TSTP Solution File: SEU174+2 by Prover9---1109a
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : SEU174+2 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Tue Jul 19 13:29:39 EDT 2022
% Result : Theorem 0.85s 1.33s
% Output : Refutation 0.85s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SEU174+2 : TPTP v8.1.0. Released v3.3.0.
% 0.07/0.13 % Command : tptp2X_and_run_prover9 %d %s
% 0.13/0.35 % Computer : n013.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 600
% 0.13/0.35 % DateTime : Mon Jun 20 00:45:44 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.79/1.11 ============================== Prover9 ===============================
% 0.79/1.11 Prover9 (32) version 2009-11A, November 2009.
% 0.79/1.11 Process 20523 was started by sandbox on n013.cluster.edu,
% 0.79/1.11 Mon Jun 20 00:45:44 2022
% 0.79/1.11 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_20370_n013.cluster.edu".
% 0.79/1.11 ============================== end of head ===========================
% 0.79/1.11
% 0.79/1.11 ============================== INPUT =================================
% 0.79/1.11
% 0.79/1.11 % Reading from file /tmp/Prover9_20370_n013.cluster.edu
% 0.79/1.11
% 0.79/1.11 set(prolog_style_variables).
% 0.79/1.11 set(auto2).
% 0.79/1.11 % set(auto2) -> set(auto).
% 0.79/1.11 % set(auto) -> set(auto_inference).
% 0.79/1.11 % set(auto) -> set(auto_setup).
% 0.79/1.11 % set(auto_setup) -> set(predicate_elim).
% 0.79/1.11 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.79/1.11 % set(auto) -> set(auto_limits).
% 0.79/1.11 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.79/1.11 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.79/1.11 % set(auto) -> set(auto_denials).
% 0.79/1.11 % set(auto) -> set(auto_process).
% 0.79/1.11 % set(auto2) -> assign(new_constants, 1).
% 0.79/1.11 % set(auto2) -> assign(fold_denial_max, 3).
% 0.79/1.11 % set(auto2) -> assign(max_weight, "200.000").
% 0.79/1.11 % set(auto2) -> assign(max_hours, 1).
% 0.79/1.11 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.79/1.11 % set(auto2) -> assign(max_seconds, 0).
% 0.79/1.11 % set(auto2) -> assign(max_minutes, 5).
% 0.79/1.11 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.79/1.11 % set(auto2) -> set(sort_initial_sos).
% 0.79/1.11 % set(auto2) -> assign(sos_limit, -1).
% 0.79/1.11 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.79/1.11 % set(auto2) -> assign(max_megs, 400).
% 0.79/1.11 % set(auto2) -> assign(stats, some).
% 0.79/1.11 % set(auto2) -> clear(echo_input).
% 0.79/1.11 % set(auto2) -> set(quiet).
% 0.79/1.11 % set(auto2) -> clear(print_initial_clauses).
% 0.79/1.11 % set(auto2) -> clear(print_given).
% 0.79/1.11 assign(lrs_ticks,-1).
% 0.79/1.11 assign(sos_limit,10000).
% 0.79/1.11 assign(order,kbo).
% 0.79/1.11 set(lex_order_vars).
% 0.79/1.11 clear(print_given).
% 0.79/1.11
% 0.79/1.11 % formulas(sos). % not echoed (122 formulas)
% 0.79/1.11
% 0.79/1.11 ============================== end of input ==========================
% 0.79/1.11
% 0.79/1.11 % From the command line: assign(max_seconds, 300).
