%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SWW379+1 : TPTP v8.1.2. Released v5.2.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n010.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Fri Sep  1 00:20:36 EDT 2023

% Result   : Theorem 123.70s 124.42s
% Output   : Proof 123.70s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.78/0.82  % Problem    : SWW379+1 : TPTP v8.1.2. Released v5.2.0.
% 0.78/0.83  % Command    : do_cvc5 %s %d
% 0.82/1.03  % Computer : n010.cluster.edu
% 0.82/1.03  % Model    : x86_64 x86_64
% 0.82/1.03  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.82/1.03  % Memory   : 8042.1875MB
% 0.82/1.03  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.82/1.03  % CPULimit   : 300
% 0.82/1.03  % WCLimit    : 300
% 0.82/1.03  % DateTime   : Sun Aug 27 21:24:50 EDT 2023
% 0.82/1.04  % CPUTime    : 
% 1.73/1.89  %----Proving TF0_NAR, FOF, or CNF
% 123.70/124.42  ------- convert to smt2 : /export/starexec/sandbox/tmp/tmp.M4URwgKLmm/cvc5---1.0.5_29318.p...
% 123.70/124.42  ------- get file name : TPTP file name is SWW379+1
% 123.70/124.42  ------- cvc5-fof : /export/starexec/sandbox/solver/bin/cvc5---1.0.5_29318.smt2...
% 123.70/124.42  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 123.70/124.42  --- Run --no-e-matching --full-saturate-quant at 5...
% 123.70/124.42  --- Run --no-e-matching --enum-inst-sum --full-saturate-quant at 5...
% 123.70/124.42  --- Run --finite-model-find --uf-ss=no-minimal at 5...
% 123.70/124.42  --- Run --multi-trigger-when-single --full-saturate-quant at 5...
% 123.70/124.42  --- Run --trigger-sel=max --full-saturate-quant at 5...
% 123.70/124.42  --- Run --multi-trigger-when-single --multi-trigger-priority --full-saturate-quant at 5...
% 123.70/124.42  --- Run --multi-trigger-cache --full-saturate-quant at 5...
% 123.70/124.42  --- Run --prenex-quant=none --full-saturate-quant at 5...
% 123.70/124.42  --- Run --enum-inst-interleave --decision=internal --full-saturate-quant at 5...
% 123.70/124.42  --- Run --relevant-triggers --full-saturate-quant at 5...
% 123.70/124.42  --- Run --finite-model-find --e-matching --sort-inference --uf-ss-fair at 5...
% 123.70/124.42  --- Run --pre-skolem-quant=on --full-saturate-quant at 10...
% 123.70/124.42  --- Run --cbqi-vo-exp --full-saturate-quant at 10...
% 123.70/124.42  --- Run --no-cbqi --full-saturate-quant at 10...
% 123.70/124.42  --- Run --macros-quant --macros-quant-mode=all --full-saturate-quant...
% 123.70/124.42  % SZS status Theorem for SWW379+1
% 123.70/124.42  % SZS output start Proof for SWW379+1
% 123.70/124.42  (
% 123.70/124.42  (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP tptp.c_Com_Ocom_OBODY tptp.v_pn)))) (let ((_let_5 (tptp.c_Option_Othe tptp.tc_Com_Ocom))) (let ((_let_6 (tptp.hAPP _let_5 (tptp.hAPP tptp.c_Com_Obody tptp.v_pn)))) (let ((_let_7 (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple tptp.tc_Com_Ostate) tptp.c_fequal))) (let ((_let_8 (tptp.hAPP _let_7 _let_6))) (let ((_let_9 (tptp.c_Set_Oinsert _let_1))) (let ((_let_10 (tptp.tc_fun tptp.tc_Com_Ostate (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool)))) (let ((_let_11 (tptp.hAPP (tptp.c_COMBB tptp.tc_Com_Ocom _let_10 tptp.tc_Com_Opname) tptp.c_Natural_Oevalc))) (let ((_let_12 (tptp.hAPP (tptp.c_COMBB tptp.tc_Com_Ocom (tptp.tc_fun _let_10 _let_1) tptp.tc_Com_Opname) _let_7))) (let ((_let_13 (tptp.c_COMBS tptp.tc_Com_Opname _let_10 _let_1))) (let ((_let_14 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) tptp.v_G) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_13 (tptp.hAPP _let_12 tptp.c_Com_Ocom_OBODY)) (tptp.hAPP _let_11 tptp.c_Com_Ocom_OBODY))) tptp.v_Procs)))) (let ((_let_15 (not (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_14 (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP _let_8 _let_4)) _let_3))))) (let ((_let_16 (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_14 (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP _let_8 (tptp.hAPP tptp.c_Natural_Oevalc _let_6))) _let_3)))) (let ((_let_17 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_Option_Ooption tptp.tc_Com_Ocom) tptp.tc_Com_Ocom tptp.tc_Com_Opname) _let_5) tptp.c_Com_Obody))) (let ((_let_18 (forall ((V_s2_2 $$unsorted) (V_s1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_Lazy__Sequence_Olazy__sequence _let_1))) (let ((_let_3 (tptp.tc_fun T_b _let_2))) (= (tptp.c_Lazy__Sequence_Oproduct T_a T_b V_s1_2 V_s2_2) (tptp.hAPP (tptp.c_Lazy__Sequence_Obind T_a _let_1 V_s1_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_a) (tptp.c_Lazy__Sequence_Obind T_b _let_1 V_s2_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_3 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 T_b) (tptp.c_Lazy__Sequence_Osingle _let_1))) (tptp.c_Product__Type_OPair T_a T_b))))))))))) (let ((_let_19 (forall ((V_P_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_New__Random__Sequence_Opos__map T_b T_a V_f_2 V_P_2) (tptp.c_New__Random__Sequence_Opos__bind T_b T_a V_P_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_a (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_a))))) T_b (tptp.c_New__Random__Sequence_Opos__single T_a)) V_f_2)))))) (let ((_let_20 (forall ((V_P_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_New__Random__Sequence_Oneg__map T_b T_a V_f_2 V_P_2) (tptp.c_New__Random__Sequence_Oneg__bind T_b T_a V_P_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_a (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_a)))))) T_b (tptp.c_New__Random__Sequence_Oneg__single T_a)) V_f_2)))))) (let ((_let_21 (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_New__Random__Sequence_Opos__empty T_a V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Opos__empty T_a))))) (let ((_let_22 (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_a))))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (= (tptp.c_New__Random__Sequence_Oneg__decr__bind T_b T_a V_R_2 V_f_2 V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Oneg__decr__bind T_b T_a (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_nrandom_2) V_size_2) V_seed_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) V_f_2) V_nrandom_2)) V_size_2)) V_seed_2))))))))) (let ((_let_23 (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_a)))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (= (tptp.c_New__Random__Sequence_Opos__decr__bind T_b T_a V_R_2 V_f_2 V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Opos__decr__bind T_b T_a (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_nrandom_2) V_size_2) V_seed_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) V_f_2) V_nrandom_2)) V_size_2)) V_seed_2))))))))) (let ((_let_24 (forall ((T_a $$unsorted)) (= (tptp.c_Lazy__Sequence_Oyieldn T_a) (tptp.c_Lazy__Sequence_Oanamorph (tptp.tc_Lazy__Sequence_Olazy__sequence T_a) T_a (tptp.c_Lazy__Sequence_Oyield T_a)))))) (let ((_let_25 (forall ((V_xq_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Lazy__Sequence_Olazy__sequence T_b))) (= (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case T_a T_b V_f_2 V_g_2 V_xq_2) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case T_a (tptp.tc_prod T_b _let_1) V_f_2 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b _let_1 T_a) V_g_2)) (tptp.hAPP (tptp.c_Lazy__Sequence_Oyield T_b) V_xq_2))))))) (let ((_let_26 (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint))) (let ((_let_27 (tptp.hAPP _let_26 tptp.c_Int_OPls))) (let ((_let_28 (tptp.hAPP _let_26 tptp.c_Int_OMin))) (let ((_let_29 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (let ((_let_30 (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__ex1 T_a V_P_2 V_xs_2) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (forall ((B_y $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_y) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_y))) (= B_y B_x))))))))) (let ((_let_31 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_32 (tptp.c_Nat_Osize__class_Osize tptp.tc_HOL_Obool))) (let ((_let_33 (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.c_FunDef_Ois__measure T_a V_a_2)))) (let ((_let_34 (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all T_a V_P_2 V_xs_2) (forall ((B_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_n))))))))) (let ((_let_35 (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__ex T_a V_P_2 V_xs_2) (exists ((B_n $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_n))))))))) (let ((_let_36 (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_R_2 $$unsorted)) (= (tptp.c_New__Random__Sequence_Opos__not__random__dseq V_R_2 V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Opos__not__seq (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_nrandom_2) V_size_2) V_seed_2)))))) (let ((_let_37 (forall ((V_fs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_List_Olist _let_1))) (= (tptp.c_List_Omeasures T_a V_fs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_2 T_a) (tptp.c_List_Olex tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_Nat_Onat) (tptp.tc_fun _let_3 _let_2) T_a) (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_COMBC _let_1 T_a tptp.tc_Nat_Onat) (tptp.c_COMBI _let_1)))) V_fs_2))))))))) (let ((_let_38 (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b (tptp.tc_List_Olist T_a)) V_f_2) V_xs_2)) (tptp.c_List_Omaps T_b T_a V_f_2 V_xs_2))))) (let ((_let_39 (forall ((V_k_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Map_Omap__comp T_b T_a T_c V_f_2 V_g_2 V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case (tptp.tc_Option_Ooption T_a) T_b (tptp.c_Option_Ooption_ONone T_a) V_f_2) (tptp.hAPP V_g_2 V_k_2)))))) (let ((_let_40 (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2))) V_xs_2))))) (let ((_let_41 (forall ((V_l_2 $$unsorted)) (= (tptp.c_List_Oembed__list V_l_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_l_2))))) (let ((_let_42 (forall ((V_l_2 $$unsorted)) (= (tptp.c_List_Oreturn__list V_l_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Int_Oint tptp.tc_Nat_Onat) tptp.c_Int_Onat) V_l_2))))) (let ((_let_43 (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Ofoldr T_b T_a V_f_2 V_xs_2 V_a_2) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b (tptp.hAPP (tptp.c_COMBC T_b T_a T_a) V_f_2) V_a_2) (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2)))))) (let ((_let_44 (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) V_n_2)) (tptp.c_List_Oupt V_m_2 V_n_2))))) (let ((_let_45 (forall ((V_A_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (= (tptp.c_List_Osublist T_a V_xs_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_1) (tptp.c_member tptp.tc_Nat_Onat)) (tptp.c_Product__Type_Osnd T_a tptp.tc_Nat_Onat))) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a tptp.tc_Nat_Onat) V_xs_2) (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))))))))))) (let ((_let_46 (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (= (tptp.c_List_Omap__filter T_b T_a V_f_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 T_a T_b (tptp.c_Option_Othe T_a)) V_f_2)) (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_f_2)) (tptp.c_Option_Ooption_ONone T_a)))) V_xs_2))))))) (let ((_let_47 (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Opartition T_a V_f_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.hAPP (tptp.c_List_Ofilter T_a V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a tptp.c_fNot) V_f_2)) V_xs_2))))))) (let ((_let_48 (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_OremoveAll T_a V_x_2) (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP tptp.c_fequal V_x_2))))))) (let ((_let_49 (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Oconcat T_a V_xss_2) (tptp.hAPP (tptp.c_List_Ofoldl _let_1 _let_1 (tptp.c_List_Oappend T_a) (tptp.c_List_Olist_ONil T_a)) V_xss_2)))))) (let ((_let_50 (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (and (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_P_2) B_x)))))))))) (let ((_let_51 (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Enum_Oenum__the T_a V_P_2) (tptp.c_HOL_OThe T_a V_P_2))))) (let ((_let_52 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Smallcheck_Oorelse T_a V_x_2 V_y_2) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case (tptp.tc_Option_Ooption T_a) T_a V_y_2 (tptp.c_Option_Ooption_OSome T_a)) V_x_2))))) (let ((_let_53 (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (= (tptp.c_Map_Odom T_a T_b V_m_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) T_a) tptp.c_fequal) V_m_2)) (tptp.c_Option_Ooption_ONone T_b))))))))) (let ((_let_54 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_55 (= tptp.c_Code__Numeral_Osubtract__code__numeral _let_54))) (let ((_let_56 (forall ((V_k $$unsorted)) (= (tptp.c_Code__Numeral_OSuc__code__numeral V_k) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_k))))))) (let ((_let_57 (forall ((V_n $$unsorted) (V_i $$unsorted)) (= (tptp.c_Code__Numeral_Onat__of__aux V_i V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_i)) V_n))))) (let ((_let_58 (= tptp.c_Code__Numeral_Oint__of (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_Nat_Onat tptp.tc_Int_Oint tptp.tc_Code__Numeral_Ocode__numeral _let_29) tptp.c_Code__Numeral_Onat__of)))) (let ((_let_59 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_60 (tptp.c_Orderings_Otop__class_Otop _let_59))) (let ((_let_61 (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_2 (tptp.tc_prod T_a _let_1))) (= (tptp.c_Random_Oselect T_a V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp _let_1 tptp.tc_Code__Numeral_Ocode__numeral _let_1 _let_2) (tptp.c_Random_Orange (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun _let_1 _let_2) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_OPair T_a _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2)) tptp.c_Code__Numeral_Onat__of))))))))) (let ((_let_62 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_63 (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_64 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_65 (= tptp.c_Code__Numeral_Oof__nat (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Code__Numeral_Ocode__numeral)))) (let ((_let_66 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_67 (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist_Olist__size T_a V_f_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a tptp.tc_Nat_Onat) V_f_2) V_xs_2))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)))))) (let ((_let_68 (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_2 (tptp.tc_prod T_a _let_1))) (= (tptp.c_Random_Oselect__weight T_a V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp _let_1 tptp.tc_Code__Numeral_Ocode__numeral _let_1 _let_2) (tptp.c_Random_Orange (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_Ofst tptp.tc_Code__Numeral_Ocode__numeral T_a)) V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun _let_1 _let_2) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_OPair T_a _let_1)) (tptp.c_Random_Opick T_a V_xs_2))))))))) (let ((_let_69 (tptp.hAPP (tptp.c_Map_Omap__of tptp.tc_Com_Opname tptp.tc_Com_Ocom) tptp.c_Com_Obodies))) (let ((_let_70 (= tptp.c_Com_Obody _let_69))) (let ((_let_71 (forall ((V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Map_Oran T_b T_a V_m_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 T_a) (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_m_2))) (tptp.c_Option_Ooption_OSome T_a)))))))))) (let ((_let_72 (forall ((V_vs_2 $$unsorted) (V_ks_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (let ((_let_2 (tptp.tc_fun T_a _let_1))) (let ((_let_3 (tptp.tc_fun T_b _let_2))) (let ((_let_4 (tptp.tc_fun T_b _let_1))) (let ((_let_5 (tptp.tc_fun _let_4 _let_3))) (let ((_let_6 (tptp.tc_fun _let_1 _let_2))) (let ((_let_7 (tptp.tc_fun T_a _let_5))) (let ((_let_8 (tptp.tc_fun T_a _let_3))) (let ((_let_9 (tptp.tc_prod T_a T_b))) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_ks_2 V_vs_2) (tptp.hAPP (tptp.c_List_Ofoldl _let_2 _let_9 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_9 _let_2) _let_2) (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_4 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_4 _let_8) _let_2) (tptp.c_COMBC T_a _let_4 _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_6) _let_7 _let_2) (tptp.hAPP (tptp.c_COMBB _let_6 _let_5 T_a) (tptp.c_COMBB _let_1 _let_2 T_b))) (tptp.c_Fun_Ofun__upd T_a _let_1)))) (tptp.c_Option_Ooption_OSome T_b))) V_m_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_ks_2) V_vs_2))))))))))))))) (let ((_let_73 (forall ((V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Orotate T_a V_n_2) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 _let_1) V_n_2) (tptp.c_List_Orotate1 T_a))))))) (let ((_let_74 (forall ((V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Obutlast T_a V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_xs))))) (let ((_let_75 (forall ((V_l_2 $$unsorted)) (= (tptp.c_List_Onat__list V_l_2) (tptp.c_Nat__Transfer_Onat__set (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) V_l_2)))))) (let ((_let_76 (= tptp.c_Com_OWT__bodies (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Com_Opname tptp.tc_Com_Ocom))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) tptp.c_Com_Obodies))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Com_Opname tptp.tc_Com_Ocom tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun tptp.tc_Com_Ocom tptp.tc_HOL_Obool) tptp.tc_Com_Opname) tptp.c_Com_OWT)) B_x)))))))) (let ((_let_77 (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Olists T_a V_A_2) (tptp.hAPP (tptp.c_Set_OCollect _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) _let_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1)) (tptp.c_List_Oset T_a))) V_A_2)))))))) (let ((_let_78 (forall ((V_XS_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun _let_2 _let_2))) (let ((_let_6 (tptp.tc_fun _let_1 _let_1))) (let ((_let_7 (tptp.tc_fun _let_6 _let_2))) (let ((_let_8 (tptp.tc_fun T_a _let_2))) (let ((_let_9 (tptp.tc_fun T_a _let_6))) (let ((_let_10 (tptp.tc_fun _let_1 _let_4))) (let ((_let_11 (tptp.tc_fun T_a _let_10))) (let ((_let_12 (tptp.tc_fun T_a _let_5))) (= (tptp.c_List_Oset__Cons T_a V_A_2 V_XS_2) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 tptp.tc_HOL_Obool _let_1) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 _let_3 _let_1) (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_8 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_8 _let_8) _let_1) (tptp.c_COMBS T_a _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 _let_1) (tptp.hAPP (tptp.c_COMBB _let_10 _let_5 T_a) (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 _let_11 _let_1) (tptp.hAPP (tptp.c_COMBB _let_2 _let_10 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 _let_1) tptp.c_fconj))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_9 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_9 _let_8) _let_1) (tptp.c_COMBB _let_6 _let_2 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_7 _let_1) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool _let_1)) tptp.c_fequal))) (tptp.c_List_Olist_OCons T_a)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_XS_2)))))))))))))))))))))) (let ((_let_79 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat _let_1))) (= (tptp.c_List_Olenlex T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_2 _let_1) (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat _let_1 tptp.c_Wellfounded_Oless__than (tptp.c_List_Olex T_a V_r_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_1 _let_2) _let_1) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat _let_1)) (tptp.c_Nat_Osize__class_Osize _let_1))) (tptp.c_COMBI _let_1))))))))) (let ((_let_80 (forall ((T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Product__Type_Ointernal__split T_a T_b T_c) (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c))))) (let ((_let_81 (forall ((T_a $$unsorted)) (= (tptp.c_Nitpick_Onum T_a) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint T_a (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.c_Nitpick_ORep__Frac T_a)))))) (let ((_let_82 (forall ((T_a $$unsorted)) (= (tptp.c_Nitpick_Odenom T_a) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint T_a (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.c_Nitpick_ORep__Frac T_a)))))) (let ((_let_83 (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Oless__eq__frac T_a V_q_2 V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nitpick_Onum T_a) (tptp.c_Nitpick_Oplus__frac T_a V_q_2 (tptp.c_Nitpick_Ouminus__frac T_a V_r_2)))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))))) (let ((_let_84 (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Oless__frac T_a V_q_2 V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nitpick_Onum T_a) (tptp.c_Nitpick_Oplus__frac T_a V_q_2 (tptp.c_Nitpick_Ouminus__frac T_a V_r_2)))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))))) (let ((_let_85 (forall ((V_q $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Ouminus__frac T_a V_q) (tptp.c_Nitpick_OAbs__Frac T_a (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nitpick_Onum T_a) V_q))) (tptp.hAPP (tptp.c_Nitpick_Odenom T_a) V_q))))))) (let ((_let_86 (forall ((V_q $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Oinverse__frac T_a V_q) (tptp.hAPP (tptp.hAPP (tptp.c_Nitpick_Ofrac T_a) (tptp.hAPP (tptp.c_Nitpick_Odenom T_a) V_q)) (tptp.hAPP (tptp.c_Nitpick_Onum T_a) V_q)))))) (let ((_let_87 (forall ((V_r $$unsorted) (V_q $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Nitpick_Onum T_a))) (= (tptp.c_Nitpick_Otimes__frac T_a V_q V_r) (tptp.hAPP (tptp.hAPP (tptp.c_Nitpick_Ofrac T_a) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_q)) (tptp.hAPP _let_3 V_r))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_q)) (tptp.hAPP _let_1 V_r)))))))))) (let ((_let_88 (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_r_2))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint)))) (let ((_let_4 (tptp.c_Nitpick_Onum T_a))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_6 (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hAPP _let_1 V_q_2))) (= (tptp.c_Nitpick_Oplus__frac T_a V_q_2 V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_a) (tptp.c_Nitpick_Oint__lcm _let_7 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint T_a) tptp.tc_Int_Oint) (tptp.c_Nitpick_Ofrac T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_q_2))) (tptp.hAPP _let_3 _let_7)))) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_r_2))) (tptp.hAPP _let_3 _let_2))))) (tptp.c_COMBI tptp.tc_Int_Oint)))))))))))))) (let ((_let_89 (forall ((T_a $$unsorted)) (= (tptp.c_Nat_Ofunpow T_a) (tptp.c_Nat_Ocompow T_a T_a))))) (let ((_let_90 (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool)) V_R_2)))))))) (let ((_let_91 (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Ototal__on T_a V_A_2 V_r_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (forall ((B_xa $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_A_2)) (=> (not (= B_x B_xa)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_xa)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_xa) B_x)) V_r_2)))))))))))))) (let ((_let_92 (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Osingle__valued T_a T_b V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) B_x) B_y)) V_r_2)) (forall ((B_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) B_x) B_z)) V_r_2)) (= B_y B_z))))))))) (let ((_let_93 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_94 (= tptp.c_FunDef_Opair__leq (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod _let_93 _let_93) tptp.tc_HOL_Obool)) tptp.c_FunDef_Opair__less) (tptp.c_Relation_OId _let_93))))) (let ((_let_95 (forall ((T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (tptp.c_Relation_OId T_a))))) (let ((_let_96 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Otrancl T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_r_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))))) (let ((_let_97 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))))))))) (let ((_let_98 (= tptp.c_Wellfounded_Oless__than (tptp.c_Transitive__Closure_Otrancl tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat)))) (let ((_let_99 (forall ((V_ts_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Hoare__Mirabelle_Ohoare__valids T_a V_Ga_2 V_ts_2) (forall ((B_n $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple T_a)) B_x) V_Ga_2)) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a B_n B_x))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple T_a)) B_x) V_ts_2)) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a B_n B_x))))))))) (let ((_let_100 (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Equiv__Relations_Ocongruent T_a T_b V_r_2 V_f_2) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) B_x) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_2) T_a) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_1 T_a) tptp.c_fequal) V_f_2))) V_f_2)) B_x))))))))))) (let ((_let_101 (forall ((T_a $$unsorted)) (= (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a tptp.tc_HOL_Obool) (tptp.c_Set_OBall T_a))))) (let ((_let_102 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Predicate_Oreflp T_a V_r_2) (tptp.c_Relation_Orefl__on T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) V_r_2))))))) (let ((_let_103 (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) V_A_2)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) V_r_2))))))))) (let ((_let_104 (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_d))) (let ((_let_2 (tptp.tc_fun T_d _let_1))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (let ((_let_4 (tptp.tc_fun T_b T_d))) (= (tptp.c_Product__Type_Omap__pair T_a T_c T_b T_d V_f_2 V_g_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_a) (tptp.c_COMBB T_d _let_1 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_2 T_a) (tptp.c_Product__Type_OPair T_c T_d)) V_f_2))) V_g_2)))))))))) (let ((_let_105 (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (= (tptp.c_Product__Type_Oapfst T_a T_c T_b V_f_2) (tptp.c_Product__Type_Omap__pair T_a T_c T_b T_b V_f_2 (tptp.c_Fun_Oid T_b)))))) (let ((_let_106 (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Oapsnd T_b T_c T_a V_f_2) (tptp.c_Product__Type_Omap__pair T_a T_a T_b T_c (tptp.c_Fun_Oid T_a) V_f_2))))) (let ((_let_107 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Owf_H T_a V_r_2) (and (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_a)) V_r_2)) (tptp.c_Nitpick_Ounknown tptp.tc_HOL_Obool))))))) (let ((_let_108 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_OwfP T_a V_r_2) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) V_r_2))))))) (let ((_let_109 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Olex T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.c_List_Olexn T_a V_r_2))))))) (let ((_let_110 (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_111 (tptp.c_Set_OCollect _let_93))) (let ((_let_112 (= tptp.c_Wellfounded_Opred__nat (tptp.hAPP _let_111 (tptp.hAPP _let_110 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_59 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.c_fequal)) tptp.c_Nat_OSuc)))))) (let ((_let_113 (forall ((V_n $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Onumber__of__frac T_a V_n) (tptp.c_Nitpick_OAbs__Frac T_a (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))) (let ((_let_114 (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (= (tptp.c_Nitpick_Oone__frac T_a) (tptp.c_Nitpick_OAbs__Frac T_a (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_1) _let_1))))))) (let ((_let_115 (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) V_P_2))) (= (tptp.c_FunDef_Oreduction__pair T_a V_P_2) (and (tptp.c_Wellfounded_Owf T_a _let_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) _let_2) (tptp.hAPP (tptp.c_Product__Type_Osnd _let_1 _let_1) V_P_2))) _let_2))))))))) (let ((_let_116 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Owf T_a V_r_2) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.c_Wellfounded_Oacc T_a V_r_2)))))))) (let ((_let_117 (forall ((T_a $$unsorted)) (= (tptp.c_Nitpick_Ozero__frac T_a) (tptp.c_Nitpick_OAbs__Frac T_a (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))) (let ((_let_118 (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (= (tptp.c_Wellfounded_Omax__extp T_a V_R_2 V_a1_2 V_a2_2) (and (tptp.hBOOL (tptp.hAPP _let_1 V_a1_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_a2_2)) (not (= V_a2_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_a1_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_a2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_xa)))))))))))) (let ((_let_119 (tptp.tc_fun _let_93 tptp.tc_HOL_Obool))) (let ((_let_120 (tptp.c_Orderings_Otop__class_Otop _let_119))) (let ((_let_121 (= tptp.c_Int_OInteg (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient _let_93) _let_120) tptp.c_Int_Ointrel)))) (let ((_let_122 (tptp.c_Orderings_Obot__class_Obot _let_119))) (let ((_let_123 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_124 (tptp.c_Set_Oinsert _let_93))) (let ((_let_125 (tptp.c_Relation_OImage _let_93 _let_93 tptp.c_Int_Ointrel))) (let ((_let_126 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_127 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_128 (forall ((V_R_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat T_a))) (= (tptp.c_Wellfounded_Omlex__prod T_a V_f_2 V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_1 T_a) (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat T_a tptp.c_Wellfounded_Oless__than V_R_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun T_a _let_1) T_a) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat T_a)) V_f_2)) (tptp.c_COMBI T_a)))))))) (let ((_let_129 (forall ((T_a $$unsorted)) (= (tptp.c_Wellfounded_Omeasure T_a) (tptp.hAPP (tptp.c_Relation_Oinv__image tptp.tc_Nat_Onat T_a) tptp.c_Wellfounded_Oless__than))))) (let ((_let_130 (= tptp.c_FunDef_Opair__less (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than tptp.c_Wellfounded_Oless__than)))) (let ((_let_131 (tptp.tc_prod _let_119 _let_119))) (let ((_let_132 (tptp.tc_fun _let_131 tptp.tc_HOL_Obool))) (let ((_let_133 (tptp.c_Product__Type_OPair _let_132 _let_132))) (let ((_let_134 (= tptp.c_FunDef_Omin__strict (tptp.c_Wellfounded_Omin__ext _let_93 tptp.c_FunDef_Opair__less)))) (let ((_let_135 (= tptp.c_FunDef_Omax__strict (tptp.c_Wellfounded_Omax__ext _let_93 tptp.c_FunDef_Opair__less)))) (let ((_let_136 (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Relation_OId__on T_a V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_Set_Oinsert _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a _let_1) (tptp.c_Product__Type_OPair T_a T_a)) (tptp.c_COMBI T_a)))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))))) (let ((_let_137 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_131) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_119 _let_119) _let_122) _let_122)) (tptp.c_Orderings_Obot__class_Obot _let_132)))) (let ((_let_138 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_132))) (let ((_let_139 (= tptp.c_FunDef_Omax__weak (tptp.hAPP (tptp.hAPP _let_138 (tptp.c_Wellfounded_Omax__ext _let_93 tptp.c_FunDef_Opair__leq)) _let_137)))) (let ((_let_140 (= tptp.c_FunDef_Omin__weak (tptp.hAPP (tptp.hAPP _let_138 (tptp.c_Wellfounded_Omin__ext _let_93 tptp.c_FunDef_Opair__leq)) _let_137)))) (let ((_let_141 (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Equiv__Relations_Oequivp T_a V_R_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.hAPP V_R_2 B_x))) (= (tptp.hBOOL (tptp.hAPP _let_1 B_y)) (= _let_1 (tptp.hAPP V_R_2 B_y))))))))) (let ((_let_142 (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_1 _let_2))) (let ((_let_4 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Wellfounded_Ofinite__psubset T_a) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_4) _let_1) (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_3 _let_1) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 _let_1) tptp.c_fconj)) (tptp.c_Orderings_Oord__class_Oless _let_1)))) (tptp.c_Finite__Set_Ofinite T_a)))))))))))) (let ((_let_143 (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oinv__image T_a T_b))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b T_a))) (let ((_let_4 (tptp.tc_prod _let_2 _let_2))) (let ((_let_5 (tptp.tc_fun _let_2 _let_4))) (let ((_let_6 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun _let_3 _let_5))) (let ((_let_8 (tptp.tc_fun _let_3 _let_2))) (let ((_let_9 (tptp.tc_fun _let_3 _let_4))) (let ((_let_10 (tptp.tc_fun _let_8 _let_9))) (let ((_let_11 (tptp.tc_fun _let_6 _let_9))) (let ((_let_12 (tptp.tc_fun _let_6 _let_8))) (= (tptp.c_FunDef_Orp__inv__image T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_6 _let_6 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_6 _let_12 _let_11) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 (tptp.tc_fun _let_12 _let_11) _let_6) (tptp.c_COMBB _let_8 _let_9 _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 _let_10 _let_6) (tptp.c_COMBS _let_3 _let_2 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 _let_7 _let_6) (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_3) (tptp.c_Product__Type_OPair _let_2 _let_2))) _let_1)))) _let_1)))))))))))))))))) (let ((_let_144 (forall ((V_p_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.c_Product__Type_Oprod_Oprod__size T_a T_b V_f_2 V_g_2 V_p_2) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) V_p_2))) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_a T_b) V_p_2)))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))) (let ((_let_145 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Oirrefl T_a V_r_2) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) V_r_2)))))))) (let ((_let_146 (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_l))) (let ((_let_3 (tptp.hAPP _let_1 V_k))) (= (tptp.c_Divides_Opdivmod