0.00/0.10 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.10 % Command : do_cvc5 %s %d 0.09/0.30 % Computer : n028.cluster.edu 0.09/0.30 % Model : x86_64 x86_64 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.09/0.30 % Memory : 8042.1875MB 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64 0.09/0.30 % CPULimit : 960 0.09/0.30 % WCLimit : 120 0.09/0.30 % DateTime : Tue Aug 9 06:14:20 EDT 2022 0.09/0.30 % CPUTime : 0.15/0.45 %----Proving TF0_NAR, FOF, or CNF 28.07/28.32 ------- cvc5-fof casc J11 : /export/starexec/sandbox2/benchmark/theBenchmark.p at /export/starexec/sandbox2/benchmark/theBenchmark.p... 28.07/28.32 --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10... 28.07/28.32 --- Run --no-e-matching --full-saturate-quant at 5... 28.07/28.32 --- Run --no-e-matching --enum-inst-sum --full-saturate-quant at 5... 28.07/28.32 --- Run --finite-model-find --uf-ss=no-minimal at 5... 28.07/28.32 --- Run --multi-trigger-when-single --full-saturate-quant at 5... 28.07/28.32 % SZS status Theorem for theBenchmark 28.07/28.32 % SZS output start Proof for theBenchmark 28.07/28.32 (let ((_let_1 (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_Ototal__on T_a V_A_2 V_r_2) (forall ((B_x $$unsorted)) (=> (c_member T_a B_x V_A_2) (forall ((B_xa $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member T_a B_xa V_A_2) (=> (not (= B_x B_xa)) (or (c_member _let_1 (c_Product__Type_OPair T_a T_a B_x B_xa) V_r_2) (c_member _let_1 (c_Product__Type_OPair T_a T_a B_xa B_x) V_r_2)))))))))))) (let ((_let_2 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Otrans T_a V_r_2) (= V_r_2 (c_Transitive__Closure_Otrancl T_a V_r_2)))))) (let ((_let_3 (forall ((B_P $$unsorted)) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (hAPP B_P B_x))) (and (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt _let_1) (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt _let_1) (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)) _let_1)))) (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt B_a B_b))) (let ((_let_2 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (=> (not (= B_a B_b)) (=> (c_member _let_2 _let_1 (hAPP B_P v_i)) (c_member _let_2 _let_1 (v_F B_P))))))))))) (let ((_let_4 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_Oirrefl T_a V_r_2) (forall ((B_x $$unsorted)) (not (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_x) V_r_2))))))) (let ((_let_5 (v_F v_P))) (let ((_let_6 (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt _let_5))) (let ((_let_7 (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt _let_5))) (let ((_let_8 (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)))) (let ((_let_9 (and (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt _let_8 _let_5) _let_7 _let_6))) (let ((_let_10 (=> (forall ((B_x $$unsorted)) (let ((_let_1 (hAPP v_P B_x))) (and (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt _let_1) (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)) _let_1) (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt _let_1)))) _let_9))) (let ((_let_11 (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) _let_1) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))))) (let ((_let_12 (forall ((B_x $$unsorted)) (let ((_let_1 (hAPP v_P B_x))) (and (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt _let_1) (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt _let_1) (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)) _let_1)))))) (let ((_let_13 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt v_a v_b))) (let ((_let_14 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (let ((_let_15 (c_member _let_14 _let_13 _let_5))) (let ((_let_16 (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (c_member T_a V_x_2 (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)))))) (let ((_let_17 (hAPP v_P v_i))) (let ((_let_18 (c_member _let_14 _let_13 _let_17))) (let ((_let_19 (not _let_18))) (let ((_let_20 (forall ((B_x $$unsorted) (BOUND_VARIABLE_10230 $$unsorted)) (let ((_let_1 (hAPP v_P v_i))) (let ((_let_2 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (let ((_let_3 (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)))) (or (not (c_member tc_Arrow__Order__Mirabelle_Oalt B_x _let_3)) (not (c_member tc_Arrow__Order__Mirabelle_Oalt BOUND_VARIABLE_10230 _let_3)) (= B_x BOUND_VARIABLE_10230) (c_member _let_2 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt B_x BOUND_VARIABLE_10230) _let_1) (c_member _let_2 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt BOUND_VARIABLE_10230 B_x) _let_1)))))))) (let ((_let_21 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt v_b v_a))) (let ((_let_22 (c_member _let_14 _let_21 _let_17))) (let ((_let_23 (= v_a v_b))) (let ((_let_24 (c_member tc_Arrow__Order__Mirabelle_Oalt v_b _let_8))) (let ((_let_25 (not _let_24))) (let ((_let_26 (c_member tc_Arrow__Order__Mirabelle_Oalt v_a _let_8))) (let ((_let_27 (not _let_26))) (let ((_let_28 (or _let_27 _let_25 _let_23 _let_18 _let_22))) (let ((_let_29 (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt _let_8 _let_17))) (let ((_let_30 (= _let_29 _let_20))) (let ((_let_31 (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_Ototal__on T_a V_A_2 V_r_2) (forall ((B_x $$unsorted) (BOUND_VARIABLE_10230 $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (or (not (c_member T_a B_x V_A_2)) (not (c_member T_a BOUND_VARIABLE_10230 V_A_2)) (= B_x BOUND_VARIABLE_10230) (c_member _let_1 (c_Product__Type_OPair T_a T_a B_x BOUND_VARIABLE_10230) V_r_2) (c_member _let_1 (c_Product__Type_OPair T_a T_a BOUND_VARIABLE_10230 B_x) V_r_2)))))))) (let ((_let_32 (EQ_RESOLVE (ASSUME :args (_let_1)) (MACRO_SR_EQ_INTRO :args (_let_1 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_33 (forall ((BOUND_VARIABLE_9106 $$unsorted)) (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)) (hAPP v_P BOUND_VARIABLE_9106))))) (let ((_let_34 (EQ_RESOLVE (ASSUME :args (_let_12)) (MACRO_SR_EQ_INTRO :args (_let_12 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_35 (AND_ELIM _let_34 :args (2)))) (let ((_let_36 (_let_33))) (let ((_let_37 (not _let_28))) (let ((_let_38 (_let_16))) (let ((_let_39 (ASSUME :args _let_38))) (let ((_let_40 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt v_a v_a))) (let ((_let_41 (c_member _let_14 _let_40 _let_5))) (let ((_let_42 (not _let_23))) (let ((_let_43 (ASSUME :args (_let_15)))) (let ((_let_44 (forall ((B_x $$unsorted)) (not (c_member (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt) (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt B_x B_x) (v_F v_P)))))) (let ((_let_45 (not _let_41))) (let ((_let_46 (= _let_7 _let_44))) (let ((_let_47 (_let_4))) (let ((_let_48 (ASSUME :args _let_47))) (let ((_let_49 (forall ((BOUND_VARIABLE_9507 $$unsorted)) (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt (hAPP v_P BOUND_VARIABLE_9507))))) (let ((_let_50 (forall ((BOUND_VARIABLE_9501 $$unsorted)) (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)) (hAPP v_P BOUND_VARIABLE_9501))))) (let ((_let_51 (forall ((BOUND_VARIABLE_9495 $$unsorted)) (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt (hAPP v_P BOUND_VARIABLE_9495))))) (let ((_let_52 (and _let_51 _let_50 _let_49))) (let ((_let_53 (forall ((BOUND_VARIABLE_9094 $$unsorted)) (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt (hAPP v_P BOUND_VARIABLE_9094))))) (let ((_let_54 (AND_ELIM _let_34 :args (0)))) (let ((_let_55 (forall ((BOUND_VARIABLE_9100 $$unsorted)) (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt (hAPP v_P BOUND_VARIABLE_9100))))) (let ((_let_56 (AND_ELIM _let_34 :args (1)))) (let ((_let_57 (MACRO_RESOLUTION_TRUST (REORDERING (IMPLIES_ELIM (EQ_RESOLVE (ASSUME :args (_let_10)) (MACRO_SR_EQ_INTRO :args (_let_10 SB_DEFAULT SBA_FIXPOINT)))) :args ((or _let_9 (not _let_52)))) (MACRO_RESOLUTION_TRUST (CNF_AND_NEG :args (_let_52)) (MACRO_RESOLUTION_TRUST (REORDERING (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_55 (= BOUND_VARIABLE_9100 BOUND_VARIABLE_9495)))) :args ((or _let_51 (not _let_55)))) _let_56 :args (_let_51 false _let_55)) (MACRO_RESOLUTION_TRUST (REORDERING (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_33 (= BOUND_VARIABLE_9106 BOUND_VARIABLE_9501)))) :args ((or _let_50 (not _let_33)))) _let_35 :args (_let_50 false _let_33)) (MACRO_RESOLUTION_TRUST (REORDERING (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_53 (= BOUND_VARIABLE_9094 BOUND_VARIABLE_9507)))) :args ((or _let_49 (not _let_53)))) _let_54 :args (_let_49 false _let_53)) :args (_let_52 false _let_51 false _let_50 false _let_49)) :args (_let_9 false _let_52)))) (let ((_let_58 (not _let_9))) (let ((_let_59 (_let_44))) (let ((_let_60 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE (ASSUME :args _let_59) :args (v_a QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_59)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_POS1 :args (_let_46)) :args ((or (not _let_7) _let_44 (not _let_46)))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_9 1)) :args ((or _let_7 _let_58))) _let_57 :args (_let_7 false _let_9)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_48 :args (_let_5 tc_Arrow__Order__Mirabelle_Oalt QUANTIFIERS_INST_E_MATCHING_SIMPLE ((c_Relation_Oirrefl T_a V_r_2)))) :args _let_47)) _let_48 :args (_let_46 false _let_4)) :args (_let_44 false _let_7 false _let_46)) :args (_let_45 false _let_44)))) (let ((_let_61 (OR))) (let ((_let_62 (MACRO_SR_PRED_INTRO :args ((= (not _let_45) _let_41))))) (let ((_let_63 (not _let_15))) (let ((_let_64 (APPLY_UF c_member))) (let ((_let_65 (REFL :args (tc_Arrow__Order__Mirabelle_Oalt)))) (let ((_let_66 (REFL :args (_let_14)))) (let ((_let_67 (ASSUME :args (_let_45)))) (let ((_let_68 (FALSE_INTRO _let_67))) (let ((_let_69 (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (NOT_AND (SCOPE (MACRO_SR_PRED_ELIM (TRANS (SYMM _let_68) (CONG _let_66 (CONG _let_65 _let_65 (REFL :args (v_a)) (SYMM (SYMM (ASSUME :args (_let_23)))) :args (APPLY_UF c_Product__Type_OPair)) (REFL :args (_let_5)) :args _let_64) (TRUE_INTRO _let_43))) :args (_let_15 _let_23 _let_45))) (CONG (REFL :args (_let_63)) (REFL :args (_let_42)) _let_62 :args _let_61)) _let_60 _let_43 :args (_let_42 true _let_41 false _let_15)))) (let ((_let_70 (c_member _let_14 _let_21 _let_5))) (let ((_let_71 (not _let_22))) (let ((_let_72 (forall ((BOUND_VARIABLE_10115 $$unsorted)) (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)) (hAPP v_P BOUND_VARIABLE_10115))))) (let ((_let_73 (not _let_72))) (let ((_let_74 (forall ((BOUND_VARIABLE_10109 $$unsorted)) (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt (hAPP v_P BOUND_VARIABLE_10109))))) (let ((_let_75 (not _let_74))) (let ((_let_76 (forall ((BOUND_VARIABLE_10103 $$unsorted)) (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt (hAPP v_P BOUND_VARIABLE_10103))))) (let ((_let_77 (not _let_76))) (let ((_let_78 (or _let_77 _let_75 _let_73 _let_23 _let_71 _let_70))) (let ((_let_79 (forall ((B_P $$unsorted) (BOUND_VARIABLE_10137 $$unsorted) (BOUND_VARIABLE_10135 $$unsorted)) (let ((_let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt BOUND_VARIABLE_10135 BOUND_VARIABLE_10137))) (let ((_let_2 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (or (not (forall ((BOUND_VARIABLE_10103 $$unsorted)) (c_Relation_Otrans tc_Arrow__Order__Mirabelle_Oalt (hAPP B_P BOUND_VARIABLE_10103)))) (not (forall ((BOUND_VARIABLE_10109 $$unsorted)) (c_Relation_Oirrefl tc_Arrow__Order__Mirabelle_Oalt (hAPP B_P BOUND_VARIABLE_10109)))) (not (forall ((BOUND_VARIABLE_10115 $$unsorted)) (c_Relation_Ototal__on tc_Arrow__Order__Mirabelle_Oalt (c_Orderings_Otop__class_Otop (tc_fun tc_Arrow__Order__Mirabelle_Oalt tc_HOL_Obool)) (hAPP B_P BOUND_VARIABLE_10115)))) (= BOUND_VARIABLE_10135 BOUND_VARIABLE_10137) (not (c_member _let_2 _let_1 (hAPP B_P v_i))) (c_member _let_2 _let_1 (v_F B_P)))))))) (let ((_let_80 (EQ_RESOLVE (ASSUME :args (_let_3)) (MACRO_SR_EQ_INTRO :args (_let_3 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_81 (c_Transitive__Closure_Otrancl tc_Arrow__Order__Mirabelle_Oalt _let_5))) (let ((_let_82 (c_member _let_14 _let_21 _let_81))) (let ((_let_83 (= _let_5 _let_81))) (let ((_let_84 (not _let_70))) (let ((_let_85 (c_member _let_14 _let_40 _let_81))) (let ((_let_86 (not _let_82))) (let ((_let_87 (or _let_63 _let_86 _let_85))) (let ((_let_88 (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (or (not (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2)) (not (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) _let_1)) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1))))))) (let ((_let_89 (EQ_RESOLVE (ASSUME :args (_let_11)) (MACRO_SR_EQ_INTRO :args (_let_11 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_90 (not _let_85))) (let ((_let_91 (not _let_6))) (let ((_let_92 (or _let_91 _let_83))) (let ((_let_93 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (or (not (c_Relation_Otrans T_a V_r_2)) (= V_r_2 (c_Transitive__Closure_Otrancl T_a V_r_2)))))) (let ((_let_94 (EQ_RESOLVE (ASSUME :args (_let_2)) (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_95 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_92)) :args ((or _let_91 _let_83 (not _let_92)))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_9 2)) :args ((or _let_6 _let_58))) _let_57 :args (_let_6 false _let_9)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_94 :args (_let_5 