%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SEU225+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n021.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  : 300s
% DateTime : Thu Aug 31 16:30:12 EDT 2023

% Result   : Theorem 0.93s 1.10s
% Output   : Proof 0.93s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14  % Problem    : SEU225+2 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.15  % Command    : do_cvc5 %s %d
% 0.16/0.36  % Computer : n021.cluster.edu
% 0.16/0.36  % Model    : x86_64 x86_64
% 0.16/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36  % Memory   : 8042.1875MB
% 0.16/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36  % CPULimit   : 300
% 0.16/0.36  % WCLimit    : 300
% 0.16/0.36  % DateTime   : Wed Aug 23 17:23:09 EDT 2023
% 0.16/0.36  % CPUTime    : 
% 0.22/0.51  %----Proving TF0_NAR, FOF, or CNF
% 0.93/1.10  ------- convert to smt2 : /export/starexec/sandbox2/tmp/tmp.B0FKEIGWE0/cvc5---1.0.5_20768.p...
% 0.93/1.10  ------- get file name : TPTP file name is SEU225+2
% 0.93/1.10  ------- cvc5-fof : /export/starexec/sandbox2/solver/bin/cvc5---1.0.5_20768.smt2...
% 0.93/1.10  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 0.93/1.10  % SZS status Theorem for SEU225+2
% 0.93/1.10  % SZS output start Proof for SEU225+2
% 0.93/1.10  (
% 0.93/1.10  (let ((_let_1 (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_dom_restriction B A) (tptp.relation_composition (tptp.identity_relation A) B)))))) (let ((_let_2 (not (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in B A) (= (tptp.apply (tptp.relation_dom_restriction C A) B) (tptp.apply C B)))))))) (let ((_let_3 (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_relation A))) (and (= (tptp.relation_dom _let_1) A) (= (tptp.relation_rng _let_1) A)))))) (let ((_let_4 (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in B A) (= (tptp.apply (tptp.identity_relation A) B) B))))) (let ((_let_5 (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in A (tptp.relation_dom B)) (= (tptp.apply (tptp.relation_composition B C) A) (tptp.apply C (tptp.apply B A)))))))))) (let ((_let_6 (tptp.relation tptp.empty_set))) (let ((_let_7 (tptp.empty tptp.empty_set))) (let ((_let_8 (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_relation A))) (and (tptp.relation _let_1) (tptp.function _let_1)))))) (let ((_let_9 (forall ((A $$unsorted)) (tptp.relation (tptp.identity_relation A))))) (let ((_let_10 (tptp.relation SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12))) (let ((_let_11 (tptp.function SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12))) (let ((_let_12 (tptp.identity_relation SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10))) (let ((_let_13 (tptp.function _let_12))) (let ((_let_14 (tptp.apply _let_12 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11))) (let ((_let_15 (tptp.relation_composition _let_12 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12))) (let ((_let_16 (= (tptp.apply _let_15 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11) (tptp.apply SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12 _let_14)))) (let ((_let_17 (not _let_11))) (let ((_let_18 (not _let_10))) (let ((_let_19 (tptp.relation_dom _let_12))) (let ((_let_20 (tptp.in SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11 _let_19))) (let ((_let_21 (not _let_20))) (let ((_let_22 (not _let_13))) (let ((_let_23 (tptp.relation _let_12))) (let ((_let_24 (not _let_23))) (let ((_let_25 (or _let_24 _let_22 _let_21 _let_18 _let_17 _let_16))) (let ((_let_26 (tptp.relation_dom_restriction SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10))) (let ((_let_27 (= (tptp.apply _let_26 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11) (tptp.apply SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11)))) (let ((_let_28 (tptp.in SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10))) (let ((_let_29 (not _let_28))) (let ((_let_30 (or _let_18 _let_17 _let_29 _let_27))) (let ((_let_31 (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (or (not (tptp.relation C)) (not (tptp.function C)) (not (tptp.in B A)) (= (tptp.apply (tptp.relation_dom_restriction C A) B) (tptp.apply C B)))))) (let ((_let_32 (not _let_30))) (let ((_let_33 (EQ_RESOLVE (ASSUME :args (_let_2)) (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_34 (or))) (let ((_let_35 (not _let_31))) (let ((_let_36 (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE _let_33) :args (_let_35))) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_35) _let_31))) (REFL :args (_let_32)) :args _let_34)) _let_33 :args (_let_32 true _let_31)))) (let ((_let_37 (REFL :args (_let_30)))) (let ((_let_38 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_30 0)) (CONG _let_37 (MACRO_SR_PRED_INTRO :args ((= (not _let_18) _let_10))) :args _let_34)) :args ((or _let_10 _let_30))) _let_36 :args (_let_10 true _let_30)))) (let ((_let_39 (forall ((BOUND_VARIABLE_4041 $$unsorted)) (tptp.function (tptp.identity_relation BOUND_VARIABLE_4041))))) (let ((_let_40 (_let_39))) (let ((_let_41 (forall ((A $$unsorted) (B $$unsorted) (BOUND_VARIABLE_4608 $$unsorted)) (or (not (tptp.relation B)) (not (tptp.function B)) (not (tptp.in A (tptp.relation_dom B))) (not (tptp.relation BOUND_VARIABLE_4608)) (not (tptp.function BOUND_VARIABLE_4608)) (= (tptp.apply (tptp.relation_composition B BOUND_VARIABLE_4608) A) (tptp.apply BOUND_VARIABLE_4608 (tptp.apply B A))))))) (let ((_let_42 (EQ_RESOLVE (ASSUME :args (_let_5)) (MACRO_SR_EQ_INTRO :args (_let_5 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_43 (_let_9))) (let ((_let_44 (ASSUME :args _let_43))) (let ((_let_45 (= _let_26 _let_15))) (let ((_let_46 (= SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11 _let_14))) (let ((_let_47 (not _let_16))) (let ((_let_48 (or _let_18 _let_45))) (let ((_let_49 (forall ((A $$unsorted) (B $$unsorted)) (or (not (tptp.relation B)) (= (tptp.relation_dom_restriction B A) (tptp.relation_composition (tptp.identity_relation A) B)))))) (let ((_let_50 (EQ_RESOLVE (ASSUME :args (_let_1)) (MACRO_SR_EQ_INTRO :args (_let_1 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_51 (or _let_29 _let_46))) (let ((_let_52 (forall ((A $$unsorted) (B $$unsorted)) (or (not (tptp.in B A)) (= B (tptp.apply (tptp.identity_relation A) B)))))) (let ((_let_53 (EQ_RESOLVE (ASSUME :args (_let_4)) (MACRO_SR_EQ_INTRO :args (_let_4 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_54 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_30 2)) (CONG _let_37 (MACRO_SR_PRED_INTRO :args ((= (not _let_29) _let_28))) :args _let_34)) :args ((or _let_28 _let_30))) _let_36 :args (_let_28 true _let_30)))) (let ((_let_55 (not _let_27))) (let ((_let_56 (and _let_55 _let_46 _let_45))) (let ((_let_57 (ASSUME :args (_let_55)))) (let ((_let_58 (APPLY_UF tptp.apply))) (let ((_let_59 (ASSUME :args (_let_46)))) (let ((_let_60 (REFL :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11)))) (let ((_let_61 (ASSUME :args (_let_45)))) (let ((_let_62 (= SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10 _let_19))) (let ((_let_63 (forall ((BOUND_VARIABLE_5374 $$unsorted)) (= BOUND_VARIABLE_5374 (tptp.relation_dom (tptp.identity_relation BOUND_VARIABLE_5374)))))) (let ((_let_64 (_let_63))) (let ((_let_65 (and _let_28 _let_62))) (let ((_let_66 (_let_28 _let_62))) (let ((_let_67 (ASSUME :args (_let_28)))) (let ((_let_68 (ASSUME :args (_let_62)))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_25)) :args ((or _let_18 _let_17 _let_24 _let_22 _let_21 _let_16 (not _let_25)))) (MACRO_RESOLUTION_TRUST (REORDERING (RESOLUTION (CNF_AND_NEG :args (_let_65)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_67 _let_68) (SCOPE (TRUE_ELIM (TRANS (CONG _let_60 (SYMM _let_68) :args (APPLY_UF tptp.in)) (TRUE_INTRO _let_67))) :args _let_66)) :args _let_66)) :args (true _let_65)) :args ((or _let_29 _let_20 (not _let_62)))) _let_54 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE (ASSUME :args _let_64) :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.identity_relation BOUND_VARIABLE_5374)))) :args _let_64)) (AND_ELIM (EQ_RESOLVE (ASSUME :args (_let_3)) (MACRO_SR_EQ_INTRO :args (_let_3 SB_DEFAULT SBA_FIXPOINT))) :args (0)) :args (_let_62 false _let_63)) :args (_let_20 false _let_28 false _let_62)) (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (RESOLUTION (CNF_AND_NEG :args (_let_56)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_61 _let_57 _let_59) (SCOPE (FALSE_ELIM (TRANS (CONG (CONG (SYMM _let_61) _let_60 :args _let_58) (CONG (REFL :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12)) (SYMM _let_59) :args _let_58) :args (=)) (FALSE_INTRO _let_57))) :args (_let_45 _let_55 _let_46))) :args (_let_55 _let_46 _let_45))) :args (true _let_56)) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_55) _let_27))) (REFL :args ((not _let_46))) (REFL :args ((not _let_45))) (REFL :args (_let_47)) :args _let_34)) (MACRO_RESOLUTION_TRUST (CNF_OR_NEG :args (_let_30 3)) _let_36 :args (_let_55 true _let_30)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_51)) :args ((or _let_29 _let_46 (not _let_51)))) _let_54 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_53 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((not (= (tptp.in B A) false))))) :args (_let_52))) _let_53 :args (_let_51 false _let_52)) :args (_let_46 false _let_28 false _let_51)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_48)) :args ((or _let_18 _let_45 (not _let_48)))) _let_38 