0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.34 % Computer : n005.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 180 0.13/0.34 % DateTime : Thu Aug 29 16:15:18 EDT 2019 0.13/0.34 % CPUTime : 0.80/1.08 ============================== Prover9 =============================== 0.80/1.08 Prover9 (32) version 2009-11A, November 2009. 0.80/1.08 Process 2386 was started by sandbox on n005.cluster.edu, 0.80/1.08 Thu Aug 29 16:15:19 2019 0.80/1.08 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 180 -f /tmp/Prover9_2221_n005.cluster.edu". 0.80/1.08 ============================== end of head =========================== 0.80/1.08 0.80/1.08 ============================== INPUT ================================= 0.80/1.08 0.80/1.08 % Reading from file /tmp/Prover9_2221_n005.cluster.edu 0.80/1.08 0.80/1.08 set(prolog_style_variables). 0.80/1.08 set(auto2). 0.80/1.08 % set(auto2) -> set(auto). 0.80/1.08 % set(auto) -> set(auto_inference). 0.80/1.08 % set(auto) -> set(auto_setup). 0.80/1.08 % set(auto_setup) -> set(predicate_elim). 0.80/1.08 % set(auto_setup) -> assign(eq_defs, unfold). 0.80/1.08 % set(auto) -> set(auto_limits). 0.80/1.08 % set(auto_limits) -> assign(max_weight, "100.000"). 0.80/1.08 % set(auto_limits) -> assign(sos_limit, 20000). 0.80/1.08 % set(auto) -> set(auto_denials). 0.80/1.08 % set(auto) -> set(auto_process). 0.80/1.08 % set(auto2) -> assign(new_constants, 1). 0.80/1.08 % set(auto2) -> assign(fold_denial_max, 3). 0.80/1.08 % set(auto2) -> assign(max_weight, "200.000"). 0.80/1.08 % set(auto2) -> assign(max_hours, 1). 0.80/1.08 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.80/1.08 % set(auto2) -> assign(max_seconds, 0). 0.80/1.08 % set(auto2) -> assign(max_minutes, 5). 0.80/1.08 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.80/1.08 % set(auto2) -> set(sort_initial_sos). 0.80/1.08 % set(auto2) -> assign(sos_limit, -1). 0.80/1.08 % set(auto2) -> assign(lrs_ticks, 3000). 0.80/1.08 % set(auto2) -> assign(max_megs, 400). 0.80/1.08 % set(auto2) -> assign(stats, some). 0.80/1.08 % set(auto2) -> clear(echo_input). 0.80/1.08 % set(auto2) -> set(quiet). 0.80/1.08 % set(auto2) -> clear(print_initial_clauses). 0.80/1.08 % set(auto2) -> clear(print_given). 0.80/1.08 assign(lrs_ticks,-1). 0.80/1.08 assign(sos_limit,10000). 0.80/1.08 assign(order,kbo). 0.80/1.08 set(lex_order_vars). 0.80/1.08 clear(print_given). 0.80/1.08 0.80/1.08 % formulas(sos). % not echoed (77 formulas) 0.80/1.08 0.80/1.08 ============================== end of input ========================== 0.80/1.08 0.80/1.08 % From the command line: assign(max_seconds, 180). 0.80/1.08 0.80/1.08 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.80/1.08 0.80/1.08 % Formulas that are not ordinary clauses: 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 10 (all P is_a_theorem(implies(P,and(P,P)))) <-> kn1 # label(kn1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 11 (all X all Y is_a_theorem(implies(and(X,Y),X))) <-> and_1 # label(and_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 12 and_2 <-> (all X all Y is_a_theorem(implies(and(X,Y),Y))) # label(and_2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 13 (all P is_a_theorem(implies(implies(not(P),P),P))) <-> cn3 # label(cn3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 14 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 15 r3 <-> (all P all Q is_a_theorem(implies(or(P,Q),or(Q,P)))) # label(r3) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 16 (all P all Q is_a_theorem(implies(P,implies(not(P),Q)))) <-> cn2 # label(cn2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 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]. 0.80/1.08 18 (all P all Q is_a_theorem(implies(Q,or(P,Q)))) <-> r2 # label(r2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 22 implies_1 <-> (all X all Y is_a_theorem(implies(X,implies(Y,X)))) # label(implies_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 23 or_1 <-> (all X all Y is_a_theorem(implies(X,or(X,Y)))) # label(or_1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 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]. 0.80/1.08 25 r1 <-> (all P is_a_theorem(implies(or(P,P),P))) # label(r1) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 26 kn2 <-> (all P all Q is_a_theorem(implies(and(P,Q),P))) # label(kn2) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 35 axiom_B <-> (all X is_a_theorem(implies(X,necessarily(possibly(X))))) # label(axiom_B) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 36 (all X is_a_theorem(strict_implies(X,possibly(X)))) <-> axiom_m6 # label(axiom_m6) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 37 (all X is_a_theorem(strict_implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_s4 # label(axiom_s4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 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]. 