% 0.79/1.11
% 0.79/1.11 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.79/1.11
% 0.79/1.11 % Formulas that are not ordinary clauses:
% 0.79/1.11 1 (all A all B (in(A,B) -> -in(B,A))) # label(antisymmetry_r2_hidden) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 2 (all A all B (proper_subset(A,B) -> -proper_subset(B,A))) # label(antisymmetry_r2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 3 (all A all B unordered_pair(A,B) = unordered_pair(B,A)) # label(commutativity_k2_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 4 (all A all B set_union2(A,B) = set_union2(B,A)) # label(commutativity_k2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 5 (all A all B set_intersection2(A,B) = set_intersection2(B,A)) # label(commutativity_k3_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 6 (all A all B (A = B <-> subset(A,B) & subset(B,A))) # label(d10_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 7 (all A all B (B = singleton(A) <-> (all C (in(C,B) <-> C = A)))) # label(d1_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 8 (all A (A = empty_set <-> (all B -in(B,A)))) # label(d1_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 9 (all A all B (B = powerset(A) <-> (all C (in(C,B) <-> subset(C,A))))) # label(d1_zfmisc_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 10 (all A all B ((-empty(A) -> (element(B,A) <-> in(B,A))) & (empty(A) -> (element(B,A) <-> empty(B))))) # label(d2_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 11 (all A all B all C (C = unordered_pair(A,B) <-> (all D (in(D,C) <-> D = A | D = B)))) # label(d2_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 12 (all A all B all C (C = set_union2(A,B) <-> (all D (in(D,C) <-> in(D,A) | in(D,B))))) # label(d2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 13 (all A all B all C (C = cartesian_product2(A,B) <-> (all D (in(D,C) <-> (exists E exists F (in(E,A) & in(F,B) & D = ordered_pair(E,F))))))) # label(d2_zfmisc_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 14 (all A all B (subset(A,B) <-> (all C (in(C,A) -> in(C,B))))) # label(d3_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 15 (all A all B all C (C = set_intersection2(A,B) <-> (all D (in(D,C) <-> in(D,A) & in(D,B))))) # label(d3_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 16 (all A all B (B = union(A) <-> (all C (in(C,B) <-> (exists D (in(C,D) & in(D,A))))))) # label(d4_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 17 (all A all B all C (C = set_difference(A,B) <-> (all D (in(D,C) <-> in(D,A) & -in(D,B))))) # label(d4_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 18 (all A all B (element(B,powerset(A)) -> subset_complement(A,B) = set_difference(A,B))) # label(d5_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 19 (all A all B ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A))) # label(d5_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 20 (all A all B (disjoint(A,B) <-> set_intersection2(A,B) = empty_set)) # label(d7_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 21 (all A all B (element(B,powerset(powerset(A))) -> (all C (element(C,powerset(powerset(A))) -> (C = complements_of_subsets(A,B) <-> (all D (element(D,powerset(A)) -> (in(D,C) <-> in(subset_complement(A,D),B))))))))) # label(d8_setfam_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 22 (all A all B (proper_subset(A,B) <-> subset(A,B) & A != B)) # label(d8_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 23 $T # label(dt_k1_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 24 $T # label(dt_k1_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 25 $T # label(dt_k1_zfmisc_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 26 $T # label(dt_k2_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 27 $T # label(dt_k2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 28 $T # label(dt_k2_zfmisc_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 29 (all A all B (element(B,powerset(A)) -> element(subset_complement(A,B),powerset(A)))) # label(dt_k3_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 30 $T # label(dt_k3_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 31 $T # label(dt_k3_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 32 $T # label(dt_k4_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 33 $T # label(dt_k4_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 34 (all A all B (element(B,powerset(powerset(A))) -> element(complements_of_subsets(A,B),powerset(powerset(A))))) # label(dt_k7_setfam_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 35 $T # label(dt_m1_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 36 (all A exists B element(B,A)) # label(existence_m1_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 37 (all A -empty(powerset(A))) # label(fc1_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 38 (all A all B -empty(ordered_pair(A,B))) # label(fc1_zfmisc_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 39 (all A all B (-empty(A) -> -empty(set_union2(A,B)))) # label(fc2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 40 (all A all B (-empty(A) -> -empty(set_union2(B,A)))) # label(fc3_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 41 (all A all B set_union2(A,A) = A) # label(idempotence_k2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 42 (all A all B set_intersection2(A,A) = A) # label(idempotence_k3_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 43 (all A all B (element(B,powerset(A)) -> subset_complement(A,subset_complement(A,B)) = B)) # label(involutiveness_k3_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 44 (all A all B (element(B,powerset(powerset(A))) -> complements_of_subsets(A,complements_of_subsets(A,B)) = B)) # label(involutiveness_k7_setfam_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 45 (all A all B -proper_subset(A,A)) # label(irreflexivity_r2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 46 (all A singleton(A) != empty_set) # label(l1_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 47 (all A all B (in(A,B) -> set_union2(singleton(A),B) = B)) # label(l23_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 48 (all A all B -(disjoint(singleton(A),B) & in(A,B))) # label(l25_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 49 (all A all B (-in(A,B) -> disjoint(singleton(A),B))) # label(l28_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 50 (all A all B (subset(singleton(A),B) <-> in(A,B))) # label(l2_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 51 (all A all B (set_difference(A,B) = empty_set <-> subset(A,B))) # label(l32_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 52 (all A all B (element(B,powerset(A)) -> (all C (in(C,B) -> in(C,A))))) # label(l3_subset_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 53 (all A all B all C (subset(A,B) -> in(C,A) | subset(A,set_difference(B,singleton(C))))) # label(l3_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 54 (all A all B (subset(A,singleton(B)) <-> A = empty_set | A = singleton(B))) # label(l4_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 55 (all A all B (in(A,B) -> subset(A,union(B)))) # label(l50_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 56 (all A all B all C all D (in(ordered_pair(A,B),cartesian_product2(C,D)) <-> in(A,C) & in(B,D))) # label(l55_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 57 (all A all B ((all C (in(C,A) -> in(C,B))) -> element(A,powerset(B)))) # label(l71_subset_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 58 (all A (-empty(A) -> (exists B (element(B,powerset(A)) & -empty(B))))) # label(rc1_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 59 (exists A empty(A)) # label(rc1_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 60 (all A exists B (element(B,powerset(A)) & empty(B))) # label(rc2_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 61 (exists A -empty(A)) # label(rc2_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 62 (all A all B subset(A,A)) # label(reflexivity_r1_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 63 (all A all B (disjoint(A,B) -> disjoint(B,A))) # label(symmetry_r1_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 64 (all A all B all C all D (in(ordered_pair(A,B),cartesian_product2(C,D)) <-> in(A,C) & in(B,D))) # label(t106_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 65 (all A all B all C all D -(unordered_pair(A,B) = unordered_pair(C,D) & A != C & A != D)) # label(t10_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 66 (all A all B all C (subset(A,B) -> subset(cartesian_product2(A,C),cartesian_product2(B,C)) & subset(cartesian_product2(C,A),cartesian_product2(C,B)))) # label(t118_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 67 (all A all B all C all D (subset(A,B) & subset(C,D) -> subset(cartesian_product2(A,C),cartesian_product2(B,D)))) # label(t119_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 68 (all A all B (subset(A,B) -> set_union2(A,B) = B)) # label(t12_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 69 (all A exists B (in(A,B) & (all C all D (in(C,B) & subset(D,C) -> in(D,B))) & (all C (in(C,B) -> in(powerset(C),B))) & (all C -(subset(C,B) & -are_equipotent(C,B) & -in(C,B))))) # label(t136_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 70 (all A all B subset(set_intersection2(A,B),A)) # label(t17_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 71 (all A all B all C (subset(A,B) & subset(A,C) -> subset(A,set_intersection2(B,C)))) # label(t19_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 72 (all A set_union2(A,empty_set) = A) # label(t1_boole) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 73 (all A all B (in(A,B) -> element(A,B))) # label(t1_subset) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 74 (all A all B all C (subset(A,B) & subset(B,C) -> subset(A,C))) # label(t1_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 75 (all A all B all C (subset(A,B) -> subset(set_intersection2(A,C),set_intersection2(B,C)))) # label(t26_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 76 (all A all B (subset(A,B) -> set_intersection2(A,B) = A)) # label(t28_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 77 (all A set_intersection2(A,empty_set) = empty_set) # label(t2_boole) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 78 (all A all B (element(A,B) -> empty(B) | in(A,B))) # label(t2_subset) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 79 (all A all B ((all C (in(C,A) <-> in(C,B))) -> A = B)) # label(t2_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 80 (all A subset(empty_set,A)) # label(t2_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 81 (all A all B all C (subset(A,B) -> subset(set_difference(A,C),set_difference(B,C)))) # label(t33_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 82 (all A all B all C all D (ordered_pair(A,B) = ordered_pair(C,D) -> A = C & B = D)) # label(t33_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 83 (all A all B subset(set_difference(A,B),A)) # label(t36_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 84 (all A all B (set_difference(A,B) = empty_set <-> subset(A,B))) # label(t37_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 85 (all A all B (subset(singleton(A),B) <-> in(A,B))) # label(t37_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 86 (all A all B all C (subset(unordered_pair(A,B),C) <-> in(A,C) & in(B,C))) # label(t38_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 87 (all A all B set_union2(A,set_difference(B,A)) = set_union2(A,B)) # label(t39_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 88 (all A all B (subset(A,singleton(B)) <-> A = empty_set | A = singleton(B))) # label(t39_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 89 (all A set_difference(A,empty_set) = A) # label(t3_boole) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 90 (all A all B (element(A,powerset(B)) <-> subset(A,B))) # label(t3_subset) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 91 (all A all B (-(-disjoint(A,B) & (all C -(in(C,A) & in(C,B)))) & -((exists C (in(C,A) & in(C,B))) & disjoint(A,B)))) # label(t3_xboole_0) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 92 (all A (subset(A,empty_set) -> A = empty_set)) # label(t3_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 93 (all A all B set_difference(set_union2(A,B),B) = set_difference(A,B)) # label(t40_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 94 (all A all B (element(B,powerset(A)) -> (all C (element(C,powerset(A)) -> (disjoint(B,C) <-> subset(B,subset_complement(A,C))))))) # label(t43_subset_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 95 (all A all B (subset(A,B) -> B = set_union2(A,set_difference(B,A)))) # label(t45_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 96 (all A all B (in(A,B) -> set_union2(singleton(A),B) = B)) # label(t46_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 97 (all A all B set_difference(A,set_difference(A,B)) = set_intersection2(A,B)) # label(t48_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 98 (all A set_difference(empty_set,A) = empty_set) # label(t4_boole) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 99 (all A all B all C (in(A,B) & element(B,powerset(C)) -> element(A,C))) # label(t4_subset) # label(axiom) # label(non_clause). [assumption].
% 0.79/1.11 100 (all A all B (-(-disjoint(A,B) & (all C -in(C,set_intersection2(A,B)))) & -((exists C in(C,set_intersection2(A,B))) & disjoint(A,B)))) # label(t4_xboole_0) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 101 (all A (A != empty_set -> (all B (element(B,powerset(A)) -> (all C (element(C,A) -> (-in(C,B) -> in(C,subset_complement(A,B))))))))) # label(t50_subset_1) # label(lemma) # label(non_clause). [assumption].