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_50 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.relation_dom_restriction B A)))) :args (_let_49))) _let_50 :args (_let_48 false _let_49)) :args (_let_45 false _let_10 false _let_48)) :args (_let_47 true _let_27 false _let_46 false _let_45)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_44 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.identity_relation A)))) :args _let_43)) _let_44 :args (_let_23 false _let_9)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_42 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_11 _let_12 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_12 QUANTIFIERS_INST_E_MATCHING ((tptp.apply (tptp.relation_composition B BOUND_VARIABLE_4608) A)))) :args (_let_41))) _let_42 :args (_let_25 false _let_41)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE (ASSUME :args _let_40) :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_10 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.identity_relation BOUND_VARIABLE_4041)))) :args _let_40)) (AND_ELIM (EQ_RESOLVE (ASSUME :args (_let_8)) (MACRO_SR_EQ_INTRO :args (_let_8 SB_DEFAULT SBA_FIXPOINT))) :args (1)) :args (_let_13 false _let_39)) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_30 1)) (CONG _let_37 (MACRO_SR_PRED_INTRO :args ((= (not _let_17) _let_11))) :args _let_34)) :args ((or _let_11 _let_30))) _let_36 :args (_let_11 true _let_30)) _let_38 :args (false false _let_20 true _let_16 false _let_23 false _let_25 false _let_13 false _let_11 false _let_10)) :args ((forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (not (tptp.in B A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.proper_subset A B) (not (tptp.proper_subset B A)))) (forall ((A $$unsorted)) (=> (tptp.empty A) (tptp.function A))) (forall ((A $$unsorted)) (=> (tptp.empty A) (tptp.relation A))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.function A))) (let ((_let_2 (tptp.relation A))) (=> (and _let_2 (tptp.empty A) _let_1) (and _let_2 _let_1 (tptp.one_to_one A)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.unordered_pair A B) (tptp.unordered_pair B A))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A B) (tptp.set_union2 B A))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_intersection2 A B) (tptp.set_intersection2 B A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (= B (tptp.identity_relation A)) (forall ((C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair C D) B) (and (tptp.in C A) (= C D))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= A B) (and (tptp.subset A B) (tptp.subset B A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (= C (tptp.relation_dom_restriction A B)) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.ordered_pair D E))) (= (tptp.in _let_1 C) (and (tptp.in D B) (tptp.in _let_1 A)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (tptp.relation C) (= (= C (tptp.relation_rng_restriction A B)) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.ordered_pair D E))) (= (tptp.in _let_1 C) (and (tptp.in E A) (tptp.in _let_1 B)))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.relation_image A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in (tptp.ordered_pair E D) A) (tptp.in E B))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.relation_inverse_image A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in (tptp.ordered_pair D E) A) (tptp.in E B))))))))) (forall ((A $$unsorted)) (= (tptp.relation A) (forall ((B $$unsorted)) (not (and (tptp.in B A) (forall ((C $$unsorted) (D $$unsorted)) (not (= B (tptp.ordered_pair C D))))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (= B (tptp.set_meet A)))) (let ((_let_2 (= A tptp.empty_set))) (and (=> (not _let_2) (= _let_1 (forall ((C $$unsorted)) (= (tptp.in C B) (forall ((D $$unsorted)) (=> (tptp.in D A) (tptp.in C D))))))) (=> _let_2 (= _let_1 (= B tptp.empty_set))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.singleton A)) (forall ((C $$unsorted)) (= (tptp.in C B) (= C A))))) (forall ((A $$unsorted)) (= (= A tptp.empty_set) (forall ((B $$unsorted)) (not (tptp.in B A))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.powerset A)) (forall ((C $$unsorted)) (= (tptp.in C B) (tptp.subset C A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (= A B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair C D))) (= (tptp.in _let_1 A) (tptp.in _let_1 B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.element B A))) (let ((_let_2 (tptp.empty A))) (and (=> (not _let_2) (= _let_1 (tptp.in B A))) (=> _let_2 (= _let_1 (tptp.empty B))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.unordered_pair A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (or (= D A) (= D