0.80/1.08 41 necessitation <-> (all X (is_a_theorem(X) -> is_a_theorem(necessarily(X)))) # label(necessitation) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 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]. 0.80/1.11 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]. 0.80/1.11 44 (all X is_a_theorem(implies(necessarily(X),necessarily(necessarily(X))))) <-> axiom_4 # label(axiom_4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 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]. 0.80/1.11 46 (all X is_a_theorem(strict_implies(possibly(X),necessarily(possibly(X))))) <-> axiom_m10 # label(axiom_m10) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 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]. 0.80/1.11 48 (all X is_a_theorem(strict_implies(possibly(possibly(X)),possibly(X)))) <-> axiom_m9 # label(axiom_m9) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 49 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 50 (all X is_a_theorem(implies(possibly(X),necessarily(possibly(X))))) <-> axiom_5 # label(axiom_5) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 51 axiom_M <-> (all X is_a_theorem(implies(necessarily(X),X))) # label(axiom_M) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 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]. 0.80/1.11 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]. 0.80/1.11 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]. 0.80/1.11 55 op_necessarily -> (all X necessarily(X) = not(possibly(not(X)))) # label(op_necessarily) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 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]. 0.80/1.11 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]. 0.80/1.11 58 op_possibly -> (all X not(necessarily(not(X))) = possibly(X)) # label(op_possibly) # label(axiom) # label(non_clause). [assumption]. 0.80/1.11 0.80/1.11 ============================== end of process non-clausal formulas === 0.80/1.11 0.80/1.11 ============================== PROCESS INITIAL CLAUSES =============== 0.80/1.11 0.80/1.11 ============================== PREDICATE ELIMINATION ================= 0.80/1.11 0.80/1.11 ============================== end predicate elimination ============= 0.80/1.11 0.80/1.11 Auto_denials: (non-Horn, no changes). 0.80/1.11 0.80/1.11 Term ordering decisions: 0.80/1.11 Function symbol KB weights: c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. c16=1. c17=1. c18=1. c19=1. c20=1. c21=1. c22=1. c23=1. c24=1. c25=1. c26=1. c27=1. c28=1. c29=1. c30=1. c31=1. c32=1. c33=1. c34=1. c35=1. c36=1. c37=1. c38=1. c39=1. c40=1. c41=1. c42=1. c43=1. c44=1. c45=1. c46=1. c47=1. c48=1. c49=1. c50=1. c51=1. c52=1. c53=1. c54=1. c55=1. c56=1. c57=1. c58=1. c59=1. c60=1. c61=1. c62=1. c63=1. c64=1. c65=1. c66=1. c67=1. c68=1. c69=1. c70=1. c71=1. c72=1. c73=1. c74=1. c75=1. c76=1. c77=1. c78=1. c79=1. c80=1. c81=1. c82=1. c83=1. c84=1. c85=1. c86=1. c87=1. c88=1. c89=1. c90=1. c91=1. c92=1. c93=1. c94=1. implies=1. and=1. strict_implies=1. or=1. equiv=1. strict_equiv=1. necessarily=1. possibly=1. not=1. 0.80/1.11 0.80/1.11 ============================== end of process initial clauses ======== 0.80/1.11 0.80/1.11 ============================== CLAUSES FOR SEARCH ==================== 0.80/1.11 179.16/179.47 ============================== end of clauses for search ============= 179.16/179.47 179.16/179.47 ============================== SEARCH ================================ 179.16/179.47 179.16/179.47 % Starting search at 0.05 seconds. 179.16/179.47 179.16/179.47 Low Water (keep): wt=55.000, iters=3355 179.16/179.47 179.16/179.47 Low Water (keep): wt=34.000, iters=3402 179.16/179.47 179.16/179.47 Low Water (keep): wt=33.000, iters=3389 179.16/179.47 179.16/179.47 Low Water (keep): wt=25.000, iters=3338 179.16/179.47 179.16/179.47 Low Water (keep): wt=23.000, iters=3425 179.16/179.47 179.16/179.47 Low Water (keep): wt=22.000, iters=3383 179.16/179.47 179.16/179.47 Low Water (keep): wt=21.000, iters=3402 179.16/179.47 179.16/179.47 Low Water (keep): wt=19.000, iters=3366 179.16/179.47 179.16/179.47 Low Water (keep): wt=18.000, iters=3398 179.16/179.47 179.16/179.47 Low Water (keep): wt=17.000, iters=3345 179.16/179.47 179.16/179.47 Low Water (keep): wt=16.000, iters=3333 179.16/179.47 179.16/179.47 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 416 (0.00 of 2.25 sec). 179.16/179.47 % back CAC tautology: 17597 equiv(not(A),B) = equiv(B,not(A)). [back_rewrite(280),rewrite([17041(4),279(4)])]. 179.16/179.47 % back CAC tautology: 17506 equiv(possibly(necessarily(not(A))),B) = equiv(B,possibly(necessarily(not(A)))). [back_rewrite(2708),rewrite([17041(8),2594(8)])]. 