36462),rewrite([40621(2),40627(3)])]. 154.46/154.76 40824 -is_a_theorem(not(A)) | is_a_theorem(implies(B,not(A))). [back_rewrite(31488),rewrite([40627(3),40627(3),40621(4)])]. 154.46/154.76 41242 implies(A,or(B,B)) = implies(A,B). [para(40621(a,1),211(a,1,1,2)),rewrite([211(3)]),flip(a)]. 154.46/154.76 41244 or(A,or(B,B)) = or(B,A). [para(40621(a,1),217(a,1,1)),rewrite([217(2),30993(3)]),flip(a)]. 154.46/154.76 41281 or(not(not(A)),not(not(B))) = or(B,not(not(A))). [para(40621(a,1),16466(a,1,1,1)),rewrite([41244(4),10572(6,R),40621(5)]),flip(a)]. 154.46/154.76 41418 implies(A,not(not(B))) = implies(A,B). [back_rewrite(12186),rewrite([41242(5)])]. 154.46/154.76 41489 or(A,not(not(B))) = or(B,A). [back_rewrite(30227),rewrite([41281(5),41244(6),41244(5),30993(4)])]. 154.46/154.76 41627 or(or(A,A),not(B)) = or(A,not(B)). [back_rewrite(30196),rewrite([41489(4)]),flip(a)]. 154.46/154.76 41781 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(C,implies(A,B))). [para(211(a,1),40824(a,1)),rewrite([211(5)])]. 154.46/154.76 41857 strict_implies(A,or(B,B)) = strict_implies(A,B). [para(41242(a,1),198(a,1,1)),rewrite([198(2)]),flip(a)]. 154.46/154.76 41926 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(not(not(B))). [back_rewrite(16396),rewrite([41857(3)])]. 154.46/154.76 41927 strict_implies(A,not(not(B))) = strict_implies(A,B). [back_rewrite(16302),rewrite([41857(5)])]. 154.46/154.76 42794 -is_a_theorem(A) | is_a_theorem(not(not(A))). [resolve(41926,b,9787,a)]. 154.46/154.76 43545 is_a_theorem(not(not(strict_implies(A,A)))). [resolve(42794,a,9787,a)]. 154.46/154.76 43622 is_a_theorem(not(not(strict_implies(and(A,B),A)))). [resolve(42794,a,202,a)]. 154.46/154.76 44192 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(43545,a,40824,a),rewrite([41418(4)])]. 154.46/154.76 44198 is_a_theorem(or(A,strict_implies(B,B))). [resolve(43545,a,40725,a),rewrite([41489(4),30993(2)])]. 154.46/154.76 44347 is_a_theorem(implies(A,implies(B,strict_implies(C,C)))). [resolve(44192,a,41781,a)]. 154.46/154.76 44387 -is_a_theorem(A) | is_a_theorem(and(A,implies(B,strict_implies(C,C)))). [resolve(44192,a,206,b)]. 154.46/154.76 44406 -is_a_theorem(strict_implies(not(A),not(strict_implies(B,B)))) | is_a_theorem(A). [resolve(44198,a,30282,a)]. 154.46/154.76 45267 is_a_theorem(implies(A,strict_implies(and(B,C),B))). [resolve(43622,a,40824,a),rewrite([41418(5)])]. 154.46/154.76 46003 is_a_theorem(and(implies(A,strict_implies(and(B,C),B)),strict_implies(D,D))). [resolve(45267,a,9878,a)]. 154.46/154.76 46552 is_a_theorem(strict_equiv(and(A,B),and(B,and(A,B)))). [para(201(a,1),18670(a,1))]. 154.46/154.76 51686 is_a_theorem(and(implies(A,strict_implies(B,B)),implies(C,strict_implies(D,D)))). [resolve(44387,a,44192,a)]. 154.46/154.76 53278 and(A,and(B,A)) = and(B,A). [resolve(46552,a,18010,a),flip(a)]. 154.46/154.76 55974 equiv(or(A,A),not(not(B))) = equiv(not(A),not(B)). [para(41627(a,1),31019(a,1,1)),rewrite([41418(6),30187(4),2593(5)]),flip(a)]. 154.46/154.76 55984 equiv(not(not(A)),or(B,B)) = equiv(not(A),not(B)). [back_rewrite(16281),rewrite([55974(8)])]. 154.46/154.76 59588 is_a_theorem(strict_implies(not(A),implies(and(B,A),C))). [resolve(31042,a,36766,a),rewrite([31018(4)])]. 154.46/154.76 62173 is_a_theorem(and(strict_implies(A,A),implies(B,strict_implies(and(C,D),C)))). [para(17041(a,1),46003(a,1))]. 154.46/154.76 62911 is_a_theorem(equiv(strict_implies(A,A),strict_implies(B,B))). [para(205(a,1),51686(a,1))]. 154.46/154.76 62913 strict_implies(A,A) = strict_implies(B,B). [resolve(62911,a,18018,a)]. 