B)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_union2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (or (tptp.in D A) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.cartesian_product2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted) (F $$unsorted)) (and (tptp.in E A) (tptp.in F B) (= D (tptp.ordered_pair E F)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (tptp.subset A B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair C D))) (=> (tptp.in _let_1 A) (tptp.in _let_1 B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset A B) (forall ((C $$unsorted)) (=> (tptp.in C A) (tptp.in C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_intersection2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D A) (tptp.in D B)))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted) (C $$unsorted)) (let ((_let_1 (= C (tptp.apply A B)))) (let ((_let_2 (tptp.in B (tptp.relation_dom A)))) (and (=> _let_2 (= _let_1 (tptp.in (tptp.ordered_pair B C) A))) (=> (not _let_2) (= _let_1 (= C tptp.empty_set))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (= B (tptp.relation_dom A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (tptp.in (tptp.ordered_pair C D) A)))))))) (forall ((A $$unsorted)) (= (tptp.cast_to_subset A) A)) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.union A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in C D) (tptp.in D A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_difference A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D A) (not (tptp.in D B))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (= B (tptp.relation_rng A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (tptp.in (tptp.ordered_pair D C) A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (= (tptp.subset_complement A B) (tptp.set_difference A B)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.ordered_pair A B) (tptp.unordered_pair (tptp.unordered_pair A B) (tptp.singleton A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_field A) (tptp.set_union2 (tptp.relation_dom A) (tptp.relation_rng A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (= B (tptp.relation_inverse A)) (forall ((C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair C D) B) (tptp.in (tptp.ordered_pair D C) A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.disjoint A B) (= (tptp.set_intersection2 A B) tptp.empty_set))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (= (tptp.one_to_one A) (forall ((B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_dom A))) (=> (and (tptp.in B _let_1) (tptp.in C _let_1) (= (tptp.apply A B) (tptp.apply A C))) (= B C))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (tptp.relation C) (= (= C (tptp.relation_composition A B)) (forall ((D $$unsorted) (E $$unsorted)) (= (tptp.in (tptp.ordered_pair D E) C) (exists ((F $$unsorted)) (and (tptp.in (tptp.ordered_pair D F) A) (tptp.in (tptp.ordered_pair F E) B)))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.powerset A))) (= (= C (tptp.complements_of_subsets A B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset A)) (= (tptp.in D C) (tptp.in (tptp.subset_complement A D) B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.proper_subset A B) (and (tptp.subset A B) (not (= A B))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (= (tptp.function_inverse A) (tptp.relation_inverse A))))) true true true true true true true (forall ((A $$unsorted)) (let ((_let_1 (tptp.function_inverse A))) (=> (and (tptp.relation A) (tptp.function A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) true (forall ((A $$unsorted)) (tptp.element (tptp.cast_to_subset A) (tptp.powerset A))) true true true true (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B _let_1) (tptp.element (tptp.subset_complement A B) _let_1)))) true true (forall ((A $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_inverse A)))) true true (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.relation_composition A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B (tptp.powerset _let_1)) (tptp.element (tptp.union_of_subsets A B) _let_1)))) _let_9 (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B (tptp.powerset _let_1)) (tptp.element (tptp.meet_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.subset_difference A B C) _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_dom_restriction A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.powerset A)))) (=> (tptp.element B _let_1) (tptp.element (tptp.complements_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.relation (tptp.relation_rng_restriction A B)))) true true (forall ((A $$unsorted)) (exists ((B $$unsorted)) (tptp.element B A))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition B A))) (=> (and (tptp.empty A) (tptp.relation B)) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (and _let_7 _let_6 (tptp.relation_empty_yielding tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction A B))) (=> (and (tptp.relation A) (tptp.relation_empty_yielding A)) (and (tptp.relation _let_1) (tptp.relation_empty_yielding _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition A B))) (=> (and (tptp.relation A) (tptp.function A) (tptp.relation B) (tptp.function B)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_intersection2 A B)))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.powerset A)))) _let_7 (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.empty (tptp.ordered_pair A B)))) _let_8 (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_union2 A B)))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.singleton A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.empty A)) (not (tptp.empty (tptp.set_union2 A B))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (and (tptp.relation A) (tptp.function A) (tptp.one_to_one A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.empty (tptp.unordered_pair A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.empty A)) (not (tptp.empty (tptp.set_union2 B A))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction A B))) (=> (and (tptp.relation A) (tptp.function A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (and _let_7 _let_6) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (not (tptp.empty B))) (not (tptp.empty (tptp.cartesian_product2 A B))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation A)) (not (tptp.empty (tptp.relation_dom A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation A)) (not (tptp.empty (tptp.relation_rng A))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_dom A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_rng A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition A B))) (=> (and (tptp.empty A) (tptp.relation B)) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A A) A)) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_intersection2 A A) A)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (= (tptp.subset_complement A (tptp.subset_complement A B)) B))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_inverse (tptp.relation_inverse A)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.complements_of_subsets A (tptp.complements_of_subsets A B)) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.proper_subset A A))) (forall ((A $$unsorted)) (not (= (tptp.singleton A) tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (= (tptp.set_union2 (tptp.singleton A) B) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.disjoint (tptp.singleton A) B) (tptp.in A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.in A B)) (tptp.disjoint (tptp.singleton A) B))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset (tptp.singleton A) B) (tptp.in A B))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A B) tptp.empty_set) (tptp.subset A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in C A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (or (tptp.in C A) (tptp.subset A (tptp.set_difference B (tptp.singleton C)))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.singleton B))) (= (tptp.subset A _let_1) (or (= A tptp.empty_set) (= A _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.subset A (tptp.union B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair A B) (tptp.cartesian_product2 C D)) (and (tptp.in A C) (tptp.in B D)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (forall ((C $$unsorted)) (=> (tptp.in C A) (tptp.in C B))) (tptp.element A (tptp.powerset B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in B (tptp.relation_dom (tptp.relation_dom_restriction C A))) (and (tptp.in B (tptp.relation_dom C)) (tptp.in B A))))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.function A))) (exists ((A $$unsorted)) (and (tptp.empty A) (tptp.relation A))) (forall ((A $$unsorted)) (=> (not (tptp.empty A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset A)) (not (tptp.empty B)))))) (exists ((A $$unsorted)) (tptp.empty A)) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.empty A) (tptp.function A))) (exists ((A $$unsorted)) (and (not (tptp.empty A)) (tptp.relation A))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset A)) (tptp.empty B)))) (exists ((A $$unsorted)) (not (tptp.empty A))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.function A) (tptp.one_to_one A))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.relation_empty_yielding A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.union_of_subsets A B) (tptp.union B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.meet_of_subsets