tc_Arrow__Order__Mirabelle_Oalt QUANTIFIERS_INST_E_MATCHING_SIMPLE ((not (= (c_Relation_Otrans T_a V_r_2) false))))) :args (_let_93))) _let_94 :args (_let_92 false _let_93)) :args (_let_83 false _let_6 false _let_92)))) (let ((_let_96 (not _let_83))) (let ((_let_97 (REFL :args (_let_96)))) (let ((_let_98 (and _let_83 _let_45))) (let ((_let_99 (ASSUME :args (_let_83)))) (let ((_let_100 (SYMM (SYMM _let_99)))) (let ((_let_101 (and _let_83 _let_86))) (let ((_let_102 (ASSUME :args (_let_86)))) (let ((_let_103 (_let_20))) (SCOPE (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE (ASSUME :args _let_103) :args (v_a v_b QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_103)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_28)) :args ((or _let_18 _let_22 _let_23 _let_25 _let_27 _let_37))) (ASSUME :args (_let_19)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_78)) :args ((or _let_70 _let_23 _let_71 _let_77 _let_75 _let_73 (not _let_78)))) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (RESOLUTION (CNF_AND_NEG :args (_let_101)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_102 _let_99) (SCOPE (FALSE_ELIM (TRANS (CONG _let_66 (REFL :args (_let_21)) _let_100 :args _let_64) (FALSE_INTRO _let_102))) :args (_let_86 _let_83))) :args (_let_83 _let_86))) :args (true _let_101)) (CONG _let_97 (MACRO_SR_PRED_INTRO :args ((= (not _let_86) _let_82))) (REFL :args (_let_84)) :args _let_61)) :args ((or _let_84 _let_96 _let_82))) _let_95 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_87)) :args ((or _let_63 _let_86 _let_85 (not _let_87)))) _let_43 (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (RESOLUTION (CNF_AND_NEG :args (_let_98)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_67 _let_99) (SCOPE (FALSE_ELIM (TRANS (CONG _let_66 (REFL :args (_let_40)) (SYMM _let_100) :args _let_64) _let_68)) :args (_let_45 _let_83))) :args (_let_83 _let_45))) :args (true _let_98)) (CONG _let_97 _let_62 (REFL :args (_let_90)) :args _let_61)) _let_95 _let_60 :args (_let_90 false _let_83 true _let_41)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_89 :args (v_a _let_5 v_b v_a tc_Arrow__Order__Mirabelle_Oalt QUANTIFIERS_INST_CBQI_PROP)) :args (_let_88))) _let_89 :args (_let_87 false _let_88)) :args (_let_86 false _let_15 true _let_85 false _let_87)) :args (_let_84 false _let_83 true _let_82)) _let_69 (MACRO_RESOLUTION_TRUST (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_53 (= BOUND_VARIABLE_9094 BOUND_VARIABLE_10103)))) _let_54 :args (_let_76 false _let_53)) (MACRO_RESOLUTION_TRUST (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_55 (= BOUND_VARIABLE_9100 BOUND_VARIABLE_10109)))) _let_56 :args (_let_74 false _let_55)) (MACRO_RESOLUTION_TRUST (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_33 (= BOUND_VARIABLE_9106 BOUND_VARIABLE_10115)))) _let_35 :args (_let_72 false _let_33)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_80 :args (v_P v_a v_b QUANTIFIERS_INST_E_MATCHING ((not (= (c_member _let_14 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt BOUND_VARIABLE_10135 BOUND_VARIABLE_10137) (hAPP B_P v_i)) false))))) :args (_let_79)))) _let_80 :args (_let_78 false _let_79)) :args (_let_71 true _let_70 true _let_23 false _let_76 false _let_74 false _let_72 false _let_78)) _let_69 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_39 :args (v_b tc_Arrow__Order__Mirabelle_Oalt QUANTIFIERS_INST_CBQI_CONFLICT)) :args _let_38)) _let_39 :args (_let_24 false _let_16)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_39 :args (v_a tc_Arrow__Order__Mirabelle_Oalt QUANTIFIERS_INST_CBQI_PROP)) :args _let_38)) _let_39 :args (_let_26 false _let_16)) :args (_let_37 true _let_18 true _let_22 true _let_23 false _let_24 false _let_26)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_POS1 :args (_let_30)) :args ((or (not _let_29) _let_20 (not _let_30)))) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE (ASSUME :args _let_36) :args (v_i QUANTIFIERS_INST_E_MATCHING_SIMPLE ((hAPP v_P BOUND_VARIABLE_9106)))) :args _let_36)) _let_35 :args (_let_29 false _let_33)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE 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$$unsorted)) (= (= V_r_2 (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_b T_a) tc_HOL_Obool))) (= (c_Relation_ORange T_b T_a V_r_2) (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (=> (c_Relation_Osingle__valued T_a T_b V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_b V_x_2 V_y_2) V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_b V_x_2 V_z_2) V_r_2) (= V_y_2 V_z_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_2)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_2)))))))) (forall ((V_x_2 $$unsorted) (V_L_2 $$unsorted)) (let ((_let_1 (tc_fun (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt) tc_HOL_Obool))) (=> (c_member _let_1 V_L_2 c_Arrow__Order__Mirabelle_OLin) (c_member _let_1 (c_Arrow__Order__Mirabelle_Omktop V_L_2 V_x_2) c_Arrow__Order__Mirabelle_OLin)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member T_a V_x_2 V_A_2) (=> (c_member T_a V_y_2 V_A_2) (= (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (= (c_Equiv__Relations_Oquotient T_a (c_Set_Oinsert T_a V_x_2 _let_1) V_r_2) (c_Equiv__Relations_Oquotient T_a (c_Set_Oinsert T_a V_y_2 _let_1) V_r_2)))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y))) (=> (class_Lattices_Osemilattice__sup T_a) (= _let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_x _let_1))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y B_z) V_r_2) (c_member _let_1 (c_Product__Type_OPair T_a T_a B_x B_z) V_r_2)))))) (c_Relation_Otrans T_a V_r_2))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_R_2))) (= _let_1 (c_Transitive__Closure_Otrancl T_a _let_1)))) (forall ((V_A_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (c_Relation_OImage T_b T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_b T_a) tc_HOL_Obool) V_R_2 V_S_2) V_A_2) (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) (c_Relation_OImage T_b T_a V_R_2 V_A_2) (c_Relation_OImage T_b T_a V_S_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (c_Set_Oinsert T_a V_a_2 V_A_2) (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) V_B_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_ORange T_a T_a (c_Transitive__Closure_Otrancl T_a V_r_2)) (c_Relation_ORange T_a T_a V_r_2))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) (=> (c_member T_a V_x_2 V_A_2) (c_member T_a V_x_2 V_B_2)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Orderings_Oord T_1) (class_Orderings_Oord (tc_fun T_2 T_1)))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (c_member T_a V_a_2 V_A_2) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_a_2) V_r_2)))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (c_Set_Othe__elem T_a (c_Set_Oinsert T_a V_x_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) V_x_2)) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_a_2 V_A_2) (= (c_Set_Oinsert T_a V_a_2 V_A_2) V_A_2))) (forall ((V_P_2 $$unsorted) (V_r_2 $$unsorted) (V_by_2 $$unsorted) (V_bx_2 $$unsorted) (V_ay_2 $$unsorted) (V_ax_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (=> (c_member (tc_prod _let_1 _let_1) (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b V_ax_2 V_ay_2) (c_Product__Type_OPair T_a T_b V_bx_2 V_by_2)) (c_Transitive__Closure_Ortrancl _let_1 V_r_2)) (=> (hBOOL (hAPP (hAPP V_P_2 V_bx_2) V_by_2)) (=> (forall ((B_a $$unsorted) (B_b $$unsorted) (B_aa $$unsorted) (B_ba $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (let ((_let_2 (c_Product__Type_OPair T_a T_b B_aa B_ba))) (let ((_let_3 (tc_prod _let_1 _let_1))) (=> (c_member _let_3 (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b B_a B_b) _let_2) V_r_2) (=> (c_member _let_3 (c_Product__Type_OPair _let_1 _let_1 _let_2 (c_Product__Type_OPair T_a T_b V_bx_2 V_by_2)) (c_Transitive__Closure_Ortrancl _let_1 V_r_2)) (=> (hBOOL (hAPP (hAPP V_P_2 B_aa) B_ba)) (hBOOL (hAPP (hAPP V_P_2 B_a) B_b))))))))) (hBOOL (hAPP (hAPP V_P_2 V_ax_2) V_ay_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (let ((_let_2 (c_Relation_OField T_a V_r_2))) (=> (c_Order__Relation_Opartial__order__on T_a _let_2 V_r_2) (=> (c_member T_a V_a_2 _let_2) (=> (c_member T_a V_b_2 _let_2) (= (= V_a_2 V_b_2) (= (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_1)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_1)))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Otrans T_a V_r_2) (c_Relation_Otrans T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (= _let_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 _let_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_member T_a V_x_2 V_A_2))) (= (and (=> _let_2 (c_Orderings_Oord__class_Oless__eq _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_x_2 (c_Orderings_Obot__class_Obot _let_1))) V_B_2)) (=> (not _let_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2))) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Set_Oinsert T_a V_x_2 V_B_2)))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2) (c_member T_b V_b_2 (c_Relation_ORange T_a T_b V_r_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_C_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (let ((_let_2 (c_Relation_OField T_a V_r_2))) (=> (c_Relation_Orefl__on T_a _let_2 V_r_2) (=> (c_Relation_Oantisym T_a V_r_2) (=> (c_member T_a V_a_2 _let_2) (=> (c_member T_a V_b_2 _let_2) (= (= (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_1)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_1))) (= V_a_2 V_b_2))))))))) (class_Lattices_Obounded__lattice__bot tc_HOL_Obool) (forall ((T_a $$unsorted)) (= (c_Relation_OId T_a) (c_Transitive__Closure_Ortrancl T_a (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Order__Relation_Ostrict__linear__order__on T_a V_A_2 V_r_2) (and (c_Relation_Oirrefl T_a V_r_2) (c_Relation_Ototal__on T_a V_A_2 V_r_2) (c_Relation_Otrans T_a V_r_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (= V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (not (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_x) (c_Transitive__Closure_Otrancl T_a V_r_2)))) (c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b)) (not (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_a) (not (c_Orderings_Oord__class_Oless__eq T_a V_x V_b))))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (c_Wellfounded_Owf T_a (c_Wellfounded_Omeasure T_a V_f_2))) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) V_z_2)) (forall ((T_a $$unsorted)) (c_Relation_Orefl__on T_a (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oorder T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (= V_x V_y))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Set_Oinsert T_a V_a_2 V_A_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot _let_1)) V_A_2)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_x)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_B_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_y)))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_x_2 (c_Orderings_Obot__class_Obot _let_1))) V_B_2) (=> (c_member T_a V_x_2 V_A_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Set_Oinsert T_a V_x_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) V_A_2))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_OField T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_s_2)) (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) (c_Relation_OField T_a V_r_2) (c_Relation_OField T_a V_s_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 V_A_2) (c_member T_a V_c_2 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) V_A_2)))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_a V_c) (=> (c_Orderings_Oord__class_Oless__eq T_a V_b V_d) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_c V_d)))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_B_2 V_A_2))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Orderings_Oord__class_Oless__eq _let_1 V_C_2 V_A_2) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2)) (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) V_C_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_2)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_2))) (=> (c_member T_a V_b_2 V_A_2) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (c_member T_a V_x_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Groups_Ominus__class_Ominus _let_1 (c_Relation_ORange T_b T_a V_A_2) (c_Relation_ORange T_b T_a V_B_2)) (c_Relation_ORange T_b T_a (c_Groups_Ominus__class_Ominus (tc_fun (tc_prod T_b T_a) tc_HOL_Obool) V_A_2 V_B_2))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_r_2) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Olinorder T_a) (=> (not (c_Orderings_Oord__class_Oless__eq T_a V_x V_y)) (c_Orderings_Oord__class_Oless__eq T_a V_y V_x)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_A_2) (c_Orderings_Obot__class_Obot _let_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Lattices_Olattice T_1) (class_Lattices_Osemilattice__inf (tc_fun T_2 T_1)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_C_2)) (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Orderings_Obot__class_Obot _let_1) V_A_2))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_r_2 V_s_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Transitive__Closure_Ortrancl T_a V_r_2) (c_Transitive__Closure_Ortrancl T_a