V_k V_l) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_3) _let_2)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_3 _let_2))))))))) (let ((_let_147 (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odivmod__nat V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)))))) (let ((_let_148 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_149 (= tptp.c_Divides_OnegateSnd (tptp.c_Product__Type_Oapsnd tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint _let_148)))) (let ((_let_150 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Orefl_H T_a V_r_2) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) V_r_2))))))) (let ((_let_151 (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.c_Code__Numeral_Odiv__mod__code__numeral V_n V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Code__Numeral_Ocode__numeral V_n V_m)))))) (let ((_let_152 (forall ((V_b_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (let ((_let_2 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_3 (tptp.tc_fun tptp.tc_Int_Oint _let_2))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_3 tptp.tc_Int_Oint) (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint)))) (let ((_let_5 (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint)) V_b_2))) (let ((_let_7 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_8 (tptp.tc_fun _let_2 _let_2))) (let ((_let_9 (tptp.tc_fun tptp.tc_Int_Oint _let_8))) (= (tptp.c_Divides_Oadjust V_b_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_3 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_3 _let_3) tptp.tc_Int_Oint) (tptp.c_COMBS tptp.tc_Int_Oint _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_9 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_2 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_2 _let_8) tptp.tc_Int_Oint) (tptp.c_If _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) _let_6)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_7 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_7 _let_3) tptp.tc_Int_Oint) (tptp.c_COMBB tptp.tc_Int_Oint _let_2 tptp.tc_Int_Oint)) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_7 tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) _let_1)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) _let_6)))) (tptp.hAPP _let_4 _let_1)))))))))))))))) (let ((_let_153 (forall ((V_q $$unsorted) (V_p $$unsorted)) (= (tptp.c_Divides_Odivmod__int V_p V_q) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_p) V_q)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_p V_q)))))) (let ((_let_154 (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_5 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_5 tptp.tc_Int_Oint) tptp.c_fconj))) (let ((_let_7 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.hAPP (tptp.hAPP _let_4 _let_1) V_b_2))) (let ((_let_9 (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint))) (let ((_let_10 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_11 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (let ((_let_12 (tptp.tc_fun tptp.tc_Int_Oint _let_5))) (= (tptp.c_Divides_Odivmod__int__rel V_a_2 V_b_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_11 _let_11) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_11 _let_11) tptp.tc_Int_Oint) _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 tptp.tc_Int_Oint) _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_11 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP tptp.c_fequal V_a_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_10 tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b_2)))))) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP tptp.c_fimplies _let_8)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_2 _let_1))) (tptp.hAPP (tptp.hAPP _let_3 _let_4) V_b_2))))) (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP tptp.c_fimplies (tptp.hAPP tptp.c_fNot _let_8))) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_4 V_b_2))) (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)))))))))))))))))))))) (let ((_let_155 (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.c_Nitpick_Oprod T_a T_b V_A_2 V_B_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_B_2)))))))))) (let ((_let_156 (forall ((V_d_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (= (tptp.c_Int_Oint__ge__less__than2 V_d_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_d_2)))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))))))))) (let ((_let_157 (forall ((V_d_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (= (tptp.c_Int_Oint__ge__less__than V_d_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) tptp.tc_Int_Oint) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_d_2)))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint)))))))))) (let ((_let_158 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (let ((_let_159 (tptp.tc_fun tptp.tc_Int_Oint _let_158))) (let ((_let_160 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_161 (tptp.hAPP _let_160 _let_126))) (let ((_let_162 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_163 (= tptp.c_Nitpick_OFrac (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_158 _let_158 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_162 tptp.tc_Int_Oint) tptp.c_fconj) _let_161))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint _let_158) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_159 _let_159 tptp.tc_Int_Oint) (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_159 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_158 tptp.tc_Int_Oint) tptp.c_fequal)) tptp.c_Nitpick_Oint__gcd))) _let_127)))))) (let ((_let_164 (tptp.c_Int_OBit1 tptp.c_Int_OPls))) (let ((_let_165 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_166 (tptp.hAPP _let_165 tptp.c_Int_OMin))) (let ((_let_167 (= tptp.c_Int_OMin (tptp.hAPP _let_148 _let_127)))) (let ((_let_168 (tptp.hAPP _let_165 tptp.c_Int_OPls))) (let ((_let_169 (tptp.hAPP _let_160 tptp.c_Int_OPls))) (let ((_let_170 (tptp.hAPP _let_160 tptp.c_Int_OMin))) (let ((_let_171 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_172 (tptp.hAPP _let_171 tptp.c_Int_OMin))) (let ((_let_173 (tptp.hAPP _let_171 tptp.c_Int_OPls))) (let ((_let_174 (tptp.hAPP tptp.c_Nat_OSuc _let_31))) (let ((_let_175 (tptp.hAPP tptp.c_Nat_OSuc _let_174))) (let ((_let_176 (tptp.c_Int_OBit0 _let_164))) (let ((_let_177 (tptp.hAPP _let_165 _let_176))) (let ((_let_178 (tptp.hAPP tptp.c_Int_Onat _let_177))) (let ((_let_179 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_180 (tptp.hAPP _let_179 _let_176))) (let ((_let_181 (tptp.hAPP _let_171 _let_126))) (let ((_let_182 (forall ((V_k $$unsorted)) (= (tptp.c_Int_OBit0 V_k) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k) V_k))))) (let ((_let_183 (tptp.c_Int_OBit1 _let_164))) (let ((_let_184 (tptp.hAPP _let_165 _let_183))) (let ((_let_185 (tptp.hAPP _let_179 _let_183))) (let ((_let_186 (tptp.hAPP _let_179 _let_164))) (let ((_let_187 (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc V_k) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))) (let ((_let_188 (forall ((V_k $$unsorted)) (= (tptp.c_Int_Opred V_k) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))) (let ((_let_189 (tptp.hAPP _let_179 tptp.c_Int_OPls))) (let ((_let_190 (= tptp.c_Int_OPls _let_126))) (let ((_let_191 (forall ((V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.c_Int_OBit1 V_k) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_k)) V_k)))))) (let ((_let_192 (forall ((V_y $$unsorted) (V_x $$unsorted)) (= (tptp.c_Nitpick_Onat__lcm V_x V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_x) V_y)) (tptp.c_Nitpick_Onat__gcd V_x V_y)))))) (let ((_let_193 (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (= (tptp.c_Nitpick_Oint__lcm V_x V_y) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.c_Nitpick_Onat__lcm (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_x)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_y))))))))) (let ((_let_194 (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_A_2)) (=> (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_f_2 B_xa)) (= B_x B_xa)))))))))) (let ((_let_195 (forall ((V_x_2 $$unsorted)) (= (tptp.c_Nat__Transfer_Ois__nat V_x_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x_2)))))) (let ((_let_196 (forall ((V_S_2 $$unsorted)) (= (tptp.c_Nat__Transfer_Onat__set V_S_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)))))))) (let ((_let_197 (tptp.c_Orderings_Obot__class_Obot _let_158))) (let ((_let_198 (forall ((V_n $$unsorted) (V_i $$unsorted)) (= (tptp.c_Int_Onat__aux V_i V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_i)) V_n))))) (let ((_let_199 (tptp.c_SetInterval_Oord__class_OatLeast tptp.tc_Nat_Onat))) (let ((_let_200 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_59) _let_60))) (let ((_let_201 (tptp.c_Orderings_Obot__class_Obot _let_59))) (let ((_let_202 (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat))) (let ((_let_203 (tptp.hAPP tptp.c_Int_Onat _let_127))) (let ((_let_204 (tptp.hAPP tptp.c_Int_Onat _let_126))) (let ((_let_205 (tptp.c_SetInterval_Oord__class_OgreaterThan tptp.tc_Nat_Onat))) (let ((_let_206 (tptp.hAPP _let_29 _let_31))) (let ((_let_207 (tptp.hAPP _let_29 _let_62))) (let ((_let_208 (forall ((V_t_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Com_Ostate _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Com_Ostate) tptp.c_fimplies))) (let ((_let_5 (tptp.tc_fun tptp.tc_Com_Ostate _let_3))) (let ((_let_6 (tptp.tc_fun tptp.tc_Com_Ostate _let_5))) (let ((_let_7 (tptp.c_COMBS tptp.tc_Com_Ostate tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_1 _let_1))) (let ((_let_9 (tptp.tc_fun tptp.tc_Com_Ostate _let_8))) (let ((_let_10 (tptp.tc_fun _let_1 _let_2))) (let ((_let_11 (tptp.tc_fun T_a _let_2))) (let ((_let_12 (tptp.tc_fun T_a _let_1))) (let ((_let_13 (tptp.tc_fun _let_12 _let_11))) (let ((_let_14 (tptp.c_HOL_OAll tptp.tc_Com_Ostate))) (let ((_let_15 (tptp.tc_fun _let_12 _let_12))) (let ((_let_16 (tptp.tc_fun T_a _let_8))) (let ((_let_17 (tptp.tc_fun _let_15 _let_15))) (let ((_let_18 (tptp.tc_fun tptp.tc_Com_Ocom _let_15))) (let ((_let_19 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_20 (tptp.tc_fun _let_12 _let_19))) (let ((_let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_20))) (let ((_let_22 (tptp.tc_fun _let_12 tptp.tc_HOL_Obool))) (= (tptp.c_Hoare__Mirabelle_Otriple__valid T_a V_n_2 V_t_2) (tptp.hBOOL (tptp.c_Hoare__Mirabelle_Otriple_Otriple__case T_a tptp.tc_HOL_Obool (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_22) _let_12) (tptp.hAPP (tptp.c_COMBB _let_20 _let_22 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_19 tptp.tc_HOL_Obool _let_12) (tptp.c_HOL_OAll T_a)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_18 _let_21 _let_12) (tptp.hAPP (tptp.c_COMBB _let_15 _let_20 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_12 _let_19 _let_12) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_12 _let_18 _let_18) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_17 (tptp.tc_fun _let_18 _let_18) _let_12) (tptp.c_COMBB _let_15 _let_15 tptp.tc_Com_Ocom)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_15 _let_17 _let_12) (tptp.c_COMBB _let_12 _let_12 _let_12)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_16 _let_15 _let_12) (tptp.c_COMBS T_a _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_5) _let_16 _let_12) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 T_a) _let_7)) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) _let_4)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 _let_15 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 _let_12) (tptp.hAPP (tptp.c_COMBB _let_2 _let_1 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool tptp.tc_Com_Ostate) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_13 tptp.tc_Com_Ocom) (tptp.c_COMBB _let_1 _let_2 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_10 tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_9 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 tptp.tc_Com_Ostate) _let_7)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 tptp.tc_Com_Ostate) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Com_Ocom tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Com_Ostate (tptp.tc_fun tptp.tc_Nat_Onat _let_1)) (tptp.tc_fun tptp.tc_Nat_Onat _let_2) tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate tptp.tc_Nat_Onat _let_1)) tptp.c_Natural_Oevaln)) V_n_2))))))))) V_t_2)))))))))))))))))))))))))))) (let ((_let_209 (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat))) (let ((_let_210 (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT tptp.c_Com_Ocom_OSKIP)))) (let ((_let_211 (forall ((V_X_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Set_Othe__elem T_a V_X_2) (tptp.c_HOL_OThe T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.hAPP tptp.c_fequal V_X_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (let ((_let_212 (forall ((V_Q_2 $$unsorted) (V_Q_H_2 $$unsorted) (V_c_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2) V_c_2))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_Q_H_2)) _let_2)) (=> (forall ((B_Z $$unsorted) (B_s $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_H_2 B_Z) B_s)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 B_Z) B_s)))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_Q_2)) _let_2))))))))))) (let ((_let_213 (forall ((V_c_2 $$unsorted)) (= (tptp.c_Hoare__Mirabelle_OMGT V_c_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple tptp.tc_Com_Ostate) tptp.c_fequal) V_c_2) (tptp.hAPP tptp.c_Natural_Oevalc V_c_2)))))) (let ((_let_214 (forall ((V_s1 $$unsorted) (V_s0 $$unsorted) (V_pn $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_pn))) V_s0) V_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn)) V_s0) V_s1)))))) (let ((_let_215 (forall ((V_s1 $$unsorted) (V_s0 $$unsorted) (V_pn $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of tptp.tc_Com_Opname tptp.tc_Com_Ocom) tptp.c_Com_Obodies) V_pn))) V_s0) V_s1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn)) V_s0) V_s1)))))) (let ((_let_216 (tptp.hAPP _let_5 (tptp.hAPP _let_69 tptp.v_pn)))) (let ((_let_217 (tptp.hAPP tptp.c_Natural_Oevalc _let_216))) (let ((_let_218 (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_217 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_202) SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_203))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_202) SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_203))))) (let ((_let_219 (TRUST_SUBS_EQ (ASSUME :args (_let_213)) :args ((= tptp.c_Hoare__Mirabelle_OMGT (lambda ((BOUND_VARIABLE_129117 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple tptp.tc_Com_Ostate) tptp.c_fequal) BOUND_VARIABLE_129117) (tptp.hAPP tptp.c_Natural_Oevalc BOUND_VARIABLE_129117)))))))) (let ((_let_220 (TRUST_SUBS_EQ (ASSUME :args (_let_211)) :args ((= tptp.c_Set_Othe__elem (lambda ((BOUND_VARIABLE_129141 $$unsorted) (BOUND_VARIABLE_129142 $$unsorted)) (let ((_let_1 (tptp.tc_fun BOUND_VARIABLE_129141 tptp.tc_HOL_Obool))) (tptp.c_HOL_OThe BOUND_VARIABLE_129141 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool BOUND_VARIABLE_129141) (tptp.hAPP tptp.c_fequal BOUND_VARIABLE_129142)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_129141 _let_1 _let_1) (tptp.c_Set_Oinsert BOUND_VARIABLE_129141)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (let ((_let_221 (TRUST_SUBS_EQ (ASSUME :args (_let_208)) :args ((= tptp.c_Hoare__Mirabelle_Otriple__valid (lambda ((BOUND_VARIABLE_129160 $$unsorted) (BOUND_VARIABLE_129161 $$unsorted) (BOUND_VARIABLE_129162 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Com_Ostate _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Com_Ostate) tptp.c_fimplies))) (let ((_let_5 (tptp.tc_fun tptp.tc_Com_Ostate _let_3))) (let ((_let_6 (tptp.tc_fun tptp.tc_Com_Ostate _let_5))) (let ((_let_7 (tptp.c_COMBS tptp.tc_Com_Ostate tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_1 _let_1))) (let ((_let_9 (tptp.tc_fun tptp.tc_Com_Ostate _let_8))) (let ((_let_10 (tptp.tc_fun _let_1 _let_2))) (let ((_let_11 (tptp.tc_fun BOUND_VARIABLE_129160 _let_2))) (let ((_let_12 (tptp.tc_fun BOUND_VARIABLE_129160 _let_1))) (let ((_let_13 (tptp.tc_fun _let_12 _let_11))) (let ((_let_14 (tptp.c_HOL_OAll tptp.tc_Com_Ostate))) (let ((_let_15 (tptp.tc_fun _let_12 _let_12))) (let ((_let_16 (tptp.tc_fun BOUND_VARIABLE_129160 _let_8))) (let ((_let_17 (tptp.tc_fun _let_15 _let_15))) (let ((_let_18 (tptp.tc_fun tptp.tc_Com_Ocom _let_15))) (let ((_let_19 (tptp.tc_fun BOUND_VARIABLE_129160 tptp.tc_HOL_Obool))) (let ((_let_20 (tptp.tc_fun _let_12 _let_19))) (let ((_let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_20))) (let ((_let_22 (tptp.tc_fun _let_12 tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.c_Hoare__Mirabelle_Otriple_Otriple__case BOUND_VARIABLE_129160 tptp.tc_HOL_Obool (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_22) _let_12) (tptp.hAPP (tptp.c_COMBB _let_20 _let_22 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_19 tptp.tc_HOL_Obool _let_12) (tptp.c_HOL_OAll BOUND_VARIABLE_129160)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_18 _let_21 _let_12) (tptp.hAPP (tptp.c_COMBB _let_15 _let_20 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_12 _let_19 _let_12) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool BOUND_VARIABLE_129160) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_12 _let_18 _let_18) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_17 (tptp.tc_fun _let_18 _let_18) _let_12) (tptp.c_COMBB _let_15 _let_15 tptp.tc_Com_Ocom)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_15 _let_17 _let_12) (tptp.c_COMBB _let_12 _let_12 _let_12)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_16 _let_15 _let_12) (tptp.c_COMBS BOUND_VARIABLE_129160 _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun BOUND_VARIABLE_129160 _let_5) _let_16 _let_12) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 BOUND_VARIABLE_129160) _let_7)) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 BOUND_VARIABLE_129160) _let_4)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 _let_15 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 _let_12) (tptp.hAPP (tptp.c_COMBB _let_2 _let_1 BOUND_VARIABLE_129160) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool tptp.tc_Com_Ostate) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_13 tptp.tc_Com_Ocom) (tptp.c_COMBB _let_1 _let_2 BOUND_VARIABLE_129160)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_10 tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_9 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 tptp.tc_Com_Ostate) _let_7)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 tptp.tc_Com_Ostate) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Com_Ocom tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Com_Ostate (tptp.tc_fun tptp.tc_Nat_Onat _let_1)) (tptp.tc_fun tptp.tc_Nat_Onat _let_2) tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate tptp.tc_Nat_Onat _let_1)) tptp.c_Natural_Oevaln)) BOUND_VARIABLE_129161))))))))) BOUND_VARIABLE_129162)))))))))))))))))))))))))))))) (let ((_let_222 (TRUST_SUBS_EQ (ASSUME :args (_let_198)) :args ((= tptp.c_Int_Onat__aux (lambda ((BOUND_VARIABLE_129241 $$unsorted) (BOUND_VARIABLE_129242 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat BOUND_VARIABLE_129241)) BOUND_VARIABLE_129242))))))) (let ((_let_223 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_196)) (MACRO_SR_EQ_INTRO :args (_let_196 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Nat__Transfer_Onat__set (lambda ((BOUND_VARIABLE_129252 $$unsorted)) (forall ((B_x $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) BOUND_VARIABLE_129252))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)))))))))) (let ((_let_224 (TRUST_SUBS_EQ (ASSUME :args (_let_195)) :args ((= tptp.c_Nat__Transfer_Ois__nat (lambda ((BOUND_VARIABLE_129439 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) BOUND_VARIABLE_129439)))))))) (let ((_let_225 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_194)) (MACRO_SR_EQ_INTRO :args (_let_194 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Fun_Oinj__on (lambda ((BOUND_VARIABLE_129521 $$unsorted) (BOUND_VARIABLE_129522 $$unsorted) (BOUND_VARIABLE_129523 $$unsorted) (BOUND_VARIABLE_129524 $$unsorted)) (forall ((B_x $$unsorted) (BOUND_VARIABLE_109330 $$unsorted)) (let ((_let_1 (tptp.c_member BOUND_VARIABLE_129521))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) BOUND_VARIABLE_129524))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 BOUND_VARIABLE_109330) BOUND_VARIABLE_129524))) (not (= (tptp.hAPP BOUND_VARIABLE_129523 B_x) (tptp.hAPP BOUND_VARIABLE_129523 BOUND_VARIABLE_109330))) (= B_x BOUND_VARIABLE_109330)))))))))) (let ((_let_226 (TRUST_SUBS_EQ (ASSUME :args (_let_193)) :args ((= tptp.c_Nitpick_Oint__lcm (lambda ((BOUND_VARIABLE_131156 $$unsorted) (BOUND_VARIABLE_131157 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.c_Nitpick_Onat__lcm (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 BOUND_VARIABLE_131156)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 BOUND_VARIABLE_131157))))))))))) (let ((_let_227 (TRUST_SUBS_EQ (ASSUME :args (_let_192)) :args ((= tptp.c_Nitpick_Onat__lcm (lambda ((BOUND_VARIABLE_131835 $$unsorted) (BOUND_VARIABLE_131836 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) BOUND_VARIABLE_131835) BOUND_VARIABLE_131836)) (tptp.c_Nitpick_Onat__gcd BOUND_VARIABLE_131835 BOUND_VARIABLE_131836)))))))) (let ((_let_228 (TRUST_SUBS_EQ (ASSUME :args (_let_191)) :args ((= tptp.c_Int_OBit1 (lambda ((BOUND_VARIABLE_131946 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) BOUND_VARIABLE_131946)) BOUND_VARIABLE_131946)))))))) (let ((_let_229 (SYMM (SYMM (ASSUME :args (_let_190)))))) (let ((_let_230 (TRUST_SUBS_EQ (ASSUME :args (_let_188)) :args ((= tptp.c_Int_Opred (lambda ((BOUND_VARIABLE_132207 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) BOUND_VARIABLE_132207) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))))) (let ((_let_231 (TRUST_SUBS_EQ (ASSUME :args (_let_187)) :args ((= tptp.c_Int_Osucc (lambda ((BOUND_VARIABLE_132396 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) BOUND_VARIABLE_132396) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))))) (let ((_let_232 (TRUST_SUBS_EQ (ASSUME :args (_let_182)) :args ((= tptp.c_Int_OBit0 (lambda ((BOUND_VARIABLE_133099 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) BOUND_VARIABLE_133099) BOUND_VARIABLE_133099))))))) (let ((_let_233 (SYMM (SYMM (ASSUME :args (_let_167)))))) (let ((_let_234 (ASSUME :args (_let_163)))) (let ((_let_235 (TRUST_SUBS_EQ (ASSUME :args (_let_157)) :args ((= tptp.c_Int_Oint__ge__less__than (lambda ((BOUND_VARIABLE_135213 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) tptp.tc_Int_Oint) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) BOUND_VARIABLE_135213)))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint)))))))))))) (let ((_let_236 (TRUST_SUBS_EQ (ASSUME :args (_let_156)) :args ((= tptp.c_Int_Oint__ge__less__than2 (lambda ((BOUND_VARIABLE_135225 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) BOUND_VARIABLE_135225)))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))))))))))) (let ((_let_237 (TRUST_SUBS_EQ (ASSUME :args (_let_155)) :args ((= tptp.c_Nitpick_Oprod (lambda ((BOUND_VARIABLE_135258 $$unsorted) (BOUND_VARIABLE_135259 $$unsorted) (BOUND_VARIABLE_135260 $$unsorted) (BOUND_VARIABLE_135261 $$unsorted)) (let ((_let_1 (tptp.tc_fun BOUND_VARIABLE_135259 tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod BOUND_VARIABLE_135258 BOUND_VARIABLE_135259)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_135258 BOUND_VARIABLE_135259 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_135258 _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_1) BOUND_VARIABLE_135258) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool BOUND_VARIABLE_135259)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 BOUND_VARIABLE_135258) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_135258 (tptp.tc_fun BOUND_VARIABLE_135258 tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member BOUND_VARIABLE_135258)) BOUND_VARIABLE_135260)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_135259 _let_1 tptp.tc_HOL_Obool) (tptp.c_member BOUND_VARIABLE_135259)) BOUND_VARIABLE_135261)))))))))))) (let ((_let_238 (TRUST_SUBS_EQ (ASSUME :args (_let_154)) :args ((= tptp.c_Divides_Odivmod__int__rel (lambda ((BOUND_VARIABLE_135298 $$unsorted) (BOUND_VARIABLE_135299 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_5 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_5 tptp.tc_Int_Oint) tptp.c_fconj))) (let ((_let_7 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.hAPP (tptp.hAPP _let_4 _let_1) BOUND_VARIABLE_135299))) (let ((_let_9 (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint))) (let ((_let_10 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_11 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (let ((_let_12 (tptp.tc_fun tptp.tc_Int_Oint _let_5))) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_11 _let_11) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_11 _let_11) tptp.tc_Int_Oint) _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 tptp.tc_Int_Oint) _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_11 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP tptp.c_fequal