A B) (tptp.set_meet B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_difference A B C) (tptp.set_difference B C))))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset A A)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.disjoint A B) (tptp.disjoint B A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair A B) (tptp.cartesian_product2 C D)) (and (tptp.in A C) (tptp.in B D)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (not (and (= (tptp.unordered_pair A B) (tptp.unordered_pair C D)) (not (= A C)) (not (= A D))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_rng (tptp.relation_rng_restriction B C))) (and (tptp.in A B) (tptp.in A (tptp.relation_rng C)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_rng_restriction A B)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng_restriction A B) B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_rng_restriction A B)) (tptp.relation_rng B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (and (tptp.subset (tptp.cartesian_product2 A C) (tptp.cartesian_product2 B C)) (tptp.subset (tptp.cartesian_product2 C A) (tptp.cartesian_product2 C B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_rng (tptp.relation_rng_restriction A B)) (tptp.set_intersection2 (tptp.relation_rng B) A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset C D)) (tptp.subset (tptp.cartesian_product2 A C) (tptp.cartesian_product2 B D)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= (tptp.set_union2 A B) B))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.subset D C)) (tptp.in D B))) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in (tptp.powerset C) B))) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (tptp.are_equipotent C B)) (not (tptp.in C B)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.relation_dom_restriction (tptp.relation_rng_restriction A C) B) (tptp.relation_rng_restriction A (tptp.relation_dom_restriction C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_image C B)) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_dom C)) (tptp.in (tptp.ordered_pair D A) C) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_image B A) (tptp.relation_rng B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_image B A) (tptp.relation_image B (tptp.set_intersection2 (tptp.relation_dom B) A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_image A (tptp.relation_dom A)) (tptp.relation_rng A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_rng (tptp.relation_composition A B)) (tptp.relation_image B (tptp.relation_rng A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_inverse_image C B)) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_rng C)) (tptp.in (tptp.ordered_pair A D) C) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_inverse_image B A) (tptp.relation_dom B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (not (and (not (= A tptp.empty_set)) (tptp.subset A (tptp.relation_rng B)) (= (tptp.relation_inverse_image B A) tptp.empty_set))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.subset A B) (tptp.subset (tptp.relation_inverse_image C A) (tptp.relation_inverse_image C B))))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset (tptp.set_intersection2 A B) A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset A C)) (tptp.subset A (tptp.set_intersection2 B C)))) (forall ((A $$unsorted)) (= (tptp.set_union2 A tptp.empty_set) A)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.element A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset B C)) (tptp.subset A C))) (= (tptp.powerset tptp.empty_set) (tptp.singleton tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A (tptp.relation_dom C)) (tptp.in B (tptp.relation_rng C)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in A (tptp.relation_dom (tptp.relation_composition C B))) (and (tptp.in A (tptp.relation_dom C)) (tptp.in (tptp.apply C A) (tptp.relation_dom B)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (tptp.subset A (tptp.cartesian_product2 (tptp.relation_dom A) (tptp.relation_rng A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.relation_composition C B))) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in A (tptp.relation_dom _let_1)) (= (tptp.apply _let_1 A) (tptp.apply B (tptp.apply C A))))))))) _let_5 (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset A B) (and (tptp.subset (tptp.relation_dom A) (tptp.relation_dom B)) (tptp.subset (tptp.relation_rng A) (tptp.relation_rng B)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (tptp.subset (tptp.set_intersection2 A C) (tptp.set_intersection2 B C)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= (tptp.set_intersection2 A B) A))) (forall ((A $$unsorted)) (= (tptp.set_intersection2 