179.16/179.47 % 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)])]. 179.16/179.47 % 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)])]. 179.16/179.47 % 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)]. 179.16/179.47 179.16/179.47 Low Water (displace): id=21011, wt=48.000 179.16/179.47 179.16/179.47 Low Water (displace): id=21017, wt=41.000 179.16/179.47 179.16/179.47 Low Water (displace): id=21015, wt=40.000 179.16/179.47 179.16/179.47 Low Water (displace): id=17845, wt=37.000 179.16/179.47 179.16/179.47 Low Water (displace): id=21022, wt=36.000 179.16/179.47 179.16/179.47 Low Water (displace): id=21039, wt=33.000 179.16/179.47 179.16/179.47 Low Water (displace): id=26840, wt=17.000 179.16/179.47 179.16/179.47 Low Water (displace): id=26961, wt=16.000 179.16/179.47 179.16/179.47 Low Water (displace): id=27445, wt=15.000 179.16/179.47 179.16/179.47 Low Water (displace): id=27844, wt=14.000 179.16/179.47 179.16/179.47 Low Water (displace): id=31656, wt=12.000 179.16/179.47 179.16/179.47 Low Water (displace): id=31713, wt=11.000 179.16/179.47 179.16/179.47 Low Water (displace): id=31718, wt=10.000 179.16/179.47 179.16/179.47 Low Water (keep): wt=15.000, iters=3341 179.16/179.47 179.16/179.47 Low Water (displace): id=45696, wt=9.000 179.16/179.47 179.16/179.47 Low Water (displace): id=45922, wt=8.000 179.16/179.47 179.16/179.47 Low Water (keep): wt=14.000, iters=3333 179.16/179.47 179.16/179.47 Low Water (keep): wt=13.000, iters=3333 179.16/179.47 179.16/179.47 Low Water (keep): wt=12.000, iters=3958 179.16/179.47 179.16/179.47 ============================== PROOF ================================= 179.16/179.47 % SZS status Theorem 179.16/179.47 % SZS output start Refutation 179.16/179.47 179.16/179.47 % Proof 1 at 173.78 (+ 4.59) seconds. 179.16/179.47 % Length of proof is 246. 179.16/179.47 % Level of proof is 56. 179.16/179.47 % Maximum clause weight is 17.000. 179.16/179.47 % Given clauses 24962. 179.16/179.47 179.16/179.47 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]. 179.16/179.47 14 or_2 <-> (all X all Y is_a_theorem(implies(Y,or(X,Y)))) # label(or_2) # label(axiom) # label(non_clause). [assumption]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 49 (all X is_a_theorem(strict_implies(X,and(X,X)))) <-> axiom_m4 # label(axiom_m4) # label(axiom) # label(non_clause). [assumption]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 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]. 179.16/179.47 63 -substitution_of_equivalents | -is_a_theorem(equiv(A,B)) | B = A # label(substitution_of_equivalents) # label(axiom). [clausify(3)]. 179.16/179.47 89 or_2 | -is_a_theorem(implies(c30,or(c29,c30))) # label(or_2) # label(axiom). [clausify(14)]. 179.16/179.47 115 -op_or | not(and(not(A),not(B))) = or(A,B) # label(op_or) # label(axiom). [clausify(28)]. 179.16/179.47 116 -op_equiv | equiv(A,B) = and(implies(A,B),implies(B,A)) # label(op_equiv) # label(axiom). [clausify(29)]. 179.16/179.47 117 -op_equiv | and(implies(A,B),implies(B,A)) = equiv(A,B). [copy(116),flip(b)]. 179.16/179.47 119 -op_implies_and | not(and(A,not(B))) = implies(A,B) # label(op_implies_and) # label(axiom). [clausify(31)]. 179.16/179.47 121 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)]. 179.16/179.47 124 -adjunction | -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(A,B)) # label(adjunction) # label(axiom). [clausify(34)]. 179.16/179.47 134 -axiom_m1 | is_a_theorem(strict_implies(and(A,B),and(B,A))) # label(axiom_m1) # label(axiom). [clausify(38)]. 179.16/179.47 138 -is_a_theorem(strict_equiv(A,B)) | B = A | -substitution_strict_equiv # label(substitution_strict_equiv) # label(axiom). [clausify(39)]. 179.16/179.47 154 -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)]. 179.16/179.47 161 is_a_theorem(strict_implies(A,and(A,A))) | -axiom_m4 # label(axiom_m4) # label(axiom). [clausify(49)]. 179.16/179.47 166 -axiom_m2 | is_a_theorem(strict_implies(and(A,B),A)) # label(axiom_m2) # label(axiom). [clausify(52)]. 179.16/179.47 170 -axiom_m3 | is_a_theorem(strict_implies(and(and(A,B),C),and(A,and(B,C)))) # label(axiom_m3) # label(axiom). [clausify(54)]. 179.16/179.47 174 -op_strict_equiv | strict_equiv(A,B) = and(strict_implies(A,B),strict_implies(B,A)) # label(op_strict_equiv) # label(axiom). [clausify(56)]. 179.16/179.47 175 -op_strict_equiv | and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B). [copy(174),flip(b)]. 179.16/179.47 176 -op_strict_implies | necessarily(implies(A,B)) = strict_implies(A,B) # label(op_strict_implies) # label(axiom). [clausify(57)]. 179.16/179.47 179 axiom_m5 # label(s1_0_axiom_m5) # label(axiom). [assumption]. 179.16/179.47 180 op_strict_implies # label(s1_0_op_strict_implies) # label(axiom). [assumption]. 179.16/179.47 181 axiom_m3 # label(s1_0_axiom_m3) # label(axiom). [assumption]. 