% 0.79/1.11 102 (all A all B all C (element(C,powerset(A)) -> -(in(B,subset_complement(A,C)) & in(B,C)))) # label(t54_subset_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 103 (all A all B all C -(in(A,B) & element(B,powerset(C)) & empty(C))) # label(t5_subset) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 104 (all A all B -(subset(A,B) & proper_subset(B,A))) # label(t60_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 105 (all A all B all C (subset(A,B) & disjoint(B,C) -> disjoint(A,C))) # label(t63_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 106 (all A all B (set_difference(A,singleton(B)) = A <-> -in(B,A))) # label(t65_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 107 (all A unordered_pair(A,A) = singleton(A)) # label(t69_enumset1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 108 (all A (empty(A) -> A = empty_set)) # label(t6_boole) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 109 (all A all B (subset(singleton(A),singleton(B)) -> A = B)) # label(t6_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 110 (all A all B -(in(A,B) & empty(B))) # label(t7_boole) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 111 (all A all B subset(A,set_union2(A,B))) # label(t7_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 112 (all A all B (disjoint(A,B) <-> set_difference(A,B) = A)) # label(t83_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 113 (all A all B -(empty(A) & A != B & empty(B))) # label(t8_boole) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 114 (all A all B all C (subset(A,B) & subset(C,B) -> subset(set_union2(A,C),B))) # label(t8_xboole_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 115 (all A all B all C (singleton(A) = unordered_pair(B,C) -> A = B)) # label(t8_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 116 (all A all B (in(A,B) -> subset(A,union(B)))) # label(t92_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 117 (all A union(powerset(A)) = A) # label(t99_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 118 (all A exists B (in(A,B) & (all C all D (in(C,B) & subset(D,C) -> in(D,B))) & (all C -(in(C,B) & (all D -(in(D,B) & (all E (subset(E,C) -> in(E,D))))))) & (all C -(subset(C,B) & -are_equipotent(C,B) & -in(C,B))))) # label(t9_tarski) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 119 (all A all B all C (singleton(A) = unordered_pair(B,C) -> B = C)) # label(t9_zfmisc_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 120 -(all A all B (element(B,powerset(powerset(A))) -> -(B != empty_set & complements_of_subsets(A,B) = empty_set))) # label(t46_setfam_1) # label(negated_conjecture) # label(non_clause). [assumption].
% 0.85/1.33
% 0.85/1.33 ============================== end of process non-clausal formulas ===
% 0.85/1.33
% 0.85/1.33 ============================== PROCESS INITIAL CLAUSES ===============
% 0.85/1.33
% 0.85/1.33 ============================== PREDICATE ELIMINATION =================
% 0.85/1.33
% 0.85/1.33 ============================== end predicate elimination =============
% 0.85/1.33
% 0.85/1.33 Auto_denials: (non-Horn, no changes).
% 0.85/1.33
% 0.85/1.33 Term ordering decisions:
% 0.85/1.33 Function symbol KB weights: empty_set=1. c1=1. c2=1. c3=1. c4=1. set_difference=1. set_union2=1. cartesian_product2=1. set_intersection2=1. unordered_pair=1. ordered_pair=1. subset_complement=1. complements_of_subsets=1. f1=1. f3=1. f11=1. f14=1. f15=1. f19=1. f23=1. f24=1. f25=1. f27=1. powerset=1. singleton=1. union=1. f2=1. f18=1. f20=1. f21=1. f22=1. f26=1. f4=1. f5=1. f8=1. f9=1. f10=1. f12=1. f13=1. f16=1. f17=1. f6=1. f7=1.
% 0.85/1.33
% 0.85/1.33 ============================== end of process initial clauses ========
% 0.85/1.33
% 0.85/1.33 ============================== CLAUSES FOR SEARCH ====================
% 0.85/1.33
% 0.85/1.33 ============================== end of clauses for search =============
% 0.85/1.33
% 0.85/1.33 ============================== SEARCH ================================
% 0.85/1.33
% 0.85/1.33 % Starting search at 0.03 seconds.
% 0.85/1.33
% 0.85/1.33 ============================== PROOF =================================
% 0.85/1.33 % SZS status Theorem
% 0.85/1.33 % SZS output start Refutation
% 0.85/1.33
% 0.85/1.33 % Proof 1 at 0.23 (+ 0.00) seconds.
% 0.85/1.33 % Length of proof is 42.
% 0.85/1.33 % Level of proof is 9.
% 0.85/1.33 % Maximum clause weight is 35.000.
% 0.85/1.33 % Given clauses 202.