A tptp.empty_set) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element A B) (or (tptp.empty B) (tptp.in A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (forall ((C $$unsorted)) (= (tptp.in C A) (tptp.in C B))) (= A B))) (forall ((A $$unsorted)) (tptp.subset tptp.empty_set A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_field C))) (=> (tptp.relation C) (=> (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A _let_1) (tptp.in B _let_1)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (tptp.subset (tptp.set_difference A C) (tptp.set_difference B C)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (= (tptp.ordered_pair A B) (tptp.ordered_pair C D)) (and (= A C) (= B D)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (= (= B (tptp.identity_relation A)) (and (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (= (tptp.apply B C) C))))))) _let_4 (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset (tptp.set_difference A B) A)) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (tptp.relation A) (and (= (tptp.relation_rng A) (tptp.relation_dom _let_1)) (= (tptp.relation_dom A) (tptp.relation_rng _let_1)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A B) tptp.empty_set) (tptp.subset A B))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset (tptp.singleton A) B) (tptp.in A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (tptp.subset (tptp.unordered_pair A B) C) (and (tptp.in A C) (tptp.in B C)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A (tptp.set_difference B A)) (tptp.set_union2 A B))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.singleton B))) (= (tptp.subset A _let_1) (or (= A tptp.empty_set) (= A _let_1))))) (forall ((A $$unsorted)) (= (tptp.set_difference A tptp.empty_set) A)) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.element A (tptp.powerset B)) (tptp.subset A B))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.disjoint A B))) (and (not (and (not _let_1) (forall ((C $$unsorted)) (not (and (tptp.in C A) (tptp.in C B)))))) (not (and (exists ((C $$unsorted)) (and (tptp.in C A) (tptp.in C B))) _let_1))))) (forall ((A $$unsorted)) (=> (tptp.subset A tptp.empty_set) (= A tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_difference (tptp.set_union2 A B) B) (tptp.set_difference A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset A)) (= (tptp.disjoint B C) (tptp.subset B (tptp.subset_complement A C))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom (tptp.relation_composition A B)) (tptp.relation_dom A)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_composition A B)) (tptp.relation_rng B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= B (tptp.set_union2 A (tptp.set_difference B A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset (tptp.relation_rng A) (tptp.relation_dom B)) (= (tptp.relation_dom (tptp.relation_composition A B)) (tptp.relation_dom A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (not (and (not (= B tptp.empty_set)) (= (tptp.complements_of_subsets A B) tptp.empty_set))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (= (tptp.set_union2 (tptp.singleton A) B) B))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset (tptp.relation_dom A) (tptp.relation_rng B)) (= (tptp.relation_rng (tptp.relation_composition B A)) (tptp.relation_rng A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.subset_difference A (tptp.cast_to_subset A) (tptp.union_of_subsets A B)) (tptp.meet_of_subsets A (tptp.complements_of_subsets A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.union_of_subsets A (tptp.complements_of_subsets A B)) (tptp.subset_difference A (tptp.cast_to_subset A) (tptp.meet_of_subsets A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_difference A (tptp.set_difference A B)) (tptp.set_intersection2 A B))) (forall ((A $$unsorted)) (= (tptp.set_difference tptp.empty_set A) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.in A B) (tptp.element B (tptp.powerset C))) (tptp.element A C))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.disjoint A B))) (and (not (and (not _let_1) (forall ((C $$unsorted)) (not (tptp.in C (tptp.set_intersection2 A B)))))) (not (and (exists ((C $$unsorted)) (tptp.in C (tptp.set_intersection2 A B))) _let_1))))) (forall ((A $$unsorted)) (=> (not (= A tptp.empty_set)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.element C A) (=> (not (tptp.in C B)) (tptp.in C (tptp.subset_complement A B))))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (forall ((B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (= (= B (tptp.function_inverse A)) (and (= (tptp.relation_dom B) (tptp.relation_rng A)) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (and (tptp.in C (tptp.relation_rng A)) (= D (tptp.apply B C))))) (let ((_let_2 (and (tptp.in D (tptp.relation_dom A)) (= C (tptp.apply A D))))) (and (=> _let_1 _let_2) (=> _let_2 _let_1)))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.powerset A)) (not (and (tptp.in B (tptp.subset_complement A C)) (tptp.in B C))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.function_inverse A))) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (and (= (tptp.relation_rng A) (tptp.relation_dom _let_1)) (= (tptp.relation_dom A) (tptp.relation_rng _let_1))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (=> (forall ((B $$unsorted) (C $$unsorted)) (not (tptp.in (tptp.ordered_pair B C) A))) (= A tptp.empty_set)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.function_inverse B))) (=> (and (tptp.relation B) (tptp.function B)) (=> (and (tptp.one_to_one B) (tptp.in A (tptp.relation_rng B))) (and (= A (tptp.apply B (tptp.apply _let_1 A))) (= A (tptp.apply (tptp.relation_composition _let_1 B) A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (not (and (tptp.in A B) (tptp.element B (tptp.powerset C)) (tptp.empty C)))) (and (= (tptp.relation_dom tptp.empty_set) tptp.empty_set) (= (tptp.relation_rng tptp.empty_set) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.subset A B) (tptp.proper_subset B A)))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (tptp.one_to_one (tptp.function_inverse A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.disjoint B C)) (tptp.disjoint A C))) (forall ((A $$unsorted)) (=> (tptp.relation A) (=> (or (= (tptp.relation_dom A) tptp.empty_set) (= (tptp.relation_rng A) tptp.empty_set)) (= A tptp.empty_set)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (= (tptp.relation_dom A) tptp.empty_set) (= (tptp.relation_rng A) tptp.empty_set)))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A (tptp.singleton B)) A) (not (tptp.in B A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (= B (tptp.relation_dom_restriction C A)) (and (= (tptp.relation_dom B) (tptp.set_intersection2 (tptp.relation_dom C) A)) (forall ((D $$unsorted)) (=> (tptp.in D (tptp.relation_dom B)) (= (tptp.apply B D) (tptp.apply C D)))))))))) (forall ((A $$unsorted)) (= (tptp.unordered_pair A A) (tptp.singleton A))) (forall ((A $$unsorted)) (=> (tptp.empty A) (= A tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset (tptp.singleton A) (tptp.singleton B)) (= A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction C A))) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in B (tptp.relation_dom _let_1)) (= (tptp.apply _let_1 B) (tptp.apply C B)))))) _let_3 _let_2 (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair A B))) (=> (tptp.relation D) (= (tptp.in _let_1 (tptp.relation_composition (tptp.identity_relation C) D)) (and (tptp.in A C) (tptp.in _let_1 D)))))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.in A B) (tptp.empty B)))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset A (tptp.set_union2 A B))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.disjoint A B) (= (tptp.set_difference A B) A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_dom (tptp.relation_dom_restriction C B))) (and (tptp.in A B) (tptp.in A (tptp.relation_dom C)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom_restriction B A) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.empty A) (not (= A B)) (tptp.empty B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A (tptp.relation_dom C)) (= B (tptp.apply C A)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset C B)) (tptp.subset (tptp.set_union2 A C) B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (= (tptp.singleton A) (tptp.unordered_pair B C)) (= A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_dom (tptp.relation_dom_restriction B A)) (tptp.set_intersection2 (tptp.relation_dom B) A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.subset A (tptp.union B)))) _let_1 (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_dom_restriction B A)) (tptp.relation_rng B)))) (forall ((A $$unsorted)) (= (tptp.union (tptp.powerset A)) A)) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.subset D C)) (tptp.in D B))) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (not (and (tptp.in D B) (forall ((E $$unsorted)) (=> (tptp.subset E C) (tptp.in E D))))))))) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (tptp.are_equipotent C B)) (not (tptp.in C B)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (= (tptp.singleton A) (tptp.unordered_pair B C)) (= B C))) true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 0.93/1.11  )
% 0.93/1.11  % SZS output end Proof for SEU225+2
% 0.93/1.11  % cvc5---1.0.5 exiting
% 0.93/1.11  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------