179.16/179.47 183 op_strict_equiv # label(s1_0_op_strict_equiv) # label(axiom). [assumption]. 179.16/179.47 184 axiom_m2 # label(s1_0_axiom_m2) # label(axiom). [assumption]. 179.16/179.47 185 axiom_m4 # label(s1_0_axiom_m4) # label(axiom). [assumption]. 179.16/179.47 186 axiom_m1 # label(s1_0_axiom_m1) # label(axiom). [assumption]. 179.16/179.47 188 op_equiv # label(s1_0_op_equiv) # label(axiom). [assumption]. 179.16/179.47 189 adjunction # label(s1_0_adjunction) # label(axiom). [assumption]. 179.16/179.47 190 modus_ponens_strict_implies # label(s1_0_modus_ponens_strict_implies) # label(axiom). [assumption]. 179.16/179.47 191 op_or # label(s1_0_op_or) # label(axiom). [assumption]. 179.16/179.47 192 substitution_strict_equiv # label(s1_0_substitution_strict_equiv) # label(axiom). [assumption]. 179.16/179.47 193 op_implies_and # label(hilbert_op_implies_and) # label(axiom). [assumption]. 179.16/179.47 194 substitution_of_equivalents # label(substitution_of_equivalents) # label(axiom). [assumption]. 179.16/179.47 195 -or_2 # label(hilbert_or_2) # label(negated_conjecture). [assumption]. 179.16/179.47 197 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)]. 179.16/179.47 198 necessarily(implies(A,B)) = strict_implies(A,B). [back_unit_del(176),unit_del(a,180)]. 179.16/179.47 199 is_a_theorem(strict_implies(and(and(A,B),C),and(A,and(B,C)))). [back_unit_del(170),unit_del(a,181)]. 179.16/179.47 201 and(strict_implies(A,B),strict_implies(B,A)) = strict_equiv(A,B). [back_unit_del(175),unit_del(a,183)]. 179.16/179.47 202 is_a_theorem(strict_implies(and(A,B),A)). [back_unit_del(166),unit_del(a,184)]. 179.16/179.47 203 is_a_theorem(strict_implies(A,and(A,A))). [back_unit_del(161),unit_del(b,185)]. 179.16/179.47 204 is_a_theorem(strict_implies(and(A,B),and(B,A))). [back_unit_del(134),unit_del(a,186)]. 179.16/179.47 205 and(implies(A,B),implies(B,A)) = equiv(A,B). [back_unit_del(117),unit_del(a,188)]. 179.16/179.47 206 -is_a_theorem(A) | -is_a_theorem(B) | is_a_theorem(and(A,B)). [back_unit_del(124),unit_del(a,189)]. 179.16/179.47 207 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(B). [back_unit_del(154),unit_del(a,190)]. 179.16/179.47 208 not(and(not(A),not(B))) = or(A,B). [back_unit_del(115),unit_del(a,191)]. 179.16/179.47 209 -is_a_theorem(strict_equiv(A,B)) | B = A. [back_unit_del(138),unit_del(c,192)]. 179.16/179.47 210 not(and(A,not(B))) = implies(A,B). [back_unit_del(119),unit_del(a,193)]. 179.16/179.47 211 -is_a_theorem(equiv(A,B)) | B = A. [back_unit_del(63),unit_del(a,194)]. 179.16/179.47 212 -is_a_theorem(implies(c30,or(c29,c30))). [back_unit_del(89),unit_del(a,195)]. 179.16/179.47 217 implies(not(A),B) = or(A,B). [back_rewrite(208),rewrite([210(4)])]. 179.16/179.47 253 -is_a_theorem(A) | is_a_theorem(and(strict_implies(and(B,C),and(C,B)),A)). [resolve(206,a,204,a)]. 179.16/179.47 254 -is_a_theorem(A) | is_a_theorem(and(strict_implies(B,and(B,B)),A)). [resolve(206,a,203,a)]. 179.16/179.47 263 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(and(B,C),B))). [resolve(206,b,202,a)]. 179.16/179.47 272 -is_a_theorem(and(strict_implies(A,B),strict_implies(B,C))) | is_a_theorem(strict_implies(A,C)). [resolve(207,b,197,a)]. 179.16/179.47 275 not(and(A,implies(B,C))) = implies(A,and(B,not(C))). [para(210(a,1),210(a,1,1,2))]. 179.16/179.47 277 necessarily(or(A,B)) = strict_implies(not(A),B). [para(217(a,1),198(a,1,1))]. 179.16/179.47 279 and(or(A,B),implies(B,not(A))) = equiv(not(A),B). [para(217(a,1),205(a,1,1))]. 179.16/179.47 281 or(and(A,not(B)),C) = implies(implies(A,B),C). [para(210(a,1),217(a,1,1)),flip(a)]. 179.16/179.47 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)]. 179.16/179.47 1136 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(and(B,C),B))). [resolve(254,a,202,a)]. 179.16/179.47 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)]. 179.16/179.47 2593 and(or(A,not(B)),or(B,not(A))) = equiv(not(A),not(B)). [para(217(a,1),279(a,1,2))]. 179.16/179.47 9787 is_a_theorem(strict_implies(A,A)). [resolve(1136,a,272,a)]. 179.16/179.47 9793 is_a_theorem(strict_equiv(A,and(A,A))). [para(201(a,1),1136(a,1))]. 179.16/179.47 9869 is_a_theorem(and(strict_implies(A,A),strict_implies(and(B,C),B))). [resolve(9787,a,263,a)]. 179.16/179.47 9876 is_a_theorem(and(strict_implies(A,and(A,A)),strict_implies(B,B))). [resolve(9787,a,254,a)]. 179.16/179.47 9878 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(B,B))). [resolve(9787,a,206,b)]. 179.16/179.47 10049 and(A,A) = A. [resolve(9793,a,209,a)]. 179.16/179.47 10055 is_a_theorem(and(strict_implies(A,A),strict_implies(B,B))). [back_rewrite(9876),rewrite([10049(1)])]. 179.16/179.47 10567 is_a_theorem(strict_implies(and(A,B),and(A,and(A,B)))). [para(10049(a,1),199(a,1,1,1))]. 179.16/179.47 10570 strict_equiv(A,A) = strict_implies(A,A). [para(10049(a,1),201(a,1)),flip(a)]. 