154.46/154.76 62921 strict_implies(A,A) = c_0. [new_symbol(62913)]. 154.46/154.76 62972 is_a_theorem(and(c_0,implies(A,strict_implies(and(B,C),B)))). [back_rewrite(62173),rewrite([62921(1)])]. 154.46/154.76 64452 -is_a_theorem(strict_implies(not(A),not(c_0))) | is_a_theorem(A). [back_rewrite(44406),rewrite([62921(2)])]. 154.46/154.76 64467 -is_a_theorem(A) | is_a_theorem(and(A,implies(B,c_0))). [back_rewrite(44387),rewrite([62921(2)])]. 154.46/154.76 64501 is_a_theorem(implies(A,implies(B,c_0))). [back_rewrite(44347),rewrite([62921(1)])]. 154.46/154.76 64782 is_a_theorem(c_0). [back_rewrite(21249),rewrite([62921(1),62921(2),10049(3),62921(2),62921(3),10049(4),10049(3)])]. 154.46/154.76 64785 -is_a_theorem(A) | is_a_theorem(and(A,c_0)). [back_rewrite(21162),rewrite([62921(2),62921(3),10049(4),62921(3),10049(4)])]. 154.46/154.76 65024 strict_equiv(A,not(not(A))) = c_0. [para(62921(a,1),31040(a,1,2)),rewrite([41927(3),62921(1),10049(3)]),flip(a)]. 154.46/154.76 65240 is_a_theorem(and(c_0,strict_implies(not(A),implies(and(A,B),C)))). [resolve(64785,a,59588,a),rewrite([17041(2),17041(6)])]. 154.46/154.76 65893 not(not(A)) = A. [para(65024(a,1),209(a,1)),unit_del(a,64782)]. 154.46/154.76 65898 or(A,A) = A. [para(65024(a,1),30994(a,1)),unit_del(a,64782)]. 154.46/154.76 65962 equiv(not(A),not(B)) = equiv(A,B). [back_rewrite(55984),rewrite([65893(2),65898(1)]),flip(a)]. 154.46/154.76 66132 strict_implies(A,not(B)) = strict_implies(B,not(A)). [back_rewrite(30448),rewrite([65898(1),65898(3)])]. 154.46/154.76 66134 or(A,not(B)) = implies(B,A). [back_rewrite(30187),rewrite([65898(1)]),flip(a)]. 154.46/154.76 66199 -is_a_theorem(strict_implies(c_0,A)) | is_a_theorem(A). [back_rewrite(64452),rewrite([66132(4),65893(3)])]. 154.46/154.76 66376 not(and(A,B)) = implies(A,not(B)). [para(65893(a,1),211(a,1,1,2))]. 154.46/154.76 66377 not(implies(A,B)) = and(A,not(B)). [para(211(a,1),65893(a,1,1))]. 154.46/154.76 68694 is_a_theorem(and(implies(A,implies(B,c_0)),implies(C,c_0))). [resolve(64467,a,64501,a)]. 154.46/154.76 70188 -is_a_theorem(equiv(A,B)) | not(A) = not(B). [para(65962(a,1),18018(a,1))]. 154.46/154.76 70216 strict_equiv(A,not(B)) = strict_equiv(B,not(A)). [para(66132(a,1),201(a,1,1)),rewrite([31040(5)])]. 154.46/154.76 70233 implies(A,not(B)) = implies(B,not(A)). [para(17041(a,1),66376(a,1,1)),rewrite([66376(2)])]. 154.46/154.76 70234 implies(A,implies(B,not(A))) = implies(B,not(A)). [para(53278(a,1),66376(a,1,1)),rewrite([66376(2),66376(4)]),flip(a)]. 154.46/154.76 70664 -is_a_theorem(strict_equiv(A,not(B))) | not(A) = B. [para(70216(a,1),209(a,1))]. 154.46/154.76 70812 -is_a_theorem(strict_equiv(A,B)) | not(A) = not(B). [para(65893(a,1),70664(a,1,2))]. 154.46/154.76 70825 -is_a_theorem(strict_equiv(A,B)) | not(B) = not(A). [para(17745(a,1),70812(a,1))]. 154.46/154.76 71923 is_a_theorem(equiv(c_0,implies(A,c_0))). [para(205(a,1),68694(a,1))]. 154.46/154.76 71929 and(A,not(c_0)) = not(c_0). [resolve(71923,a,70188,a),rewrite([66377(5)]),flip(a)]. 154.46/154.76 71951 implies(A,c_0) = c_0. [resolve(71923,a,18018,a),flip(a)]. 154.46/154.76 71957 strict_implies(A,c_0) = necessarily(c_0). [para(71951(a,1),198(a,1,1)),flip(a)]. 154.46/154.76 71958 and(c_0,implies(c_0,A)) = equiv(A,c_0). [para(71951(a,1),205(a,1,1))]. 154.46/154.76 72289 and(necessarily(c_0),strict_implies(c_0,A)) = strict_equiv(A,c_0). [para(71957(a,1),201(a,1,1))]. 154.46/154.76 72326 necessarily(c_0) = c_0. [para(71957(a,1),62921(a,1))]. 154.46/154.76 72335 and(c_0,strict_implies(c_0,A)) = strict_equiv(A,c_0). [back_rewrite(72289),rewrite([72326(2)])]. 