% 0.85/1.33
% 0.85/1.33 18 (all A all B (element(B,powerset(A)) -> subset_complement(A,B) = set_difference(A,B))) # label(d5_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 21 (all A all B (element(B,powerset(powerset(A))) -> (all C (element(C,powerset(powerset(A))) -> (C = complements_of_subsets(A,B) <-> (all D (element(D,powerset(A)) -> (in(D,C) <-> in(subset_complement(A,D),B))))))))) # label(d8_setfam_1) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 44 (all A all B (element(B,powerset(powerset(A))) -> complements_of_subsets(A,complements_of_subsets(A,B)) = B)) # label(involutiveness_k7_setfam_1) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 59 (exists A empty(A)) # label(rc1_xboole_0) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 60 (all A exists B (element(B,powerset(A)) & empty(B))) # label(rc2_subset_1) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 89 (all A set_difference(A,empty_set) = A) # label(t3_boole) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 102 (all A all B all C (element(C,powerset(A)) -> -(in(B,subset_complement(A,C)) & in(B,C)))) # label(t54_subset_1) # label(lemma) # label(non_clause). [assumption].
% 0.85/1.33 108 (all A (empty(A) -> A = empty_set)) # label(t6_boole) # label(axiom) # label(non_clause). [assumption].
% 0.85/1.33 120 -(all A all B (element(B,powerset(powerset(A))) -> -(B != empty_set & complements_of_subsets(A,B) = empty_set))) # label(t46_setfam_1) # label(negated_conjecture) # label(non_clause). [assumption].
% 0.85/1.33 185 -element(A,powerset(B)) | subset_complement(B,A) = set_difference(B,A) # label(d5_subset_1) # label(axiom). [clausify(18)].
% 0.85/1.33 193 -element(A,powerset(powerset(B))) | -element(C,powerset(powerset(B))) | complements_of_subsets(B,A) = C | in(f17(B,A,C),C) | in(subset_complement(B,f17(B,A,C)),A) # label(d8_setfam_1) # label(axiom). [clausify(21)].
% 0.85/1.33 210 -element(A,powerset(powerset(B))) | complements_of_subsets(B,complements_of_subsets(B,A)) = A # label(involutiveness_k7_setfam_1) # label(axiom). [clausify(44)].
% 0.85/1.33 236 empty(c1) # label(rc1_xboole_0) # label(axiom). [clausify(59)].
% 0.85/1.33 237 element(f21(A),powerset(A)) # label(rc2_subset_1) # label(axiom). [clausify(60)].
% 0.85/1.33 238 empty(f21(A)) # label(rc2_subset_1) # label(axiom). [clausify(60)].
% 0.85/1.33 276 set_difference(A,empty_set) = A # label(t3_boole) # label(axiom). [clausify(89)].
% 0.85/1.33 297 -element(A,powerset(B)) | -in(C,subset_complement(B,A)) | -in(C,A) # label(t54_subset_1) # label(lemma). [clausify(102)].
% 0.85/1.33 304 -empty(A) | empty_set = A # label(t6_boole) # label(axiom). [clausify(108)].
% 0.85/1.33 322 element(c4,powerset(powerset(c3))) # label(t46_setfam_1) # label(negated_conjecture). [clausify(120)].
% 0.85/1.33 323 empty_set != c4 # label(t46_setfam_1) # label(negated_conjecture). [clausify(120)].
% 0.85/1.33 324 c4 != empty_set. [copy(323),flip(a)].
% 0.85/1.33 325 complements_of_subsets(c3,c4) = empty_set # label(t46_setfam_1) # label(negated_conjecture). [clausify(120)].
% 0.85/1.33 326 empty_set = complements_of_subsets(c3,c4). [copy(325),flip(a)].
% 0.85/1.33 362 -element(A,powerset(powerset(B))) | complements_of_subsets(B,A) = A | in(f17(B,A,A),A) | in(subset_complement(B,f17(B,A,A)),A). [factor(193,a,b)].
% 0.85/1.33 403 complements_of_subsets(c3,c4) != c4. [back_rewrite(324),rewrite([326(2)]),flip(a)].