179.16/179.47 10572 not(not(A)) = or(A,A). [para(10049(a,1),210(a,1,1)),rewrite([217(4)])]. 179.16/179.47 10585 implies(or(A,A),B) = or(not(A),B). [para(10049(a,1),281(a,1,1)),rewrite([217(4)]),flip(a)]. 179.16/179.47 10985 strict_implies(not(not(A)),B) = strict_implies(or(A,A),B). [para(10572(a,1),277(a,2,1)),rewrite([277(3)])]. 179.16/179.47 11003 not(not(not(A))) = not(or(A,A)). [para(10572(a,1),10572(a,1,1)),rewrite([10572(5,R)]),flip(a)]. 179.16/179.47 11214 is_a_theorem(and(and(strict_implies(A,A),strict_implies(B,B)),strict_implies(C,C))). [resolve(10055,a,9878,a)]. 179.16/179.47 11267 -is_a_theorem(A) | is_a_theorem(and(A,and(strict_implies(B,B),strict_implies(C,C)))). [resolve(10055,a,206,b)]. 179.16/179.47 12186 implies(A,not(not(B))) = implies(A,or(B,B)). [para(11003(a,1),210(a,1,1,2)),rewrite([210(4)]),flip(a)]. 179.16/179.47 12187 or(not(not(A)),B) = or(or(A,A),B). [para(11003(a,1),217(a,1,1)),rewrite([217(3)]),flip(a)]. 179.16/179.47 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)]. 179.16/179.47 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))]. 179.16/179.47 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)]. 179.16/179.47 16303 or(A,not(not(B))) = or(A,or(B,B)). [para(12186(a,1),217(a,1)),rewrite([217(3)]),flip(a)]. 179.16/179.47 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))]. 179.16/179.47 16466 or(not(not(A)),or(B,B)) = or(or(A,A),not(not(B))). [para(16303(a,1),12187(a,1))]. 179.16/179.47 16467 or(not(not(A)),not(not(B))) = or(or(A,A),or(B,B)). [para(16303(a,1),12187(a,2))]. 179.16/179.47 17014 is_a_theorem(strict_equiv(and(A,B),and(B,A))). [para(201(a,1),1024(a,1))]. 179.16/179.47 17041 and(A,B) = and(B,A). [resolve(17014,a,209,a)]. 179.16/179.47 17607 is_a_theorem(strict_implies(and(A,and(B,C)),and(B,and(C,A)))). [back_rewrite(199),rewrite([17041(2)])]. 179.16/179.47 17745 strict_equiv(A,B) = strict_equiv(B,A). [para(17041(a,1),201(a,1)),rewrite([201(3)])]. 179.16/179.47 17746 is_a_theorem(strict_implies(and(A,B),B)). [para(17041(a,1),202(a,1,1))]. 179.16/179.47 17747 equiv(A,B) = equiv(B,A). [para(17041(a,1),205(a,1)),rewrite([205(3)])]. 179.16/179.47 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))]. 179.16/179.47 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)])]. 179.16/179.47 17984 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(and(B,C),C))). [resolve(17746,a,206,b)]. 179.16/179.47 18010 -is_a_theorem(strict_equiv(A,B)) | A = B. [para(17745(a,1),209(a,1))]. 179.16/179.47 18018 -is_a_theorem(equiv(A,B)) | A = B. [para(17747(a,1),211(a,1))]. 179.16/179.47 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)]. 179.16/179.47 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)]. 179.16/179.47 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)]. 179.16/179.47 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)])]. 179.16/179.47 24238 not(and(not(A),or(B,C))) = or(A,and(not(B),not(C))). [para(2588(a,1),217(a,1))]. 179.16/179.47 24250 not(and(A,or(B,C))) = not(and(A,or(C,B))). [para(17041(a,1),2588(a,1,2)),rewrite([2588(4)])]. 179.16/179.47 25899 not(and(or(A,B),or(B,A))) = not(or(A,B)). [para(10049(a,1),24250(a,1,1)),flip(a)]. 179.16/179.47 26478 not(or(A,B)) = not(or(B,A)). [para(17041(a,1),25899(a,1,1)),rewrite([25899(4)])]. 179.16/179.47 26483 implies(A,or(B,C)) = implies(A,or(C,B)). [para(26478(a,1),210(a,1,1,2)),rewrite([210(4)])]. 179.16/179.47 26508 strict_implies(A,or(B,C)) = strict_implies(A,or(C,B)). [para(26483(a,1),198(a,1,1)),rewrite([198(3)])]. 179.16/179.47 26757 and(strict_implies(A,or(B,C)),strict_implies(or(C,B),A)) = strict_equiv(A,or(C,B)). [para(26508(a,1),201(a,1,1))]. 179.16/179.47 30110 or(A,and(not(B),not(C))) = implies(or(B,C),A). [para(17041(a,1),24238(a,1,1)),rewrite([210(4)]),flip(a)]. 179.16/179.47 30121 or(not(A),B) = or(B,not(A)). [para(10049(a,1),30110(a,1,2)),rewrite([10585(4)]),flip(a)]. 179.16/179.47 30125 implies(or(A,B),C) = implies(or(B,A),C). [para(17041(a,1),30110(a,1,2)),rewrite([30110(4)])]. 179.16/179.47 30171 implies(or(A,A),B) = or(B,not(A)). [back_rewrite(10585),rewrite([30121(4)])]. 179.16/179.47 30177 strict_implies(not(not(A)),B) = strict_implies(not(B),not(A)). [para(30121(a,1),277(a,1,1)),rewrite([277(3)]),flip(a)]. 179.16/179.47 30211 or(not(not(A)),not(not(B))) = or(or(B,B),or(A,A)). [para(30121(a,1),16467(a,1))]. 179.16/179.47 30267 -is_a_theorem(or(A,A)) | -is_a_theorem(strict_implies(not(B),not(A))) | is_a_theorem(B). [back_rewrite(15904),rewrite([30177(5)])]. 179.16/179.47 30272 strict_implies(not(A),not(B)) = strict_implies(or(B,B),A). [back_rewrite(10985),rewrite([30177(3)])]. 179.16/179.47 30326 strict_implies(or(A,B),C) = strict_implies(or(B,A),C). [para(30125(a,1),198(a,1,1)),rewrite([198(3)])]. 