154.46/154.76 72493 is_a_theorem(equiv(c_0,strict_implies(and(A,B),A))). [para(71958(a,1),62972(a,1)),rewrite([17747(4)])]. 154.46/154.76 72858 strict_implies(and(A,B),A) = c_0. [resolve(72493,a,18018,a),flip(a)]. 154.46/154.76 72866 strict_implies(and(A,B),B) = c_0. [para(17041(a,1),72858(a,1,1))]. 154.46/154.76 72875 -is_a_theorem(and(c_0,strict_implies(A,B))) | is_a_theorem(strict_implies(and(A,C),B)). [para(72866(a,1),272(a,1,1)),rewrite([17041(5)])]. 154.46/154.76 75828 implies(A,implies(A,B)) = implies(A,B). [para(217(a,1),70234(a,1,2)),rewrite([66134(2),70233(5),65893(4)])]. 154.46/154.76 75841 strict_implies(A,implies(A,B)) = strict_implies(A,B). [para(75828(a,1),198(a,1,1)),rewrite([198(2)]),flip(a)]. 154.46/154.76 75848 -is_a_theorem(strict_implies(c_0,A)) | is_a_theorem(implies(c_0,A)). [para(75841(a,1),66199(a,1))]. 154.46/154.76 75853 -is_a_theorem(strict_implies(A,not(c_0))) | is_a_theorem(implies(A,not(c_0))). [para(66132(a,1),75848(a,1)),rewrite([70233(7)])]. 154.46/154.76 83270 is_a_theorem(strict_implies(and(A,not(B)),implies(and(B,C),D))). [resolve(72875,a,65240,a),rewrite([17041(2)])]. 154.46/154.76 83363 is_a_theorem(strict_implies(and(A,not(A)),B)). [para(75841(a,1),83270(a,1))]. 154.46/154.76 83367 is_a_theorem(implies(c_0,implies(A,A))). [resolve(83363,a,75853,a),rewrite([70233(5),66376(4),65893(3)])]. 154.46/154.76 83613 is_a_theorem(equiv(c_0,implies(A,A))). [resolve(83367,a,64785,a),rewrite([17041(5),71958(5),17747(3)])]. 154.46/154.76 83847 and(A,not(A)) = not(c_0). [resolve(83613,a,70188,a),rewrite([66377(4)]),flip(a)]. 154.46/154.76 83878 is_a_theorem(strict_implies(c_0,implies(A,implies(B,A)))). [para(83847(a,1),22802(a,1,2,2)),rewrite([17041(2),71929(6),66132(6),66376(5),66376(4),65893(3)])]. 154.46/154.76 83897 is_a_theorem(strict_equiv(c_0,implies(A,implies(B,A)))). [resolve(83878,a,64785,a),rewrite([17041(6),72335(6),17745(4)])]. 154.46/154.76 84158 and(A,and(B,not(A))) = not(c_0). [resolve(83897,a,70825,a),rewrite([66377(3),66377(2)])]. 154.50/154.76 84190 and(not(A),and(A,B)) = not(c_0). [para(65893(a,1),84158(a,1,2,2)),rewrite([17041(2)])]. 154.50/154.76 84206 is_a_theorem(strict_implies(c_0,implies(and(A,B),implies(C,A)))). [para(84190(a,1),17607(a,1,2,2)),rewrite([71929(7),66132(7),66376(6),66376(5),65893(4)])]. 154.50/154.76 84465 is_a_theorem(implies(and(A,B),implies(C,A))). [resolve(84206,a,66199,a)]. 154.50/154.76 84508 is_a_theorem(implies(and(A,B),A)). [para(75828(a,1),84465(a,1))]. 154.50/154.76 84509 $F. [resolve(84508,a,212,a)]. 154.50/154.76 154.50/154.76 % SZS output end Refutation 154.50/154.76 ============================== end of proof ========================== 154.50/154.76 154.50/154.76 ============================== STATISTICS ============================ 154.50/154.76 154.50/154.76 Given=24841. Generated=8346453. Kept=84446. proofs=1. 154.50/154.76 Usable=12468. Sos=5879. Demods=473. Limbo=27, Disabled=66206. Hints=0. 154.50/154.76 Megabytes=88.31. 154.50/154.76 User_CPU=149.20, System_CPU=4.48, Wall_clock=154. 154.50/154.76 154.50/154.76 ============================== end of statistics ===================== 154.50/154.76 154.50/154.76 ============================== end of search ========================= 154.50/154.76 154.50/154.76 THEOREM PROVED 154.50/154.76 % SZS status Theorem 154.50/154.76 154.50/154.76 Exiting with 1 proof. 154.50/154.76 154.50/154.76 Process 12421 exit (max_proofs) Thu Aug 29 15:48:10 2019 154.50/154.76 Prover9 interrupted 154.50/154.76 EOF