V_s_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)))) (class_Lattices_Osemilattice__sup tc_HOL_Obool) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (not (c_member T_a V_x_2 V_A_2)) (= (c_Groups_Ominus__class_Ominus _let_1 (c_Set_Oinsert T_a V_x_2 V_A_2) (c_Set_Oinsert T_a V_x_2 (c_Orderings_Obot__class_Obot _let_1))) V_A_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Set_Oinsert T_a V_a_2 (c_Set_Oinsert T_a V_b_2 (c_Orderings_Obot__class_Obot _let_1))) (c_Relation_OField T_a V_r_2)) (c_Relation_OField T_a (c_Set_Oinsert (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2))))) (forall ((V_x $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) V_x) (not (=> (c_Orderings_Oord__class_Oless__eq T_a V_a V_x) (not (c_Orderings_Oord__class_Oless__eq T_a V_b V_x))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b))) (=> (class_Lattices_Osemilattice__sup T_a) (= _let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_a _let_1))))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (c_member T_a V_c_2 V_B_2))) (let ((_let_2 (c_member T_a V_c_2 V_A_2))) (=> (= V_B_2 V_A_2) (=> (=> _let_2 (not _let_1)) (not (=> (not _let_2) _let_1))))))) (forall ((T_a $$unsorted)) (c_Wellfounded_Owf T_a (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Equiv__Relations_Oquotient T_a (c_Orderings_Obot__class_Obot _let_1) V_r_2) (c_Orderings_Obot__class_Obot (tc_fun _let_1 tc_HOL_Obool))))) (forall ((T_a $$unsorted)) (c_Relation_Osingle__valued T_a T_a (c_Relation_OId T_a))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_member T_a V_a_2 V_A_2) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 V_B_2)) (c_Set_Oinsert T_a V_a_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_C_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (= (= _let_2 V_A_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_2))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Transitive__Closure_Ortrancl T_a V_R_2) (c_Transitive__Closure_Ortrancl T_a V_S_2)) (c_Transitive__Closure_Ortrancl T_a (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_R_2 V_S_2))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (=> (c_Orderings_Oord__class_Oless__eq T_a V_z V_x) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z) V_x))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (c_Set_Oinsert T_a V_x_2 _let_2))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_3) (or (= _let_3 V_A_2) (= V_A_2 _let_2))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Opreorder T_a) (=> (= V_y V_x) (c_Orderings_Oord__class_Oless__eq T_a V_x V_y)))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2) V_A_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_A_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_C_2 V_A_2))))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Relation_Oinv__image T_b T_a V_r_2 V_f_2)) (c_member (tc_prod T_b T_b) (c_Product__Type_OPair T_b T_b (hAPP V_f_2 V_x_2) (hAPP V_f_2 V_y_2)) V_r_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) (c_Relation_ODomain T_a T_b V_A_2) (c_Relation_ODomain T_a T_b V_B_2)) (c_Relation_ODomain T_a T_b (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_b) tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_b V_a) (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (V_b_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (= (or (and (= V_a_2 V_a_H_2) (c_member (tc_prod T_b T_b) (c_Product__Type_OPair T_b T_b V_b_2 V_b_H_2) V_s_2)) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_a_H_2) V_r_2)) (c_member (tc_prod _let_1 _let_1) (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b V_a_2 V_b_2) (c_Product__Type_OPair T_a T_b V_a_H_2 V_b_H_2)) (c_Wellfounded_Olex__prod T_a T_b V_r_2 V_s_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (=> (= V_a_2 V_b_2) (=> (c_member T_a V_a_2 V_A_2) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (c_Relation_OId__on T_a V_A_2))))) (forall ((V_z_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_y_2 V_z_2) _let_1) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_x_2 V_z_2) _let_1)))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_Relation_Otrans T_a V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_r_2) (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) V_r_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Transitive__Closure_Otrancl T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a))) (c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (and (c_Wellfounded_Owf T_a (c_Groups_Ominus__class_Ominus (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a))) (c_Order__Relation_Olinear__order__on T_a V_A_2 V_r_2)) (c_Order__Relation_Owell__order__on T_a V_A_2 V_r_2))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_C_2))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (c_Orderings_Oord__class_Oless__eq T_a V_y (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y)))) (class_Orderings_Opreorder tc_HOL_Obool) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) V_y))) (forall ((V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (c_member T_a V_x_2 (c_Wellfounded_Oacc T_a V_R_2))) (not (forall ((B_z $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_z V_x_2) V_R_2) (c_member T_a B_z (c_Wellfounded_Oacc T_a V_R_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Groups_Ominus__class_Ominus _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_C_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) V_C_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_x) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_A_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_A_2) V_A_2)))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oorder T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2) (= (= V_x_2 V_y_2) (c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 _let_1 (c_Transitive__Closure_Otrancl T_a V_r_2)) (c_member _let_2 _let_1 (c_Transitive__Closure_Ortrancl T_a V_r_2)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (c_member (tc_prod T_b T_a) (c_Product__Type_OPair T_b T_a V_a_2 V_b_2) V_r_2) (c_member T_a V_b_2 (c_Relation_OImage T_b T_a V_r_2 (c_Set_Oinsert T_b V_a_2 (c_Orderings_Obot__class_Obot (tc_fun T_b tc_HOL_Obool))))))) (forall ((V_b_H $$unsorted) (V_a_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (= (c_Product__Type_OPair T_a T_b V_a V_b) (c_Product__Type_OPair T_a T_b V_a_H V_b_H)) (not (=> (= V_a V_a_H) (not (= V_b V_b_H)))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2) V_A_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_A_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_C_2 V_A_2))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))) (= (c_Transitive__Closure_Ortrancl T_a (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_R_2 V_S_2)) (c_Transitive__Closure_Ortrancl T_a (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Transitive__Closure_Ortrancl T_a V_R_2) (c_Transitive__Closure_Ortrancl T_a V_S_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))) (= _let_2 (c_Groups_Ominus__class_Ominus _let_1 _let_2 V_B_2))))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_Wellfounded_Owf T_a V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_x_2) V_r_2) (not (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 V_a_2) V_r_2)))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_a V_x_2 V_y_2))) (let ((_let_2 (tc_prod T_a T_a))) (= (or (= V_y_2 V_x_2) (and (not (= V_x_2 V_y_2)) (c_member _let_2 _let_1 (c_Transitive__Closure_Otrancl T_a V_R_2)))) (c_member _let_2 _let_1 (c_Transitive__Closure_Ortrancl T_a V_R_2)))))) (forall ((V_r_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_a V_x_2 V_z_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 _let_1 (c_Transitive__Closure_Otrancl T_a V_r_2)) (=> (not (c_member _let_2 _let_1 V_r_2)) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 B_y) V_r_2) (not (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y V_z_2) (c_Transitive__Closure_Otrancl T_a V_r_2)))))))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_b V_c)) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_b (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_c))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Relation_ORange T_b T_a (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun (tc_prod T_b T_a) tc_HOL_Obool) V_A_2 V_B_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Relation_ORange T_b T_a V_A_2) (c_Relation_ORange T_b T_a V_B_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (not (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_x) (c_Transitive__Closure_Otrancl T_a V_r_2)))) (c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Lattices_Obounded__lattice T_1) (class_Lattices_Obounded__lattice__bot (tc_fun T_2 T_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_C_2 V_A_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_C_2 V_B_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_C_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2)))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (c_Transitive__Closure_Ortrancl T_a V_R_2) (c_Transitive__Closure_Ortrancl T_a (c_Transitive__Closure_Otrancl T_a V_R_2)))) (forall ((V_s_2 $$unsorted) (V_B_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (c_Relation_Orefl__on T_a V_B_2 V_s_2) (c_Relation_Orefl__on T_a (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_s_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Obot T_a) (c_Orderings_Oord__class_Oless__eq T_a (c_Orderings_Obot__class_Obot T_a) V_x))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (=> (= (c_Set_Oinsert T_a V_b_2 _let_1) (c_Set_Oinsert T_a V_a_2 _let_1)) (= V_b_2 V_a_2)))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Transitive__Closure_Otrancl T_a V_R_2)) (exists ((B_z $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (and (c_member _let_1 (c_Product__Type_OPair T_a T_a B_z V_y_2) (c_Transitive__Closure_Ortrancl T_a V_R_2)) (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 B_z) V_R_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_OImage T_a T_a (c_Relation_OId T_a) V_A_2) V_A_2)) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Obounded__lattice__top T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Orderings_Otop__class_Otop T_a) V_x) V_x))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Orderings_Opreorder T_1) (class_Orderings_Opreorder (tc_fun T_2 T_1)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_L_2 $$unsorted)) (let ((_let_1 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (=> (c_member (tc_fun _let_1 tc_HOL_Obool) V_L_2 c_Arrow__Order__Mirabelle_OLin) (=> (not (= V_y_2 V_x_2)) (= (not (c_member _let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_y_2) V_L_2)) (c_member _let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_y_2 V_x_2) V_L_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (hAPP (hAPP V_f1_2 V_a_2) V_b_2) (c_Product__Type_Oprod_Oprod__rec T_b T_c T_a V_f1_2 (c_Product__Type_OPair T_b T_c V_a_2 V_b_2)))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_a V_x) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b) V_x)))) (forall ((T_a $$unsorted)) (c_Relation_Oantisym T_a (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((V_R2_2 $$unsorted) (V_R1_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_R1_2 V_R2_2) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) (c_Wellfounded_Oacc T_a V_R2_2) (c_Wellfounded_Oacc T_a V_R1_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y)))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (c_member T_a V_a_2 (c_Set_Oinsert T_a V_b_2 V_A_2)) (or (c_member T_a V_a_2 V_A_2) (= V_a_2 V_b_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (not (= V_a_2 V_b_2)) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 B_y) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (not (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y V_b_2) V_r_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Relation_ODomain T_a T_b (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun (tc_prod T_a T_b) tc_HOL_Obool) V_A_2 V_B_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Relation_ODomain T_a T_b V_A_2) (c_Relation_ODomain T_a T_b V_B_2))))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_D_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_C_2 V_D_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_A_2 V_A_2)) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Orderings_Oord__class_Oless__eq _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) V_C_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))) (let ((_let_3 (c_Groups_Ominus__class_Ominus _let_1 (c_Set_Oinsert T_a V_x_2 V_A_2) V_B_2))) (let ((_let_4 (c_member T_a V_x_2 V_B_2))) (and (=> (not _let_4) (= (c_Set_Oinsert T_a V_x_2 _let_2) _let_3)) (=> _let_4 (= _let_3 _let_2)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2)) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_C_2 V_A_2)) (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_C_2 V_A_2))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))) (=> (= V_A_2 (c_Orderings_Obot__class_Obot _let_1)) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (not (c_member T_a V_a_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member T_a V_a_2 V_A_2) (c_member T_a V_a_2 (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Relation_OImage T_b T_a V_R_2 (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_b tc_HOL_Obool) V_A_2 V_B_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Relation_OImage T_b T_a V_R_2 V_A_2) (c_Relation_OImage T_b T_a V_R_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (hBOOL (hAPP V_B_2 V_x_2)) (hBOOL (hAPP (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) (c_Relation_ODomain T_a T_b V_r_2)) (= (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_b) tc_HOL_Obool)) V_r_2))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) V_x))) (forall ((V_x_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_P_2 V_Q_2) (=> (hBOOL (hAPP V_P_2 V_x_2)) (hBOOL (hAPP V_Q_2 V_x_2))))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_D_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_C_2 V_D_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_member T_a V_a_2 V_C_2) (= (c_Set_Oinsert T_a V_a_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Set_Oinsert T_a V_a_2 V_B_2) V_C_2))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z)) (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_z))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Set_Oinsert T_a V_a_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 V_B_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) V_x))) _let_19 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (= (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (and (= (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_x_2 _let_1)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_y_2 _let_1))) (c_member T_a V_y_2 V_A_2) (c_member T_a V_x_2 V_A_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (= (c_Groups_Ominus__class_Ominus _let_1 V_B_2 (c_Groups_Ominus__class_Ominus _let_1 V_C_2 V_A_2)) V_A_2))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Orderings_Oord__class_Oless__eq T_a V_x_2 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y_2 V_z_2)) (and (c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_z_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (and (c_member T_b B_x V_A_2) (c_member (tc_prod T_b T_a) (c_Product__Type_OPair T_b T_a B_x V_b_2) V_r_2))) (c_member T_a V_b_2 (c_Relation_OImage T_b T_a V_r_2 V_A_2)))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (forall ((B_y $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_y V_x_2) V_r_2) (= (hAPP V_g_2 B_y) (hAPP V_f_2 B_y)))) (= (c_Recdef_Ocut T_a T_b V_f_2 V_r_2 V_x_2) (c_Recdef_Ocut T_a T_b V_g_2 V_r_2 V_x_2)))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) _let_1) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_R_2))) (let ((_let_2 (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_2 V_R_2 V_S_2) (=> (c_Orderings_Oord__class_Oless__eq _let_2 V_S_2 _let_1) (= (c_Transitive__Closure_Ortrancl T_a V_S_2) _let_1)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_L_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_y_2))) (let ((_let_2 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (let ((_let_3 (= V_y_2 V_a_2))) (=> (not (= V_b_2 V_a_2)) (=> (c_member (tc_fun _let_2 tc_HOL_Obool) V_L_2 c_Arrow__Order__Mirabelle_OLin) (= (c_member _let_2 _let_1 (c_Arrow__Order__Mirabelle_Obelow V_L_2 V_a_2 V_b_2)) (and (=> _let_3 (c_member _let_2 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_b_2) V_L_2)) (=> (not _let_3) (and (=> (= V_x_2 V_a_2) (or (c_member _let_2 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_b_2 V_y_2) V_L_2) (= V_b_2 V_y_2))) (=> (not (= V_a_2 V_x_2)) (c_member _let_2 _let_1 V_L_2)))) (not (= V_y_2 V_x_2)))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_b V_a)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_z)) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_x) V_x))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (c_Relation_OField T_a V_r_2))) (=> (c_Order__Relation_Opreorder__on T_a _let_3 V_r_2) (=> (c_member T_a V_a_2 _let_3) (=> (c_member T_a V_b_2 _let_3) (= (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_b_2 V_a_2) V_r_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_2)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_2))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_b_2 $$unsorted) (V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Wellfounded_Oacc T_a V_r_2))) (=> (c_member T_a V_a_2 _let_1) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_b_2 V_a_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (c_member T_a V_b_2 _let_1))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y)) V_x))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Lattices_Odistrib__lattice T_1) (class_Lattices_Odistrib__lattice (tc_fun T_2 T_1)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (= V_y_2 (c_Lattices_Osemilattice__sup__class_Osup T_a V_x_2 V_y_2))))) (forall ((V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Wellfounded_Owf T_a V_R_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Relation_OImage T_a T_a V_R_2 V_A_2)) (= (c_Orderings_Obot__class_Obot _let_1) V_A_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_Oantisym T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a))) (c_Relation_Oantisym T_a V_r_2))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oorder T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (=> (c_Orderings_Oord__class_Oless__eq T_a V_z V_y) (c_Orderings_Oord__class_Oless__eq T_a V_z V_x))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (c_Wellfounded_Owf T_a (c_Transitive__Closure_Otrancl T_a V_r_2)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_z)) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_Relation_Osingle__valued T_a T_a V_r_2) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) _let_1) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_x_2 V_z_2) _let_1) (or (c_member _let_2 (c_Product__Type_OPair T_a T_a V_z_2 V_y_2) _let_1) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_y_2 V_z_2) _let_1)))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_a_2) V_r_2) (= (hAPP (c_Recdef_Ocut T_a T_b V_f_2 V_r_2 V_a_2) V_x_2) (hAPP V_f_2 V_x_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 _let_1 (c_Transitive__Closure_Ortrancl T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun _let_2 tc_HOL_Obool) V_P_2 V_Q_2))) (=> (forall ((B_x $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 B_x) (c_Transitive__Closure_Ortrancl T_a V_P_2)) (forall ((B_y $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_y) V_Q_2) (= B_y B_x))))) (c_member _let_2 _let_1 (c_Transitive__Closure_Ortrancl T_a V_P_2))))))) (forall ((V_x_2 $$unsorted) (V_Fa_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (c_Nitpick_Owf__wfrec_H T_b T_a V_R_2 V_Fa_2 V_x_2) (hAPP (hAPP V_Fa_2 (c_Recdef_Ocut T_b T_a (c_Nitpick_Owf__wfrec T_b T_a V_R_2 V_Fa_2) V_R_2 V_x_2)) V_x_2))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_y $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_y V_x_2) V_r_2) (c_member T_a B_y (c_Wellfounded_Oacc T_a V_r_2)))) (c_member T_a V_x_2 (c_Wellfounded_Oacc T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x V_a_2) V_r_2) (c_member T_a B_x (c_Wellfounded_Oacc T_a V_r_2)))) (c_member T_a V_a_2 (c_Wellfounded_Oacc T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) (c_Relation_ODomain T_a T_a V_r_2) (c_Relation_ORange T_a T_a V_r_2)) (c_Relation_OField T_a V_r_2))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) (c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (= (c_Wellfounded_Owf T_a (c_Set_Oinsert _let_1 (c_Product__Type_OPair T_a T_a V_y_2 V_x_2) V_r_2)) (and (not (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Transitive__Closure_Ortrancl T_a V_r_2))) (c_Wellfounded_Owf T_a V_r_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y))) (=> (class_Lattices_Olattice T_a) (= _let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_x _let_1))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (c_Relation_OImage T_b T_a V_R_2 (c_Orderings_Obot__class_Obot (tc_fun T_b tc_HOL_Obool))) (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Transitive__Closure_Otrancl T_a V_R_2)) (exists ((B_z $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (and (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 B_z) (c_Transitive__Closure_Ortrancl T_a V_R_2)) (c_member _let_1 (c_Product__Type_OPair T_a T_a B_z V_y_2) V_R_2)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_a) V_a))) (forall ((V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 V_p_2 V_r_2) (c_member _let_1 V_p_2 (c_Transitive__Closure_Otrancl T_a V_r_2))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Set_Oinsert T_a V_b_2 V_B_2))))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Lattices_Obounded__lattice T_1) (class_Lattices_Obounded__lattice (tc_fun T_2 T_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_B_2)))) (forall ((V_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) (=> (c_member T_a V_c_2 V_A_2) (c_member T_a V_c_2 V_B_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (= V_x_2 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x_2 V_y_2))))) (class_Orderings_Otop tc_HOL_Obool) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_A_2)) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y))) (=> (class_Lattices_Osemilattice__inf T_a) (= _let_1 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x _let_1))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (c_Relation_Osingle__valued T_a T_b V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (=> (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b B_x B_z) V_r_2) (= B_z B_y))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_ODomain T_a T_a V_r_2) (c_Relation_ODomain T_a T_a (c_Transitive__Closure_Otrancl T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y B_x) V_r_2) (= B_y B_x))))) (c_Relation_Oantisym T_a V_r_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (c_member T_a V_c_2 V_B_2))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= V_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_a)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (class_Orderings_Oord T_b) (= (forall ((B_x $$unsorted)) (c_Orderings_Oord__class_Oless__eq T_b (hAPP V_f_2 B_x) (hAPP V_g_2 B_x))) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a T_b) V_f_2 V_g_2)))) (forall ((T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (= (c_Transitive__Closure_Otrancl T_a _let_1) _let_1))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_Relation_Oantisym T_a V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_b_2 V_a_2) V_r_2) (= V_b_2 V_a_2)))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) _let_1) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2) (= V_A_2 V_B_2))))) (forall ((V_p_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) V_p_2 (c_Relation_OId T_a)) (not (forall ((B_x $$unsorted)) (not (= V_p_2 (c_Product__Type_OPair T_a T_a B_x B_x))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Order__Relation_Olinear__order__on T_a V_A_2 V_r_2) (c_Order__Relation_Ostrict__linear__order__on T_a V_A_2 (c_Groups_Ominus__class_Ominus (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a))))) (forall ((V_y_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_member T_a V_x_2 V_A_2) (=> (c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_x_2 (c_Orderings_Obot__class_Obot _let_1))) V_y_2) (c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 V_A_2 (hAPP (hAPP V_f_2 V_x_2) V_y_2)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z)) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_z))))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (and (= V_x_2 V_y_2) (c_member T_a V_x_2 V_A_2)) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Relation_OId__on T_a V_A_2)))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (= V_A_2 V_B_2) (not (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (not (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_A_2)) (c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_C_2))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_y (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_z)) (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z))))) (forall ((T_a $$unsorted)) (c_Relation_Orefl__on T_a (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)) (c_Relation_OId T_a))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (c_Wellfounded_Owf T_a (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_H_2 V_r_2)))) (forall ((V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (and (c_member T_a B_x (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (hBOOL (hAPP V_P_2 B_x)))) (or (exists ((B_x $$unsorted)) (and (c_member T_a B_x V_A_2) (hBOOL (hAPP V_P_2 B_x)))) (exists ((B_x $$unsorted)) (and (hBOOL (hAPP V_P_2 B_x)) (c_member T_a B_x V_B_2)))))) (forall ((T_a $$unsorted)) (= (c_Relation_OId__on T_a (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) V_y)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (c_member T_a V_a_2 (c_Relation_ORange T_b T_a V_r_2)) (exists ((B_y $$unsorted)) (c_member (tc_prod T_b T_a) (c_Product__Type_OPair T_b T_a B_y V_a_2) V_r_2)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Odistrib__lattice T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_z)) (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) V_A_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= V_A_2 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_A_2))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (= V_B_2 