179.16/179.47 30328 strict_equiv(A,or(B,C)) = strict_equiv(A,or(C,B)). [back_rewrite(26757),rewrite([30326(4),201(5)])]. 179.16/179.47 30429 is_a_theorem(strict_implies(or(A,A),A)). [para(30272(a,1),9787(a,1))]. 179.16/179.47 30433 strict_implies(or(A,A),not(B)) = strict_implies(or(B,B),not(A)). [para(10572(a,1),30272(a,1,1))]. 179.16/179.47 30434 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(B,B))). [para(30272(a,1),10055(a,1,1))]. 179.16/179.47 30441 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(and(B,C),B))). [para(30272(a,1),9869(a,1,1))]. 179.16/179.47 30794 -is_a_theorem(A) | is_a_theorem(and(A,strict_implies(or(B,B),B))). [resolve(30429,a,206,b)]. 179.16/179.47 30844 is_a_theorem(and(strict_implies(or(A,A),A),strict_implies(or(B,B),B))). [para(30272(a,1),30434(a,1,2))]. 179.16/179.47 30949 -is_a_theorem(strict_equiv(A,or(B,C))) | or(C,B) = A. [para(30328(a,1),209(a,1))]. 179.16/179.47 30951 strict_equiv(A,not(not(B))) = strict_equiv(A,or(B,B)). [para(10572(a,2),30328(a,1,2))]. 179.16/179.47 30978 or(A,B) = or(B,A). [para(10570(a,1),30949(a,1)),unit_del(a,9787)]. 179.16/179.47 30979 -is_a_theorem(strict_equiv(A,not(not(B)))) | or(B,B) = A. [para(10572(a,2),30949(a,1,2))]. 179.16/179.47 31003 strict_implies(not(A),B) = strict_implies(not(B),A). [para(30978(a,1),277(a,1,1)),rewrite([277(2)])]. 179.16/179.47 31025 and(strict_implies(A,not(B)),strict_implies(not(A),B)) = strict_equiv(A,not(B)). [para(31003(a,1),201(a,1,1)),rewrite([17041(5),17745(7)])]. 179.16/179.47 31027 -is_a_theorem(and(strict_implies(not(A),B),strict_implies(A,C))) | is_a_theorem(strict_implies(not(B),C)). [para(31003(a,1),272(a,1,1))]. 179.16/179.47 31085 strict_equiv(A,not(not(not(B)))) = strict_equiv(A,not(or(B,B))). [para(10572(a,1),30951(a,1,2,1)),rewrite([10572(6,R)]),flip(a)]. 179.16/179.47 31124 is_a_theorem(strict_implies(not(A),not(and(A,B)))). [resolve(30441,a,272,a),rewrite([10572(3,R),31003(4)])]. 179.16/179.47 31203 -is_a_theorem(not(A)) | is_a_theorem(not(and(A,B))). [resolve(31124,a,207,b)]. 179.16/179.47 31210 is_a_theorem(strict_implies(not(A),implies(A,B))). [para(210(a,1),31124(a,1,2))]. 179.16/179.47 31253 is_a_theorem(and(strict_implies(not(A),implies(A,B)),strict_implies(or(C,C),C))). [resolve(31210,a,30794,a)]. 179.16/179.47 31261 is_a_theorem(and(strict_implies(not(A),implies(A,B)),strict_implies(and(C,D),D))). [resolve(31210,a,17984,a)]. 179.16/179.47 31473 -is_a_theorem(not(not(not(not(not(A)))))) | is_a_theorem(implies(B,not(or(A,A)))). [para(12217(a,2),31203(a,1)),rewrite([17041(10),210(11)])]. 179.16/179.47 32882 is_a_theorem(strict_implies(not(A),not(or(A,A)))). [resolve(30844,a,17751,a),rewrite([10572(3,R),31003(4)])]. 179.16/179.47 32923 is_a_theorem(strict_implies(not(A),not(not(not(A))))). [para(10572(a,2),32882(a,1,2,1))]. 179.16/179.47 33069 is_a_theorem(and(strict_implies(not(A),not(not(not(A)))),strict_implies(B,B))). [resolve(32923,a,9878,a)]. 179.16/179.47 36311 -is_a_theorem(strict_equiv(A,not(not(not(B))))) | not(or(B,B)) = A. [para(31085(a,2),209(a,1))]. 179.16/179.47 36331 is_a_theorem(strict_implies(not(not(not(A))),implies(A,B))). [resolve(31253,a,17751,a),rewrite([10572(3,R)])]. 179.16/179.47 36341 -is_a_theorem(not(not(not(A)))) | is_a_theorem(implies(A,B)). [resolve(36331,a,207,b)]. 179.16/179.47 36372 -is_a_theorem(not(not(not(or(A,A))))) | is_a_theorem(or(B,not(A))). [para(11003(a,1),36341(a,1,1,1)),rewrite([217(8),30978(7)])]. 179.16/179.47 40282 is_a_theorem(strict_equiv(not(A),not(not(not(A))))). [para(31025(a,1),33069(a,1))]. 179.16/179.47 40671 not(or(A,A)) = not(A). [resolve(40282,a,36311,a)]. 179.16/179.47 40677 not(not(not(A))) = not(A). [resolve(40282,a,30979,a),rewrite([10572(3,R)])]. 179.16/179.47 40787 -is_a_theorem(not(A)) | is_a_theorem(or(B,not(A))). [back_rewrite(36372),rewrite([40671(2),40677(3)])]. 179.16/179.47 40877 -is_a_theorem(not(A)) | is_a_theorem(implies(B,not(A))). [back_rewrite(31473),rewrite([40677(3),40677(3),40671(4)])]. 179.16/179.47 41292 implies(A,or(B,B)) = implies(A,B). [para(40671(a,1),210(a,1,1,2)),rewrite([210(3)]),flip(a)]. 179.16/179.47 41294 or(A,or(B,B)) = or(B,A). [para(40671(a,1),217(a,1,1)),rewrite([217(2),30978(3)]),flip(a)]. 179.16/179.47 41331 or(not(not(A)),not(not(B))) = or(B,not(not(A))). [para(40671(a,1),16466(a,1,1,1)),rewrite([41294(4),10572(6,R),40671(5)]),flip(a)]. 179.16/179.47 41467 implies(A,not(not(B))) = implies(A,B). [back_rewrite(12186),rewrite([41292(5)])]. 179.16/179.47 41540 or(A,not(not(B))) = or(B,A). [back_rewrite(30211),rewrite([41331(5),41294(6),41294(5),30978(4)])]. 179.16/179.47 41833 -is_a_theorem(implies(A,B)) | is_a_theorem(implies(C,implies(A,B))). [para(210(a,1),40877(a,1)),rewrite([210(5)])]. 179.16/179.47 41909 strict_implies(A,or(B,B)) = strict_implies(A,B). [para(41292(a,1),198(a,1,1)),rewrite([198(2)]),flip(a)]. 