% 0.85/1.33 404 -empty(A) | complements_of_subsets(c3,c4) = A. [back_rewrite(304),rewrite([326(2)])].
% 0.85/1.33 408 set_difference(A,complements_of_subsets(c3,c4)) = A. [back_rewrite(276),rewrite([326(1)])].
% 0.85/1.33 961 subset_complement(A,f21(A)) = set_difference(A,f21(A)). [resolve(237,a,185,a)].
% 0.85/1.33 1181 -in(A,set_difference(B,f21(B))) | -in(A,f21(B)). [resolve(297,a,237,a),rewrite([961(2)])].
% 0.85/1.33 1299 complements_of_subsets(c3,complements_of_subsets(c3,c4)) = c4. [resolve(322,a,210,a)].
% 0.85/1.33 1983 complements_of_subsets(A,f21(powerset(A))) = f21(powerset(A)) | in(f17(A,f21(powerset(A)),f21(powerset(A))),f21(powerset(A))) | in(subset_complement(A,f17(A,f21(powerset(A)),f21(powerset(A)))),f21(powerset(A))). [resolve(362,a,237,a)].
% 0.85/1.33 2732 f21(A) = complements_of_subsets(c3,c4). [resolve(404,a,238,a),flip(a)].
% 0.85/1.33 2733 complements_of_subsets(c3,c4) = c1. [resolve(404,a,236,a)].
% 0.85/1.33 2734 f21(A) = c1. [back_rewrite(2732),rewrite([2733(4)])].
% 0.85/1.33 2774 complements_of_subsets(c3,c1) = c4. [back_rewrite(1299),rewrite([2733(4)])].
% 0.85/1.33 2782 set_difference(A,c1) = A. [back_rewrite(408),rewrite([2733(3)])].
% 0.85/1.33 2786 c4 != c1. [back_rewrite(403),rewrite([2733(3)]),flip(a)].
% 0.85/1.33 2795 complements_of_subsets(A,c1) = c1 | in(f17(A,c1,c1),c1) | in(subset_complement(A,f17(A,c1,c1)),c1). [back_rewrite(1983),rewrite([2734(2),2734(4),2734(6),2734(7),2734(9),2734(11),2734(12),2734(15)])].
% 0.85/1.33 2800 -in(A,B) | -in(A,c1). [back_rewrite(1181),rewrite([2734(1),2782(2),2734(2)])].
% 0.85/1.33 2825 -in(A,c1). [factor(2800,a,b)].
% 0.85/1.33 2830 complements_of_subsets(A,c1) = c1. [back_unit_del(2795),unit_del(b,2825),unit_del(c,2825)].
% 0.85/1.33 2833 $F. [back_rewrite(2774),rewrite([2830(3)]),flip(a),unit_del(a,2786)].
% 0.85/1.33
% 0.85/1.33 % SZS output end Refutation
% 0.85/1.33 ============================== end of proof ==========================
% 0.85/1.33
% 0.85/1.33 ============================== STATISTICS ============================
% 0.85/1.33
% 0.85/1.33 Given=202. Generated=3818. Kept=2688. proofs=1.
% 0.85/1.33 Usable=185. Sos=2227. Demods=39. Limbo=3, Disabled=473. Hints=0.
% 0.85/1.33 Megabytes=4.21.
% 0.85/1.33 User_CPU=0.23, System_CPU=0.00, Wall_clock=1.
% 0.85/1.33
% 0.85/1.33 ============================== end of statistics =====================
% 0.85/1.33
% 0.85/1.33 ============================== end of search =========================
% 0.85/1.33
% 0.85/1.33 THEOREM PROVED
% 0.85/1.33 % SZS status Theorem
% 0.85/1.33
% 0.85/1.33 Exiting with 1 proof.
% 0.85/1.33
% 0.85/1.33 Process 20523 exit (max_proofs) Mon Jun 20 00:45:45 2022
% 0.85/1.33 Prover9 interrupted
%------------------------------------------------------------------------------