V_A_2) (and (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (c_Relation_ODomain T_a T_b (c_Set_Oinsert (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2)) (c_Set_Oinsert T_a V_a_2 (c_Relation_ODomain T_a T_b V_r_2)))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (c_Set_Oinsert T_a V_x_2 (c_Set_Oinsert T_a V_y_2 V_A_2)) (c_Set_Oinsert T_a V_y_2 (c_Set_Oinsert T_a V_x_2 V_A_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (hBOOL (hAPP (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2)) (=> (not (hBOOL (hAPP V_A_2 V_x_2))) (hBOOL (hAPP V_B_2 V_x_2))))) (forall ((V_z_2 $$unsorted) (V_L_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (= V_y_2 V_z_2))) (let ((_let_2 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_y_2))) (let ((_let_3 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (= (c_member _let_3 _let_2 (c_Arrow__Order__Mirabelle_Omktop V_L_2 V_z_2)) (and (not (= V_x_2 V_z_2)) (=> (not _let_1) (c_member _let_3 _let_2 V_L_2)) (=> _let_1 (not (= V_x_2 V_y_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Otop__class_Otop T_a))) (=> (class_Lattices_Obounded__lattice__top T_a) (= (= _let_1 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x_2 V_y_2)) (and (= V_x_2 _let_1) (= _let_1 V_y_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Transitive__Closure_Ortrancl T_a (c_Groups_Ominus__class_Ominus (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a))) (c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Otrans T_a V_r_2) (=> (c_Relation_Oantisym T_a V_r_2) (c_Relation_Otrans T_a (c_Groups_Ominus__class_Ominus (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Wellfounded_Oacc T_a V_r_2))) (=> (c_member T_a V_b_2 _let_1) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (c_member T_a V_a_2 _let_1))))) (class_Lattices_Odistrib__lattice tc_HOL_Obool) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) V_z) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Wellfounded_Owf T_a V_r_2) (forall ((B_x $$unsorted)) (c_member T_a B_x (c_Wellfounded_Oacc T_a V_r_2))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_r_2) (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) (c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (class_Orderings_Oord T_b) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a T_b) V_f_2 V_g_2) (c_Orderings_Oord__class_Oless__eq T_b (hAPP V_f_2 V_x_2) (hAPP V_g_2 V_x_2))))) (forall ((V_a_2 $$unsorted) (V_H_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_f_2 $$unsorted)) (=> (= (c_Recdef_Owfrec T_a T_b V_r_2 V_H_2) V_f_2) (=> (c_Wellfounded_Owf T_a V_r_2) (= (hAPP V_f_2 V_a_2) (hAPP (hAPP V_H_2 (c_Recdef_Ocut T_a T_b V_f_2 V_r_2 V_a_2)) V_a_2))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (hBOOL (hAPP V_A_2 V_x_2)) (=> (hBOOL (hAPP V_B_2 V_x_2)) (hBOOL (hAPP (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) V_c) (c_Lattices_Osemilattice__sup__class_Osup T_a V_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_b V_c))))) (forall ((V_B_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_D_2 V_B_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) (c_Groups_Ominus__class_Ominus _let_1 V_C_2 V_D_2)))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Otop__class_Otop _let_1))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 _let_2 V_B_2) _let_2)))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_B_2 (c_Set_Oinsert T_a V_a_2 V_B_2))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_C_2)) (c_Groups_Ominus__class_Ominus _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) V_C_2)))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_a_2 (c_Set_Oinsert T_a V_b_2 V_A_2)) (=> (not (= V_b_2 V_a_2)) (c_member T_a V_a_2 V_A_2)))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (or (c_member _let_1 V_x_2 (c_Transitive__Closure_Ortrancl T_a V_R_2)) (c_member _let_1 V_x_2 (c_Transitive__Closure_Ortrancl T_a V_S_2))) (c_member _let_1 V_x_2 (c_Transitive__Closure_Ortrancl T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun _let_1 tc_HOL_Obool) V_R_2 V_S_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (= (and (not (c_member T_a V_c_2 V_B_2)) (c_member T_a V_c_2 V_A_2)) (c_member T_a V_c_2 (c_Groups_Ominus__class_Ominus (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (= V_B_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_A_2)))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (c_Transitive__Closure_Ortrancl T_a V_R_2) (c_Transitive__Closure_Ortrancl T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_R_2 (c_Relation_OId T_a))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Orefl__on T_a V_A_2 (c_Relation_OId__on T_a V_A_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Orderings_Otop__class_Otop _let_1)))) (class_Lattices_Olattice tc_HOL_Obool) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Set_Oinsert T_a V_a_2 V_B_2) V_C_2) (c_Set_Oinsert T_a V_a_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2))))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (c_Wellfounded_Owf T_a (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_r_H_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot T_a))) (=> (class_Lattices_Obounded__lattice__bot T_a) (= (= _let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_x_2 V_y_2)) (and (= _let_1 V_y_2) (= V_x_2 _let_1)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_b_2 (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (= V_b_2 V_a_2))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Set_Oinsert T_a V_x_2 V_A_2))) (= _let_1 (c_Set_Oinsert T_a V_x_2 _let_1)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (= (hBOOL (hAPP (c_Set_Oinsert T_a V_y_2 V_A_2) V_x_2)) (or (hBOOL (hAPP V_A_2 V_x_2)) (= V_x_2 V_y_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (= (= _let_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)) (and (= _let_2 V_A_2) (= _let_2 V_B_2)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z)) (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_z)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_Otrans T_a V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y B_z) V_r_2) (c_member _let_1 (c_Product__Type_OPair T_a T_a B_x B_z) V_r_2)))))))) (forall ((V_Q_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (hBOOL (hAPP V_P_2 V_x_2)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_P_2 V_Q_2) (hBOOL (hAPP V_Q_2 V_x_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_A_2) V_A_2)) (forall ((V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Equiv__Relations_Oquotient T_a V_A_2 V_r_2))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member _let_1 V_X_2 _let_2) (=> (c_member _let_1 V_Y_2 _let_2) (or (= V_Y_2 V_X_2) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_X_2 V_Y_2) (c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (class_Groups_Oordered__ab__group__add T_a) (=> (= (c_Groups_Ominus__class_Ominus T_a V_a_2 V_b_2) (c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (c_Orderings_Oord__class_Oless__eq T_a V_c_2 V_d_2) (c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 _let_1 (c_Transitive__Closure_Ortrancl T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun _let_2 tc_HOL_Obool) V_P_2 V_Q_2))) (=> (forall ((B_x $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x V_b_2) (c_Transitive__Closure_Ortrancl T_a V_P_2)) (forall ((B_y $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_y B_x) V_Q_2) (= B_x B_y))))) (c_member _let_2 _let_1 (c_Transitive__Closure_Ortrancl T_a V_P_2))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (= (and (c_Wellfounded_Oacyclic T_a V_r_2) (not (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)))) (c_Wellfounded_Oacyclic T_a (c_Set_Oinsert _let_1 (c_Product__Type_OPair T_a T_a V_y_2 V_x_2) V_r_2))))) (forall ((V_R2_2 $$unsorted) (T_b $$unsorted) (V_R1_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Otrans T_a V_R1_2) (=> (c_Relation_Otrans T_b V_R2_2) (c_Relation_Otrans (tc_prod T_a T_b) (c_Wellfounded_Olex__prod T_a T_b V_R1_2 V_R2_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_b) (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_L_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (let ((_let_2 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_y_2))) (=> (not (= V_a_2 V_b_2)) (=> (c_member (tc_fun _let_1 tc_HOL_Obool) V_L_2 c_Arrow__Order__Mirabelle_OLin) (= (c_member _let_1 _let_2 (c_Arrow__Order__Mirabelle_Oabove V_L_2 V_a_2 V_b_2)) (and (=> (not (= V_b_2 V_x_2)) (and (=> (= V_y_2 V_b_2) (or (= V_x_2 V_a_2) (c_member _let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_a_2) V_L_2))) (=> (not (= V_b_2 V_y_2)) (c_member _let_1 _let_2 V_L_2)))) (=> (= V_x_2 V_b_2) (c_member _let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_a_2 V_y_2) V_L_2)) (not (= V_y_2 V_x_2))))))))) (forall ((T_a $$unsorted)) (c_Relation_Otrans T_a (c_Relation_OId T_a))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Otop T_a) (= (c_Orderings_Otop__class_Otop T_a) (hAPP (c_Orderings_Otop__class_Otop (tc_fun T_b T_a)) V_x_2)))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (c_member T_a V_b_2 (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (= V_b_2 V_a_2))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Otop__class_Otop T_a))) (=> (class_Lattices_Obounded__lattice__top T_a) (= _let_1 (c_Lattices_Osemilattice__sup__class_Osup T_a V_x _let_1))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (c_Set_Oinsert T_a V_b_2 (c_Relation_ORange T_b T_a V_r_2)) (c_Relation_ORange T_b T_a (c_Set_Oinsert (tc_prod T_b T_a) (c_Product__Type_OPair T_b T_a V_a_2 V_b_2) V_r_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Otrans T_a (c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_g_2 $$unsorted) (V_Fa_2 $$unsorted) (T_b $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_z $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_z V_x_2) V_R_2) (c_Recdef_Owfrec__rel T_a T_b V_R_2 V_Fa_2 B_z (hAPP V_g_2 B_z)))) (c_Recdef_Owfrec__rel T_a T_b V_R_2 V_Fa_2 V_x_2 (hAPP (hAPP V_Fa_2 V_g_2) V_x_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (c_member T_a V_a_2 (c_Set_Oinsert T_a V_a_2 V_B_2))) (forall ((V_x_2 $$unsorted) (V_L_2 $$unsorted)) (let ((_let_1 (tc_fun (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt) tc_HOL_Obool))) (=> (c_member _let_1 V_L_2 c_Arrow__Order__Mirabelle_OLin) (c_member _let_1 (c_Arrow__Order__Mirabelle_Omkbot V_L_2 V_x_2) c_Arrow__Order__Mirabelle_OLin)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) (=> (c_member T_a V_x_2 V_A_2) (c_member T_a V_x_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (= (and (c_member T_a V_c_2 V_B_2) (c_member T_a V_c_2 V_A_2)) (c_member T_a V_c_2 (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Oantisym T_a (c_Relation_OId__on T_a V_A_2))) (forall ((V_L_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tc_fun (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt) tc_HOL_Obool))) (=> (not (= V_x_2 V_y_2)) (=> (c_member _let_1 V_L_2 c_Arrow__Order__Mirabelle_OLin) (c_member _let_1 (c_Arrow__Order__Mirabelle_Oabove V_L_2 V_x_2 V_y_2) c_Arrow__Order__Mirabelle_OLin))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Otrans T_a (c_Transitive__Closure_Otrancl T_a V_r_2))) (forall ((V_s_2 $$unsorted) (V_B_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (c_Relation_Orefl__on T_a V_B_2 V_s_2) (c_Relation_Orefl__on T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_s_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 _let_1 (c_Transitive__Closure_Otrancl T_a V_r_2)) (=> (not (c_member _let_2 _let_1 V_r_2)) (not (forall ((B_c $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 B_c) (c_Transitive__Closure_Otrancl T_a V_r_2)) (not (c_member _let_1 (c_Product__Type_OPair T_a T_a B_c V_b_2) V_r_2))))))))))) (forall ((T_a $$unsorted)) (= (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)) (c_Relation_ODomain T_a T_a (c_Relation_OId T_a)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 V_B_2)))) (let ((_let_3 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2))) (let ((_let_4 (c_member T_a V_a_2 V_A_2))) (and (=> (not _let_4) (= _let_2 _let_3)) (=> _let_4 (= (c_Set_Oinsert T_a V_a_2 _let_3) _let_2)))))))) (forall ((T_a $$unsorted) (V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_y_H_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (=> (hBOOL (hAPP V_P_2 V_x_2)) (=> (c_member (tc_prod T_b T_b) (c_Product__Type_OPair T_b T_b V_y_H_2 V_y_2) (hAPP V_R_2 V_x_2)) (c_member (tc_prod _let_1 _let_1) (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b V_x_2 V_y_H_2) (c_Product__Type_OPair T_a T_b V_x_2 V_y_2)) (c_Recdef_Osame__fst T_a T_b V_P_2 V_R_2)))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_ODomain T_a T_a (c_Transitive__Closure_Ortrancl T_a V_R_2)) (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 (c_Groups_Ominus__class_Ominus (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (not (=> (c_member T_a V_c_2 V_A_2) (c_member T_a V_c_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (=> (not (c_member T_a V_c_2 V_B_2)) (c_member T_a V_c_2 V_A_2)) (c_member T_a V_c_2 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2) (c_member T_a V_a_2 (c_Relation_ODomain T_a T_b V_r_2)))) _let_16 (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (not (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_a_2) V_r_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (hBOOL (hAPP (c_fequal V_x_2) V_y_2))) (= V_y_2 V_x_2))) _let_15 (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 V_A_2) (=> (c_member T_a V_c_2 V_B_2) (c_member T_a V_c_2 (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_x)))) (forall ((T_a $$unsorted)) (c_Order__Relation_Owell__order__on T_a (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((V_a $$unsorted) (V_x $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_b V_x) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b) V_x)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (hBOOL (hAPP (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2)) (not (=> (hBOOL (hAPP V_A_2 V_x_2)) (not (hBOOL (hAPP V_B_2 V_x_2))))))) (forall ((V_p_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_p_2 V_r_2) (c_Wellfounded_Owf T_a V_p_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (c_member T_a V_x_2 V_A_2)) (=> (not (c_member T_a V_x_2 V_B_2)) (= (= V_A_2 V_B_2) (= (c_Set_Oinsert T_a V_x_2 V_A_2) (c_Set_Oinsert T_a V_x_2 V_B_2)))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_b (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_c)) (c_Lattices_Osemilattice__sup__class_Osup T_a V_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_b V_c))))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Orderings_Otop T_1) (class_Orderings_Otop (tc_fun T_2 T_1)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_x)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Order__Relation_Opartial__order__on T_a V_A_2 V_r_2) (and (c_Relation_Oantisym T_a V_r_2) (c_Order__Relation_Opreorder__on T_a V_A_2 V_r_2)))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (c_member T_a V_x_2 V_A_2) (hBOOL (hAPP V_A_2 V_x_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_x_2 V_A_2) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) (c_member T_a V_x_2 V_B_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Orefl__on T_a (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)) (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a)))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Odistrib__lattice T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z) V_x) (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_x) (c_Lattices_Osemilattice__sup__class_Osup T_a V_z V_x))))) (forall ((V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 V_p_2 V_r_2) (c_member _let_1 V_p_2 (c_Transitive__Closure_Ortrancl T_a V_r_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_a) (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (class_Groups_Oab__group__add T_a) (=> (= (c_Groups_Ominus__class_Ominus T_a V_a_2 V_b_2) (c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (= V_b_2 V_a_2) (= V_c_2 V_d_2))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Obot T_a) (= (c_Orderings_Obot__class_Obot T_a) (hAPP (c_Orderings_Obot__class_Obot (tc_fun T_b T_a)) V_x_2)))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (hBOOL (hAPP V_A_2 V_x_2)) (hBOOL (hAPP (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (c_Relation_OId T_a)) (= V_b_2 V_a_2))) (forall ((V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (and (forall ((B_x $$unsorted)) (=> (c_member T_a B_x V_A_2) (hBOOL (hAPP V_P_2 B_x)))) (forall ((B_x $$unsorted)) (=> (c_member T_a B_x V_B_2) (hBOOL (hAPP V_P_2 B_x))))) (forall ((B_x $$unsorted)) (=> (c_member T_a B_x (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (hBOOL (hAPP V_P_2 B_x)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2)) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_C_2))))) _let_12 (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Wellfounded_Oacc T_a V_r_2))) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_b_2 V_a_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (c_member T_a V_a_2 _let_1) (c_member T_a V_b_2 _let_1))))) (forall ((V_r_2 $$unsorted) (V_zb_2 $$unsorted) (V_za_2 $$unsorted) (V_xb_2 $$unsorted) (V_xa_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_b V_xa_2 V_xb_2))) (let ((_let_2 (c_Product__Type_OPair T_a T_b V_za_2 V_zb_2))) (let ((_let_3 (tc_prod T_a T_b))) (=> (c_member (tc_prod _let_3 _let_3) (c_Product__Type_OPair _let_3 _let_3 _let_1 _let_2) (c_Transitive__Closure_Ortrancl _let_3 V_r_2)) (=> (not (= _let_2 _let_1)) (not (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (let ((_let_2 (c_Product__Type_OPair T_a T_b B_a B_b))) (let ((_let_3 (tc_prod _let_1 _let_1))) (=> (c_member _let_3 (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b V_xa_2 V_xb_2) _let_2) V_r_2) (not (c_member _let_3 (c_Product__Type_OPair _let_1 _let_1 _let_2 (c_Product__Type_OPair T_a T_b V_za_2 V_zb_2)) (c_Transitive__Closure_Ortrancl _let_1 V_r_2))))))))))))))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Lattices_Olattice T_1) (class_Lattices_Olattice (tc_fun T_2 T_1)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Order__Relation_Olinear__order__on T_a V_A_2 V_r_2) (and (c_Order__Relation_Opartial__order__on T_a V_A_2 V_r_2) (c_Relation_Ototal__on T_a V_A_2 V_r_2)))) (forall ((T_b $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (B_f $$unsorted) (B_R $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_a_2) B_R) (= (hAPP (c_Recdef_Ocut T_a T_b B_f B_R V_a_2) V_x_2) (hAPP B_f V_x_2)))) (class_Lattices_Obounded__lattice__top tc_HOL_Obool) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_z) (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z))))) (exists ((B_L $$unsorted)) (c_member (tc_fun (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt) tc_HOL_Obool) B_L c_Arrow__Order__Mirabelle_OLin)) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (=> (= V_y_2 V_x_2) (= (c_member T_a V_x_2 V_A_2) (c_member T_a V_y_2 V_A_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_s_2) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) (c_Relation_OField T_a V_r_2) (c_Relation_OField T_a V_s_2)))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (c_Nitpick_Ofold__graph_H T_a T_b V_a1_2 V_a2_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) V_a2_2)) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (class_Orderings_Oord T_b) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a T_b) V_f_2 V_g_2) (c_Orderings_Oord__class_Oless__eq T_b (hAPP V_f_2 V_x_2) (hAPP V_g_2 V_x_2))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (=> (not (c_member T_a V_a_2 V_B_2)) (= V_a_2 V_b_2)) (c_member T_a V_a_2 (c_Set_Oinsert T_a V_b_2 V_B_2)))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_a V_x) (=> (c_Orderings_Oord__class_Oless__eq T_a V_b V_x) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) V_x))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Orderings_Otop__class_Otop _let_1)) (c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_r_2))) (= (c_Transitive__Closure_Ortrancl T_a _let_1) _let_1))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Otrans T_a (c_Relation_OId__on T_a V_A_2))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot T_a))) (=> (class_Lattices_Obounded__lattice__bot T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a _let_1 V_x) _let_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot T_a))) (=> (class_Lattices_Obounded__lattice__bot T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x _let_1) _let_1)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_x)))) (forall ((V_c_2 $$unsorted) (V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_R_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_R_2) (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) (c_Transitive__Closure_Otrancl T_a V_R_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b))) (=> (class_Lattices_Osemilattice__inf T_a) (= _let_1 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 _let_2) _let_2)))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) V_c_2 (c_Relation_OId__on T_a V_A_2)) (not (forall ((B_x $$unsorted)) (=> (c_member T_a B_x V_A_2) (not (= (c_Product__Type_OPair T_a T_a B_x B_x) V_c_2))))))) (forall ((V_P_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (hBOOL (hAPP V_P_2 V_b_2)) (=> (forall ((B_y $$unsorted) (B_z $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y B_z) V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_z V_b_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (hBOOL (hAPP V_P_2 B_z)) (hBOOL (hAPP V_P_2 B_y))))))) (hBOOL (hAPP V_P_2 V_a_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (c_member T_a B_x V_A_2)) (not (= (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) V_A_2)))) (forall ((T_a $$unsorted)) (= (c_Relation_OField T_a (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))) (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Groups_Ominus__class_Ominus _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_C_2 V_A_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_C_2 V_B_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_C_2 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Opreorder T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y V_z) (c_Orderings_Oord__class_Oless__eq T_a V_x V_z))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (c_member T_a V_y_2 V_A_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_C_2)) (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Orderings_Oord__class_Oless__eq _let_1 (c_Set_Oinsert T_a V_x_2 V_A_2) V_B_2) (and (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (c_member T_a V_x_2 V_B_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_b) tc_HOL_Obool) V_r_2 V_s_2) (=> (c_Relation_Osingle__valued T_a T_b V_s_2) (c_Relation_Osingle__valued T_a T_b V_r_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (not (c_member T_a V_x_2 V_A_2)) (= (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (c_Set_Oinsert T_a V_x_2 V_B_2)) (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_ORange T_a T_a (c_Transitive__Closure_Ortrancl T_a V_R_2)) (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_member T_a V_x_2 V_B_2) (= (c_Groups_Ominus__class_Ominus _let_1 (c_Set_Oinsert T_a V_x_2 V_A_2) V_B_2) (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= V_A_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Orderings_Otop__class_Otop _let_1))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (=> (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b B_x B_z) V_r_2) (= B_z B_y))))) (c_Relation_Osingle__valued T_a T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Orefl__on T_a (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)) (c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Orderings_Oorder T_1) (class_Orderings_Oorder (tc_fun T_2 T_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Opreorder T_a) (c_Orderings_Oord__class_Oless__eq T_a V_x V_x))) (class_Orderings_Oord tc_HOL_Obool) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (c_member T_a V_c_2 V_A_2))) (forall ((T_a $$unsorted)) (c_Relation_Oantisym T_a (c_Relation_OId T_a))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_A_2)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Obounded__lattice__top T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Orderings_Otop__class_Otop T_a)) V_x))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Otop__class_Otop T_a))) (=> (class_Lattices_Obounded__lattice__top T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a _let_1 V_x) _let_1)))) (forall ((T_a $$unsorted)) (c_Order__Relation_Opreorder__on T_a (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y))))) (forall ((V_y_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (c_member T_a V_x_2 (c_Relation_ODomain T_a T_a V_R_2))) (= (= V_y_2 V_x_2) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (c_Transitive__Closure_Ortrancl T_a V_R_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) V_y))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_a_2 V_B_2) (c_member T_a V_a_2 (c_Set_Oinsert T_a V_b_2 V_B_2)))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted)) (not (c_member T_a V_c_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (not (=> (c_member T_a V_c_2 V_A_2) (not (c_member T_a V_c_2 V_B_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (= V_y_2 V_x_2) (c_member T_a V_x_2 (c_fequal V_y_2)))) (forall ((V_a_2 $$unsorted) (V_H_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Recdef_Owfrec T_a T_b V_r_2 V_H_2))) (=> (c_Wellfounded_Owf T_a V_r_2) (= (hAPP _let_1 V_a_2) (hAPP (hAPP V_H_2 (c_Recdef_Ocut T_a T_b _let_1 V_r_2 V_a_2)) V_a_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Transitive__Closure_Ortrancl T_a V_r_2) (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) (c_Transitive__Closure_Otrancl T_a V_r_2) (c_Relation_OId T_a)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) V_z) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z))))) _let_11 (class_Lattices_Osemilattice__inf tc_HOL_Obool) (forall ((V_b_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (c_Product__Type_OPair T_a T_b V_a_H_2 V_b_H_2) (c_Product__Type_OPair T_a T_b V_a_2 V_b_2)) (and (= V_a_H_2 V_a_2) (= V_b_2 V_b_H_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_a_2 V_A_2) (=> (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2) (c_member T_b V_b_2 (c_Relation_OImage T_a T_b V_r_2 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 (c_Groups_Ominus__class_Ominus (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (not (c_member T_a V_c_2 