179.16/179.47 41979 -is_a_theorem(A) | -is_a_theorem(strict_implies(A,B)) | is_a_theorem(not(not(B))). [back_rewrite(16396),rewrite([41909(3)])]. 179.16/179.47 41980 strict_implies(A,not(not(B))) = strict_implies(A,B). [back_rewrite(16302),rewrite([41909(5)])]. 179.16/179.47 42869 -is_a_theorem(A) | is_a_theorem(not(not(A))). [resolve(41979,b,9787,a)]. 179.16/179.47 43593 is_a_theorem(not(not(and(strict_implies(A,A),strict_implies(B,B))))). [resolve(42869,a,10055,a)]. 179.16/179.47 43616 is_a_theorem(not(not(strict_implies(A,A)))). [resolve(42869,a,9787,a)]. 179.16/179.47 43693 is_a_theorem(not(not(strict_implies(and(A,B),A)))). [resolve(42869,a,202,a)]. 179.16/179.47 44266 is_a_theorem(implies(A,strict_implies(B,B))). [resolve(43616,a,40877,a),rewrite([41467(4)])]. 179.16/179.47 44272 is_a_theorem(or(A,strict_implies(B,B))). [resolve(43616,a,40787,a),rewrite([41540(4),30978(2)])]. 179.16/179.47 44461 -is_a_theorem(A) | is_a_theorem(and(A,implies(B,strict_implies(C,C)))). [resolve(44266,a,206,b)]. 179.16/179.47 44480 -is_a_theorem(strict_implies(not(A),not(strict_implies(B,B)))) | is_a_theorem(A). [resolve(44272,a,30267,a)]. 179.16/179.47 45341 is_a_theorem(implies(A,strict_implies(and(B,C),B))). [resolve(43693,a,40877,a),rewrite([41467(5)])]. 179.16/179.47 47238 is_a_theorem(strict_equiv(and(A,B),and(B,and(A,B)))). [para(201(a,1),18670(a,1))]. 179.16/179.47 51389 is_a_theorem(implies(A,and(strict_implies(B,B),strict_implies(C,C)))). [resolve(43593,a,40877,a),rewrite([41467(6)])]. 179.16/179.47 51784 is_a_theorem(and(implies(A,strict_implies(B,B)),implies(C,strict_implies(D,D)))). [resolve(44461,a,44266,a)]. 179.16/179.47 53411 and(A,and(B,A)) = and(B,A). [resolve(47238,a,18010,a),flip(a)]. 179.16/179.47 54346 is_a_theorem(implies(A,implies(B,and(strict_implies(C,C),strict_implies(D,D))))). [resolve(51389,a,41833,a)]. 179.16/179.47 58722 is_a_theorem(strict_implies(not(A),implies(and(B,A),C))). [resolve(31027,a,31261,a),rewrite([31003(4)])]. 179.16/179.47 62230 is_a_theorem(equiv(strict_implies(A,A),strict_implies(B,B))). [para(205(a,1),51784(a,1))]. 179.16/179.47 62232 strict_implies(A,A) = strict_implies(B,B). [resolve(62230,a,18018,a)]. 179.16/179.47 62240 strict_implies(A,A) = c_0. [new_symbol(62232)]. 179.16/179.47 62993 is_a_theorem(implies(A,implies(B,c_0))). [back_rewrite(54346),rewrite([62240(1),62240(2),10049(3)])]. 179.16/179.47 63752 -is_a_theorem(strict_implies(not(A),not(c_0))) | is_a_theorem(A). [back_rewrite(44480),rewrite([62240(2)])]. 179.16/179.47 63766 -is_a_theorem(A) | is_a_theorem(and(A,implies(B,c_0))). [back_rewrite(44461),rewrite([62240(2)])]. 179.16/179.47 64071 is_a_theorem(c_0). [back_rewrite(21249),rewrite([62240(1),62240(2),10049(3),62240(2),62240(3),10049(4),10049(3)])]. 179.16/179.47 64076 -is_a_theorem(A) | is_a_theorem(and(A,c_0)). [back_rewrite(21162),rewrite([62240(2),62240(3),10049(4),62240(3),10049(4)])]. 179.16/179.47 64331 strict_equiv(A,not(not(A))) = c_0. [para(62240(a,1),31025(a,1,2)),rewrite([41980(3),62240(1),10049(3)]),flip(a)]. 179.16/179.47 64541 is_a_theorem(and(c_0,strict_implies(not(A),implies(and(A,B),C)))). [resolve(64076,a,58722,a),rewrite([17041(2),17041(6)])]. 179.16/179.47 64679 is_a_theorem(and(c_0,implies(A,strict_implies(and(B,C),B)))). [resolve(64076,a,45341,a),rewrite([17041(5)])]. 179.16/179.47 65218 not(not(A)) = A. [para(64331(a,1),209(a,1)),unit_del(a,64071)]. 179.16/179.47 65223 or(A,A) = A. [para(64331(a,1),30979(a,1)),unit_del(a,64071)]. 179.16/179.47 65479 strict_implies(A,not(B)) = strict_implies(B,not(A)). [back_rewrite(30433),rewrite([65223(1),65223(3)])]. 179.16/179.47 65481 or(A,not(B)) = implies(B,A). [back_rewrite(30171),rewrite([65223(1)]),flip(a)]. 179.16/179.47 65544 -is_a_theorem(strict_implies(c_0,A)) | is_a_theorem(A). [back_rewrite(63752),rewrite([65479(4),65218(3)])]. 179.16/179.47 65705 equiv(not(A),not(B)) = equiv(A,B). [back_rewrite(2593),rewrite([65481(2),65481(3),17041(3),205(3)]),flip(a)]. 179.16/179.47 65716 not(and(A,B)) = implies(A,not(B)). [para(65218(a,1),210(a,1,1,2))]. 179.16/179.47 65717 not(implies(A,B)) = and(A,not(B)). [para(210(a,1),65218(a,1,1))]. 179.16/179.47 68086 is_a_theorem(and(implies(A,implies(B,c_0)),implies(C,c_0))). [resolve(63766,a,62993,a)]. 179.16/179.47 69611 strict_equiv(A,not(B)) = strict_equiv(B,not(A)). [para(65479(a,1),201(a,1,1)),rewrite([31025(5)])]. 179.16/179.47 69627 -is_a_theorem(equiv(A,B)) | not(A) = not(B). [para(65705(a,1),18018(a,1))]. 179.16/179.47 69644 implies(A,not(B)) = implies(B,not(A)). [para(17041(a,1),65716(a,1,1)),rewrite([65716(2)])]. 179.16/179.47 69645 implies(A,implies(B,not(A))) = implies(B,not(A)). [para(53411(a,1),65716(a,1,1)),rewrite([65716(2),65716(4)]),flip(a)]. 179.16/179.47 69648 and(not(A),not(B)) = not(or(A,B)). [para(217(a,1),65717(a,1,1)),flip(a)]. 