V_B_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (= (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_1)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_1))))))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Orderings_Obot T_1) (class_Orderings_Obot (tc_fun T_2 T_1)))) _let_10 (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (forall ((B_x $$unsorted)) (=> (c_member T_a B_x V_A_2) (forall ((B_xa $$unsorted)) (=> (c_member T_a B_xa V_B_2) (not (= B_x B_xa)))))) (= (c_Orderings_Obot__class_Obot _let_1) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (c_Wellfounded_Owf T_b (c_Relation_Oinv__image T_a T_b V_r_2 V_f_2)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a V_b) V_c) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_b V_c))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (class_Groups_Ominus T_a) (= (hAPP (c_Groups_Ominus__class_Ominus (tc_fun T_b T_a) V_A_2 V_B_2) V_x_2) (c_Groups_Ominus__class_Ominus T_a (hAPP V_A_2 V_x_2) (hAPP V_B_2 V_x_2))))) (forall ((V_rb_2 $$unsorted) (T_b $$unsorted) (V_ra_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_ra_2) (=> (c_Wellfounded_Owf T_b V_rb_2) (c_Wellfounded_Owf (tc_prod T_a T_b) (c_Wellfounded_Olex__prod T_a T_b V_ra_2 V_rb_2))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) _let_1) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_P_2 $$unsorted) (V_r_2 $$unsorted) (V_by_2 $$unsorted) (V_bx_2 $$unsorted) (V_ay_2 $$unsorted) (V_ax_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (=> (c_member (tc_prod _let_1 _let_1) (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b V_ax_2 V_ay_2) (c_Product__Type_OPair T_a T_b V_bx_2 V_by_2)) (c_Transitive__Closure_Ortrancl _let_1 V_r_2)) (=> (hBOOL (hAPP (hAPP V_P_2 V_ax_2) V_ay_2)) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (=> (c_member (tc_prod _let_1 _let_1) (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b V_ax_2 V_ay_2) (c_Product__Type_OPair T_a T_b B_a B_b)) (c_Transitive__Closure_Ortrancl _let_1 V_r_2)) (forall ((B_aa $$unsorted) (B_ba $$unsorted)) (let ((_let_1 (tc_prod T_a T_b))) (=> (c_member (tc_prod _let_1 _let_1) (c_Product__Type_OPair _let_1 _let_1 (c_Product__Type_OPair T_a T_b B_a B_b) (c_Product__Type_OPair T_a T_b B_aa B_ba)) V_r_2) (=> (hBOOL (hAPP (hAPP V_P_2 B_a) B_b)) (hBOOL (hAPP (hAPP V_P_2 B_aa) B_ba))))))))) (hBOOL (hAPP (hAPP V_P_2 V_bx_2) V_by_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) V_C_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (c_member T_a V_x_2 (c_Wellfounded_Oacc T_a V_r_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_C_2)) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (hBOOL (hAPP (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2)) (hBOOL (hAPP V_B_2 V_x_2)))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (c_Set_Oinsert T_a V_a_2 V_A_2) (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Otrans T_a V_r_2) (=> (c_Relation_Otrans T_a V_s_2) (c_Relation_Otrans T_a (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_s_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (= (c_Orderings_Obot__class_Obot _let_1) V_A_2) (= (c_Equiv__Relations_Oquotient T_a V_A_2 V_r_2) (c_Orderings_Obot__class_Obot (tc_fun _let_1 tc_HOL_Obool)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 _let_2) _let_2)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oorder T_a) (=> (= V_a V_b) (=> (c_Orderings_Oord__class_Oless__eq T_a V_c V_b) (c_Orderings_Oord__class_Oless__eq T_a V_c V_a))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Odistrib__lattice T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_x) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_z V_x)) (c_Lattices_Osemilattice__inf__class_Oinf T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z) V_x)))) _let_4 (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Lattices_Olattice T_1) (class_Lattices_Osemilattice__sup (tc_fun T_2 T_1)))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_r_2) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Groups_Ominus__class_Ominus _let_1 (c_Relation_ODomain T_a T_b V_A_2) (c_Relation_ODomain T_a T_b V_B_2)) (c_Relation_ODomain T_a T_b (c_Groups_Ominus__class_Ominus (tc_fun (tc_prod T_a T_b) tc_HOL_Obool) V_A_2 V_B_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y))) (=> (class_Lattices_Olattice T_a) (= _let_1 (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x _let_1))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Set_Oinsert T_a V_a_2 V_A_2) (c_Set_Oinsert T_a V_a_2 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot _let_1))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (c_Orderings_Oord__class_Oless__eq T_a V_y (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (c_member T_a V_a_2 (c_Relation_ODomain T_a T_b V_r_2)) (exists ((B_y $$unsorted)) (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b V_a_2 B_y) V_r_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oorder T_a) (= (= V_x_2 V_y_2) (and (c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2) (c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_b_2 (c_Relation_ORange T_b T_a V_r_2)) (not (forall ((B_x $$unsorted)) (not (c_member (tc_prod T_b T_a) (c_Product__Type_OPair T_b T_a B_x V_b_2) V_r_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (c_FunDef_Oin__rel T_a T_b V_R_2 V_x_2 V_y_2) (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b V_x_2 V_y_2) V_R_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_a_2 (c_Relation_ODomain T_a T_b V_r_2)) (not (forall ((B_y $$unsorted)) (not (c_member (tc_prod T_a T_b) (c_Product__Type_OPair T_a T_b V_a_2 B_y) V_r_2)))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_Fa_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (c_Recdef_Owfrec__rel T_a T_b V_R_2 V_Fa_2 V_a1_2 V_a2_2) (exists ((B_g $$unsorted)) (and (= V_a2_2 (hAPP (hAPP V_Fa_2 B_g) V_a1_2)) (forall ((B_z $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_z V_a1_2) V_R_2) (c_Recdef_Owfrec__rel T_a T_b V_R_2 V_Fa_2 B_z (hAPP B_g B_z)))))))) (class_Orderings_Obot tc_HOL_Obool) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_B_2 V_A_2) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2))) (forall ((T_a $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (=> (not (hBOOL (hAPP V_B_2 V_x_2))) (hBOOL (hAPP V_A_2 V_x_2))) (hBOOL (hAPP (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (c_Orderings_Oord__class_Oless__eq T_a V_b V_c) (c_Orderings_Oord__class_Oless__eq T_a V_a V_c))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (c_Relation_OImage T_b T_a V_R_2 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_b tc_HOL_Obool) V_A_2 V_B_2)) (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) (c_Relation_OImage T_b T_a V_R_2 V_A_2) (c_Relation_OImage T_b T_a V_R_2 V_B_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_s_2))) (let ((_let_2 (tc_fun (tc_prod T_a T_a) tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_2 V_r_2 _let_1) (c_Orderings_Oord__class_Oless__eq _let_2 (c_Transitive__Closure_Ortrancl T_a V_r_2) _let_1))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member T_a V_x_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_2)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_2)))) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_s_2) (=> (c_Relation_Oantisym T_a V_s_2) (c_Relation_Oantisym T_a V_r_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (c_Product__Type_Ointernal__split T_b T_c T_a V_c_2 (c_Product__Type_OPair T_b T_c V_a_2 V_b_2)) (hAPP (hAPP V_c_2 V_a_2) V_b_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_L_2 $$unsorted)) (let ((_let_1 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (=> (c_member (tc_fun _let_1 tc_HOL_Obool) V_L_2 c_Arrow__Order__Mirabelle_OLin) (=> (c_member _let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_a_2 V_b_2) V_L_2) (not (c_member _let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_b_2 V_a_2) V_L_2)))))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Lattices_Obounded__lattice T_1) (class_Lattices_Obounded__lattice__top (tc_fun T_2 T_1)))) (forall ((T_a $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (hBOOL (hAPP V_P_2 B_x)) (c_Wellfounded_Owf T_b (hAPP V_R_2 B_x)))) (c_Wellfounded_Owf (tc_prod T_a T_b) (c_Recdef_Osame__fst T_a T_b V_P_2 V_R_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 V_B_2) (c_member T_a V_c_2 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 V_p_2 (c_Transitive__Closure_Otrancl T_a V_r_2)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun _let_1 tc_HOL_Obool) V_r_2 V_s_2) (c_member _let_1 V_p_2 (c_Transitive__Closure_Otrancl T_a V_s_2)))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (hBOOL (hAPP (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2) V_x_2)) (hBOOL (hAPP V_A_2 V_x_2)))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (c_member _let_2 (c_Product__Type_OPair T_a T_a V_b_2 V_c_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (c_member _let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (and (c_Relation_Otrans T_a V_r_2) (c_Relation_Orefl__on T_a V_A_2 V_r_2)) (c_Order__Relation_Opreorder__on T_a V_A_2 V_r_2))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Groups_Ominus__class_Ominus _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot _let_1))) (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= V_B_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Orderings_Obot__class_Obot _let_1) V_B_2)))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_Wellfounded_Owf T_a V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_x_2) V_r_2) (not (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 V_a_2) V_r_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (not (c_member T_a V_a_2 V_A_2)) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_B_2) (c_Orderings_Obot__class_Obot _let_1)) (= V_A_2 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_b) (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (c_Relation_ODomain T_a T_b (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_b) tc_HOL_Obool))) (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Otop__class_Otop _let_1))) (= _let_2 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 _let_2))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2))) (let ((_let_3 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Set_Oinsert T_a V_a_2 V_B_2) V_C_2))) (let ((_let_4 (c_member T_a V_a_2 V_C_2))) (and (=> (not _let_4) (= _let_2 _let_3)) (=> _let_4 (= _let_3 (c_Set_Oinsert T_a V_a_2 _let_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (= (c_member T_a V_c_2 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (or (c_member T_a V_c_2 V_A_2) (c_member T_a V_c_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 (c_Groups_Ominus__class_Ominus (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (c_member T_a V_c_2 V_A_2))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_A_2)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_y (c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_z)) (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z))))) (forall ((V_P_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (hBOOL (hAPP V_P_2 V_a_2)) (=> (forall ((B_y $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 B_y) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (forall ((B_z $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_y B_z) V_r_2) (=> (hBOOL (hAPP V_P_2 B_y)) (hBOOL (hAPP V_P_2 B_z))))))) (hBOOL (hAPP V_P_2 V_b_2)))))) (forall ((V_r_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_z_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (not (= V_z_2 V_x_2)) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a V_x_2 B_y) V_r_2) (not (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y V_z_2) (c_Transitive__Closure_Ortrancl T_a V_r_2)))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (=> (= V_a_2 V_b_2) (=> (c_member T_a V_b_2 V_A_2) (c_member T_a V_a_2 V_A_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_C_2) (c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_C_2)) (c_Groups_Ominus__class_Ominus _let_1 (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_C_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (= (c_Orderings_Obot__class_Obot _let_1) V_A_2) (= (c_Orderings_Obot__class_Obot (tc_fun _let_1 tc_HOL_Obool)) (c_Equiv__Relations_Oquotient T_a V_A_2 V_r_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Obounded__lattice__bot T_a) (= V_x (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Orderings_Obot__class_Obot T_a) V_x)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (hAPP (c_Lattices_Osemilattice__inf__class_Oinf (tc_fun T_b T_a) V_f_2 V_g_2) V_x_2) (c_Lattices_Osemilattice__inf__class_Oinf T_a (hAPP V_f_2 V_x_2) (hAPP V_g_2 V_x_2))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_a_2) (c_Relation_OId T_a))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (= (or (and (= V_d_2 V_b_2) (= V_c_2 V_a_2)) (and (= V_b_2 V_c_2) (= V_d_2 V_a_2))) (= (c_Set_Oinsert T_a V_c_2 (c_Set_Oinsert T_a V_d_2 _let_1)) (c_Set_Oinsert T_a V_a_2 (c_Set_Oinsert T_a V_b_2 _let_1)))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 V_C_2))) (= (c_Groups_Ominus__class_Ominus _let_1 _let_2 (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2)) (c_Groups_Ominus__class_Ominus _let_1 _let_2 V_B_2))))) (forall ((V_L_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tc_fun (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt) tc_HOL_Obool))) (=> (not (= V_x_2 V_y_2)) (=> (c_member _let_1 V_L_2 c_Arrow__Order__Mirabelle_OLin) (c_member _let_1 (c_Arrow__Order__Mirabelle_Obelow V_L_2 V_x_2 V_y_2) c_Arrow__Order__Mirabelle_OLin))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tc_prod T_a T_a))) (=> (c_member _let_2 _let_1 V_r_2) (c_member _let_2 _let_1 (c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_Oantisym T_a V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tc_prod T_a T_a))) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (=> (c_member _let_1 (c_Product__Type_OPair T_a T_a B_y B_x) V_r_2) (= B_y B_x))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (= (c_Groups_Ominus__class_Ominus _let_1 _let_2 V_A_2) _let_2)))) (forall ((V_a_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_C_2 V_D_2) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Set_Oinsert T_a V_a_2 V_C_2) (c_Set_Oinsert T_a V_a_2 V_D_2))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))) (not (c_member T_a V_a_2 V_A_2)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Odistrib__lattice T_a) (= (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x (c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z)) (c_Lattices_Osemilattice__sup__class_Osup T_a (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_y) (c_Lattices_Osemilattice__inf__class_Oinf T_a V_x V_z))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Relation_OField T_a V_r_2))) (=> (c_Order__Relation_Opreorder__on T_a _let_2 V_r_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_2) (=> (c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 _let_2) (= (forall ((B_x $$unsorted)) (=> (c_member T_a B_x V_A_2) (exists ((B_xa $$unsorted)) (and (c_member T_a B_xa V_B_2) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_xa B_x) V_r_2))))) (c_Orderings_Oord__class_Oless__eq _let_1 (c_Relation_OImage T_a T_a V_r_2 V_A_2) (c_Relation_OImage T_a T_a V_r_2 V_B_2))))))))) _let_3 (forall ((T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (not (= (c_Orderings_Obot__class_Obot _let_1) (c_Orderings_Otop__class_Otop _let_1))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (c_member T_a V_x_2 (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Obounded__lattice__bot T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a V_x (c_Orderings_Obot__class_Obot T_a)) V_x))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2)) (=> (not (c_member T_a V_c_2 V_A_2)) (c_member T_a V_c_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (let ((_let_2 (c_Orderings_Obot__class_Obot _let_1))) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_A_2 _let_2) _let_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_ORange T_a T_a (c_Relation_OId__on T_a V_A_2)) V_A_2)) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_Fa_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (exists ((B_g $$unsorted)) (and (forall ((B_z $$unsorted)) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_z V_a1_2) V_R_2) (c_Recdef_Owfrec__rel T_a T_b V_R_2 V_Fa_2 B_z (hAPP B_g B_z)))) (= (hAPP (hAPP V_Fa_2 V_a2_2) V_a1_2) (hAPP (hAPP V_Fa_2 B_g) V_a1_2)))) (c_Recdef_Owfrec__rel T_a T_b V_R_2 V_Fa_2 V_a1_2 (hAPP (hAPP V_Fa_2 V_a2_2) V_a1_2)))) _let_2 (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_a_2) (c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Relation_Otrans T_a V_r_2) (c_Relation_Otrans T_b (c_Relation_Oinv__image T_a T_b V_r_2 V_f_2)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (class_Groups_Ominus T_1) (class_Groups_Ominus (tc_fun T_2 T_1)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (= V_x_2 V_y_2)) (hBOOL (hAPP (c_fequal V_x_2) V_y_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_b) tc_HOL_Obool) V_r_2 V_s_2) (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) (c_Relation_ODomain T_a T_b V_r_2) (c_Relation_ODomain T_a T_b V_s_2)))) _let_1 (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (= (hAPP V_g_2 B_x) (hAPP V_f_2 B_x))) (= V_f_2 V_g_2))) (forall ((V_a5_2 $$unsorted) (V_a1_2 $$unsorted) (V_a2_2 $$unsorted) (V_a4_2 $$unsorted) (V_a3_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (hAPP (hAPP V_a3_2 V_a1_2) V_a5_2))) (= (c_Nitpick_Ofold__graph_H T_a T_b V_a3_2 V_a4_2 V_a2_2 _let_1) (or (and (= V_a2_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))) (= _let_1 V_a4_2)) (exists ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (and (= (hAPP (hAPP V_a3_2 V_a1_2) V_a5_2) (hAPP (hAPP V_a3_2 B_x) B_y)) (c_Nitpick_Ofold__graph_H T_a T_b V_a3_2 V_a4_2 (c_Groups_Ominus__class_Ominus _let_1 V_a2_2 (c_Set_Oinsert T_a B_x (c_Orderings_Obot__class_Obot _let_1))) B_y) (c_member T_a B_x V_a2_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Olinorder T_a) (or (c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (c_Orderings_Oord__class_Oless__eq T_a V_y V_x)))) (forall ((V_X_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) (c_Relation_OImage T_a T_a V_r_2 V_X_2) V_X_2) (= V_X_2 (c_Relation_OImage T_a T_a (c_Transitive__Closure_Ortrancl T_a V_r_2) V_X_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (not (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_x) (c_Transitive__Closure_Otrancl T_a V_r_2)))) (=> (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (not (= V_y_2 V_x_2))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (= (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 V_B_2)) (c_Groups_Ominus__class_Ominus _let_1 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot _let_1))) V_B_2)))) (forall ((V_r_2 $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Oacyclic T_a V_s_2) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 V_s_2) (c_Wellfounded_Oacyclic T_a V_r_2)))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (=> (= (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_b_2 _let_1)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_a_2 _let_1))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member T_a V_b_2 V_A_2) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2)))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oorder T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_b V_a) (=> (= V_c V_b) (c_Orderings_Oord__class_Oless__eq T_a V_c V_a))))) (class_Orderings_Oorder tc_HOL_Obool) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (not (c_member T_a B_x V_A_2))) (= V_A_2 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_c_2 V_A_2) (=> (not (c_member T_a V_c_2 V_B_2)) (c_member T_a V_c_2 (c_Groups_Ominus__class_Ominus (tc_fun T_a tc_HOL_Obool) V_A_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (c_member T_a V_b_2 (c_Relation_OImage T_b T_a V_r_2 V_A_2)) (not (forall ((B_x $$unsorted)) (=> (c_member (tc_prod T_b T_a) (c_Product__Type_OPair T_b T_a B_x V_b_2) V_r_2) (not (c_member T_b B_x V_A_2))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (hBOOL (hAPP (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)) V_x_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)))) (=> (c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (c_member T_a V_x_2 V_A_2) (=> (c_member T_a V_y_2 V_A_2) (= (= (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_x_2 _let_1)) (c_Relation_OImage T_a T_a V_r_2 (c_Set_Oinsert T_a V_y_2 _let_1))) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2))))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__sup T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_a V_c) (=> (c_Orderings_Oord__class_Oless__eq T_a V_b V_d) (c_Orderings_Oord__class_Oless__eq T_a (c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) (c_Lattices_Osemilattice__sup__class_Osup T_a V_c V_d)))))) (forall ((T_a $$unsorted)) (c_Order__Relation_Opartial__order__on T_a (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((T_b $$unsorted) (T_a $$unsorted) (B_M $$unsorted) (B_R $$unsorted)) (=> (c_Wellfounded_Owf T_a B_R) (forall ((B_x $$unsorted)) (let ((_let_1 (c_Recdef_Owfrec T_a T_b B_R B_M))) (= (hAPP _let_1 B_x) (hAPP (hAPP B_M (c_Recdef_Ocut T_a T_b _let_1 B_R B_x)) B_x)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Osingle__valued T_a T_a (c_Relation_OId__on T_a V_A_2))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Osemilattice__inf T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_z) (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Lattices_Osemilattice__inf__class_Oinf T_a V_y V_z)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (not (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a V_a_2 V_a_2) V_r_2)))) (forall ((T_a $$unsorted)) (= (c_Orderings_Otop__class_Otop (tc_fun T_a tc_HOL_Obool)) (c_Relation_ORange T_a T_a (c_Relation_OId T_a)))) (forall ((V_A_2 $$unsorted) (V_A_H_2 $$unsorted) (V_r_2 $$unsorted) (V_r_H_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun (tc_prod T_a T_b) tc_HOL_Obool) V_r_H_2 V_r_2) (=> (c_Orderings_Oord__class_Oless__eq (tc_fun T_a tc_HOL_Obool) V_A_H_2 V_A_2) (c_Orderings_Oord__class_Oless__eq (tc_fun T_b tc_HOL_Obool) (c_Relation_OImage T_a T_b V_r_H_2 V_A_H_2) (c_Relation_OImage T_a T_b V_r_2 V_A_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Otop T_a) (c_Orderings_Oord__class_Oless__eq T_a V_x (c_Orderings_Otop__class_Otop T_a)))) (class_Groups_Ominus tc_HOL_Obool) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (c_member T_a V_a_2 V_A_2) (= V_A_2 (c_Set_Oinsert T_a V_a_2 (c_Groups_Ominus__class_Ominus _let_1 V_A_2 (c_Set_Oinsert T_a V_a_2 (c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Ototal__on T_a (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) V_r_2)) (forall ((V_x_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_B_2 V_A_2) (= (c_member T_a V_x_2 V_A_2) (c_member T_a V_x_2 V_B_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oorder T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (=> (c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (= V_y V_x))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (class_Lattices_Olattice T_a) (= (c_Lattices_Osemilattice__sup__class_Osup T_a (hAPP V_f_2 V_x_2) (hAPP V_g_2 V_x_2)) (hAPP (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_b T_a) V_f_2 V_g_2) V_x_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (c_member (tc_prod T_a T_a) (c_Product__Type_OPair T_a T_a B_x B_x) V_r_2)) (c_Nitpick_Orefl_H T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (c_Relation_Oirrefl T_a (c_Groups_Ominus__class_Ominus (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) V_r_2 (c_Relation_OId T_a)))) (forall ((V_z_2 $$unsorted) (V_L_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (c_Product__Type_OPair tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_y_2))) (let ((_let_2 (tc_prod tc_Arrow__Order__Mirabelle_Oalt tc_Arrow__Order__Mirabelle_Oalt))) (= (and (not (= V_y_2 V_z_2)) (=> (not (= V_x_2 V_z_2)) (c_member _let_2 _let_1 V_L_2)) (=> (= V_z_2 V_x_2) (not (= V_y_2 V_x_2)))) (c_member _let_2 _let_1 (c_Arrow__Order__Mirabelle_Omkbot V_L_2 V_z_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (c_Lattices_Osemilattice__sup__class_Osup (tc_fun T_a tc_HOL_Obool) (c_Relation_ORange T_b T_a V_A_2) (c_Relation_ORange T_b T_a V_B_2)) (c_Relation_ORange T_b T_a (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_b T_a) tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (c_Transitive__Closure_Ortrancl T_a V_R_2))) (= _let_1 (c_Lattices_Osemilattice__sup__class_Osup (tc_fun (tc_prod T_a T_a) tc_HOL_Obool) _let_1 (c_Relation_OId T_a))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (c_Relation_ODomain T_a T_a (c_Relation_OId__on T_a V_A_2)) V_A_2)) (forall ((T_a $$unsorted)) (c_Order__Relation_Olinear__order__on T_a (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_a T_a) tc_HOL_Obool)))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_P_2 $$unsorted)) (= (forall ((B_x1 $$unsorted)) (hBOOL (hAPP V_P_2 B_x1))) (forall ((B_a $$unsorted) (B_b $$unsorted)) (hBOOL (hAPP V_P_2 (c_Product__Type_OPair T_a T_b B_a B_b)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Oacyclic T_a V_r_2) (c_Relation_Oantisym T_a (c_Transitive__Closure_Ortrancl T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (c_Wellfounded_Owf T_a V_r_2) (c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((T_a $$unsorted) (T_b $$unsorted)) (= (c_Orderings_Obot__class_Obot (tc_fun T_a tc_HOL_Obool)) (c_Relation_ORange T_b T_a (c_Orderings_Obot__class_Obot (tc_fun (tc_prod T_b T_a) tc_HOL_Obool))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (class_Orderings_Oord T_a) (=> (c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (=> (= V_c V_b) (c_Orderings_Oord__class_Oless__eq T_a V_a V_c))))) (forall ((V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (= V_b_2 V_a_2))) (let ((_let_2 (c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_3 (tc_prod T_a T_a))) (=> (c_member _let_3 _let_2 (c_Transitive__Closure_Ortrancl T_a V_R_2)) (or (and (not _let_1) (c_member _let_3 _let_2 (c_Transitive__Closure_Otrancl T_a V_R_2))) _let_1)))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tc_fun T_a tc_HOL_Obool))) (=> (not (c_member T_a V_a_2 V_C_2)) (= (c_Lattices_Osemilattice__inf__class_Oinf _let_1 (c_Set_Oinsert T_a V_a_2 V_B_2) V_C_2) (c_Lattices_Osemilattice__inf__class_Oinf _let_1 V_B_2 V_C_2))))) true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 28.07/28.32 % SZS output end Proof for theBenchmark 28.07/28.32 EOF