179.16/179.47 70019 -is_a_theorem(strict_equiv(A,not(B))) | not(A) = B. [para(69611(a,1),209(a,1))]. 179.16/179.47 70210 -is_a_theorem(strict_equiv(A,B)) | not(A) = not(B). [para(65218(a,1),70019(a,1,2))]. 179.16/179.47 70254 -is_a_theorem(strict_equiv(A,B)) | not(B) = not(A). [para(17745(a,1),70210(a,1))]. 179.16/179.47 71406 is_a_theorem(equiv(c_0,implies(A,c_0))). [para(205(a,1),68086(a,1))]. 179.16/179.47 71412 and(A,not(c_0)) = not(c_0). [resolve(71406,a,69627,a),rewrite([65717(5)]),flip(a)]. 179.16/179.47 71434 implies(A,c_0) = c_0. [resolve(71406,a,18018,a),flip(a)]. 179.16/179.47 71440 strict_implies(A,c_0) = necessarily(c_0). [para(71434(a,1),198(a,1,1)),flip(a)]. 179.16/179.47 71441 and(c_0,implies(c_0,A)) = equiv(A,c_0). [para(71434(a,1),205(a,1,1))]. 179.16/179.47 71742 and(necessarily(c_0),strict_implies(c_0,A)) = strict_equiv(A,c_0). [para(71440(a,1),201(a,1,1))]. 179.16/179.47 71778 necessarily(c_0) = c_0. [para(71440(a,1),62240(a,1))]. 179.16/179.47 71787 and(c_0,strict_implies(c_0,A)) = strict_equiv(A,c_0). [back_rewrite(71742),rewrite([71778(2)])]. 179.16/179.47 71946 is_a_theorem(equiv(c_0,strict_implies(and(A,B),A))). [para(71441(a,1),64679(a,1)),rewrite([17747(4)])]. 179.16/179.47 72253 strict_implies(and(A,B),A) = c_0. [resolve(71946,a,18018,a),flip(a)]. 179.16/179.47 72261 strict_implies(and(A,B),B) = c_0. [para(17041(a,1),72253(a,1,1))]. 179.16/179.47 72270 -is_a_theorem(and(c_0,strict_implies(A,B))) | is_a_theorem(strict_implies(and(A,C),B)). [para(72261(a,1),272(a,1,1)),rewrite([17041(5)])]. 179.16/179.47 75078 implies(A,implies(A,B)) = implies(A,B). [para(217(a,1),69645(a,1,2)),rewrite([65481(2),69644(5),65218(4)])]. 179.16/179.47 75085 strict_implies(A,implies(A,B)) = strict_implies(A,B). [para(75078(a,1),198(a,1,1)),rewrite([198(2)]),flip(a)]. 179.16/179.47 75092 -is_a_theorem(strict_implies(c_0,A)) | is_a_theorem(implies(c_0,A)). [para(75085(a,1),65544(a,1))]. 179.16/179.47 75097 -is_a_theorem(strict_implies(A,not(c_0))) | is_a_theorem(implies(A,not(c_0))). [para(65479(a,1),75092(a,1)),rewrite([69644(7)])]. 179.16/179.47 82395 is_a_theorem(strict_implies(and(A,not(B)),implies(and(B,C),D))). [resolve(72270,a,64541,a),rewrite([17041(2)])]. 179.16/179.47 82490 is_a_theorem(strict_implies(and(A,not(A)),B)). [para(75085(a,1),82395(a,1))]. 179.16/179.47 82494 is_a_theorem(implies(c_0,implies(A,A))). [resolve(82490,a,75097,a),rewrite([69644(5),65716(4),65218(3)])]. 179.16/179.47 82745 is_a_theorem(equiv(c_0,implies(A,A))). [resolve(82494,a,64076,a),rewrite([17041(5),71441(5),17747(3)])]. 179.16/179.47 82979 and(A,not(A)) = not(c_0). [resolve(82745,a,69627,a),rewrite([65717(4)]),flip(a)]. 179.16/179.47 83009 is_a_theorem(strict_implies(c_0,implies(A,implies(B,A)))). [para(82979(a,1),22802(a,1,2,2)),rewrite([17041(2),71412(6),65479(6),65716(5),65716(4),65218(3)])]. 179.16/179.47 83028 is_a_theorem(strict_equiv(c_0,implies(A,implies(B,A)))). [resolve(83009,a,64076,a),rewrite([17041(6),71787(6),17745(4)])]. 179.16/179.47 83284 and(A,and(B,not(A))) = not(c_0). [resolve(83028,a,70254,a),rewrite([65717(3),65717(2)])]. 179.16/179.47 83317 and(not(A),and(A,B)) = not(c_0). [para(65218(a,1),83284(a,1,2,2)),rewrite([17041(2)])]. 179.16/179.47 83333 is_a_theorem(strict_implies(c_0,implies(and(A,B),implies(C,A)))). [para(83317(a,1),17607(a,1,2,2)),rewrite([71412(7),65479(7),65716(6),65716(5),65218(4)])]. 179.16/179.47 83587 is_a_theorem(implies(and(A,B),implies(C,A))). [resolve(83333,a,65544,a)]. 179.16/179.47 83630 is_a_theorem(implies(and(A,B),A)). [para(75078(a,1),83587(a,1))]. 179.16/179.47 83691 is_a_theorem(implies(and(A,B),B)). [para(17041(a,1),83630(a,1,1))]. 179.16/179.47 83750 is_a_theorem(implies(A,or(B,A))). [para(69648(a,1),83691(a,1,1)),rewrite([69644(4),65218(3)])]. 179.16/179.47 83751 $F. [resolve(83750,a,212,a)]. 179.16/179.47 179.16/179.47 % SZS output end Refutation 179.16/179.47 ============================== end of proof ========================== 179.16/179.47 179.16/179.47 ============================== STATISTICS ============================ 179.16/179.47 179.16/179.47 Given=24962. Generated=8351386. Kept=83688. proofs=1. 179.16/179.47 Usable=12568. Sos=5936. Demods=422. Limbo=55, Disabled=65263. Hints=0. 179.16/179.47 Megabytes=88.42. 179.16/179.47 User_CPU=173.79, System_CPU=4.59, Wall_clock=179. 179.16/179.47 179.16/179.47 ============================== end of statistics ===================== 179.16/179.47 179.16/179.47 ============================== end of search ========================= 179.16/179.47 179.16/179.47 THEOREM PROVED 179.16/179.47 % SZS status Theorem 179.16/179.47 179.16/179.47 Exiting with 1 proof. 179.16/179.47 179.16/179.47 Process 2386 exit (max_proofs) Thu Aug 29 16:18:18 2019 179.16/179.47 Prover9 interrupted 179.16/179.47 EOF