BOUND_VARIABLE_135298))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_10 tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) BOUND_VARIABLE_135299)))))) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP tptp.c_fimplies _let_8)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_2 _let_1))) (tptp.hAPP (tptp.hAPP _let_3 _let_4) BOUND_VARIABLE_135299))))) (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP tptp.c_fimplies (tptp.hAPP tptp.c_fNot _let_8))) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_4 BOUND_VARIABLE_135299))) (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)))))))))))))))))))))))) (let ((_let_239 (TRUST_SUBS_EQ (ASSUME :args (_let_153)) :args ((= tptp.c_Divides_Odivmod__int (lambda ((BOUND_VARIABLE_135737 $$unsorted) (BOUND_VARIABLE_135738 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) BOUND_VARIABLE_135737) BOUND_VARIABLE_135738)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint BOUND_VARIABLE_135737 BOUND_VARIABLE_135738)))))))) (let ((_let_240 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_152)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_152 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Divides_Oadjust (lambda ((BOUND_VARIABLE_136078 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)) _let_1))) (let ((_let_5 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP _let_3 _let_4) _let_4))))) (let ((_let_6 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_7 (tptp.tc_fun tptp.tc_Int_Oint _let_6))) (let ((_let_8 (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_7 tptp.tc_Int_Oint) (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint)))) (let ((_let_9 (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_10 (tptp.hAPP (tptp.hAPP _let_9 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint)) BOUND_VARIABLE_136078))) (let ((_let_11 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_12 (tptp.tc_fun _let_6 _let_6))) (let ((_let_13 (tptp.tc_fun tptp.tc_Int_Oint _let_12))) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_7 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_7 _let_7) tptp.tc_Int_Oint) (tptp.c_COMBS tptp.tc_Int_Oint _let_6 _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 _let_13 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_6 _let_12) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_6 _let_12) tptp.tc_Int_Oint) (tptp.c_If _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1)) _let_10)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_11 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_11 _let_7) tptp.tc_Int_Oint) (tptp.c_COMBB tptp.tc_Int_Oint _let_6 tptp.tc_Int_Oint)) (tptp.hAPP _let_8 (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_11 tptp.tc_Int_Oint) _let_3) _let_5)) _let_2)))) _let_10)))) (tptp.hAPP _let_8 _let_5)))))))))))))))))))))) (let ((_let_241 (TRUST_SUBS_EQ (ASSUME :args (_let_151)) :args ((= tptp.c_Code__Numeral_Odiv__mod__code__numeral (lambda ((BOUND_VARIABLE_136094 $$unsorted) (BOUND_VARIABLE_136095 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Code__Numeral_Ocode__numeral) BOUND_VARIABLE_136094) BOUND_VARIABLE_136095)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Code__Numeral_Ocode__numeral BOUND_VARIABLE_136094 BOUND_VARIABLE_136095)))))))) (let ((_let_242 (TRUST_SUBS_EQ (ASSUME :args (_let_150)) :args ((= tptp.c_Nitpick_Orefl_H (lambda ((BOUND_VARIABLE_136278 $$unsorted) (BOUND_VARIABLE_136279 $$unsorted)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod BOUND_VARIABLE_136278 BOUND_VARIABLE_136278)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair BOUND_VARIABLE_136278 BOUND_VARIABLE_136278) B_x) B_x)) BOUND_VARIABLE_136279))))))))) (let ((_let_243 (ASSUME :args (_let_149)))) (let ((_let_244 (TRUST_SUBS_EQ (ASSUME :args (_let_147)) :args ((= tptp.c_Divides_Odivmod__nat (lambda ((BOUND_VARIABLE_136537 $$unsorted) (BOUND_VARIABLE_136538 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) BOUND_VARIABLE_136537) BOUND_VARIABLE_136538)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat BOUND_VARIABLE_136537 BOUND_VARIABLE_136538)))))))) (let ((_let_245 (TRUST_SUBS_EQ (ASSUME :args (_let_146)) :args ((= tptp.c_Divides_Opdivmod (lambda ((BOUND_VARIABLE_136670 $$unsorted) (BOUND_VARIABLE_136671 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 BOUND_VARIABLE_136671))) (let ((_let_3 (tptp.hAPP _let_1 BOUND_VARIABLE_136670))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_3) _let_2)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_3 _let_2))))))))))) (let ((_let_246 (TRUST_SUBS_EQ (ASSUME :args (_let_145)) :args ((= tptp.c_Relation_Oirrefl (lambda ((BOUND_VARIABLE_136941 $$unsorted) (BOUND_VARIABLE_136942 $$unsorted)) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod BOUND_VARIABLE_136941 BOUND_VARIABLE_136941)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair BOUND_VARIABLE_136941 BOUND_VARIABLE_136941) B_x) B_x)) BOUND_VARIABLE_136942)))))))))) (let ((_let_247 (TRUST_SUBS_EQ (ASSUME :args (_let_144)) :args ((= tptp.c_Product__Type_Oprod_Oprod__size (lambda ((BOUND_VARIABLE_136988 $$unsorted) (BOUND_VARIABLE_136989 $$unsorted) (BOUND_VARIABLE_136990 $$unsorted) (BOUND_VARIABLE_136991 $$unsorted) (BOUND_VARIABLE_136992 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP BOUND_VARIABLE_136990 (tptp.hAPP (tptp.c_Product__Type_Ofst BOUND_VARIABLE_136988 BOUND_VARIABLE_136989) BOUND_VARIABLE_136992))) (tptp.hAPP BOUND_VARIABLE_136991 (tptp.hAPP (tptp.c_Product__Type_Osnd BOUND_VARIABLE_136988 BOUND_VARIABLE_136989) BOUND_VARIABLE_136992)))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))) (let ((_let_248 (TRUST_SUBS_EQ (ASSUME :args (_let_143)) :args ((= tptp.c_FunDef_Orp__inv__image (lambda ((BOUND_VARIABLE_137039 $$unsorted) (BOUND_VARIABLE_137040 $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oinv__image BOUND_VARIABLE_137039 BOUND_VARIABLE_137040))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod BOUND_VARIABLE_137040 BOUND_VARIABLE_137040) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun BOUND_VARIABLE_137040 BOUND_VARIABLE_137039))) (let ((_let_4 (tptp.tc_prod _let_2 _let_2))) (let ((_let_5 (tptp.tc_fun _let_2 _let_4))) (let ((_let_6 (tptp.tc_fun (tptp.tc_prod BOUND_VARIABLE_137039 BOUND_VARIABLE_137039) tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun _let_3 _let_5))) (let ((_let_8 (tptp.tc_fun _let_3 _let_2))) (let ((_let_9 (tptp.tc_fun _let_3 _let_4))) (let ((_let_10 (tptp.tc_fun _let_8 _let_9))) (let ((_let_11 (tptp.tc_fun _let_6 _let_9))) (let ((_let_12 (tptp.tc_fun _let_6 _let_8))) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_6 _let_6 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_6 _let_12 _let_11) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 (tptp.tc_fun _let_12 _let_11) _let_6) (tptp.c_COMBB _let_8 _let_9 _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 _let_10 _let_6) (tptp.c_COMBS _let_3 _let_2 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 _let_7 _let_6) (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_3) (tptp.c_Product__Type_OPair _let_2 _let_2))) _let_1)))) _let_1)))))))))))))))))))) (let ((_let_249 (TRUST_SUBS_EQ (ASSUME :args (_let_142)) :args ((= tptp.c_Wellfounded_Ofinite__psubset (lambda ((BOUND_VARIABLE_137081 $$unsorted)) (let ((_let_1 (tptp.tc_fun BOUND_VARIABLE_137081 tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_1 _let_2))) (let ((_let_4 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_4) _let_1) (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_3 _let_1) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 _let_1) tptp.c_fconj)) (tptp.c_Orderings_Oord__class_Oless _let_1)))) (tptp.c_Finite__Set_Ofinite BOUND_VARIABLE_137081)))))))))))))) (let ((_let_250 (TRUST_SUBS_EQ (ASSUME :args (_let_141)) :args ((= tptp.c_Equiv__Relations_Oequivp (lambda ((BOUND_VARIABLE_137109 $$unsorted) (BOUND_VARIABLE_137110 $$unsorted)) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.hAPP BOUND_VARIABLE_137110 B_x))) (= (tptp.hBOOL (tptp.hAPP _let_1 B_y)) (= _let_1 (tptp.hAPP BOUND_VARIABLE_137110 B_y))))))))))) (let ((_let_251 (ASSUME :args (_let_140)))) (let ((_let_252 (ASSUME :args (_let_139)))) (let ((_let_253 (TRUST_SUBS_EQ (ASSUME :args (_let_136)) :args ((= tptp.c_Relation_OId__on (lambda ((BOUND_VARIABLE_137175 $$unsorted) (BOUND_VARIABLE_137176 $$unsorted)) (let ((_let_1 (tptp.tc_prod BOUND_VARIABLE_137175 BOUND_VARIABLE_137175))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR BOUND_VARIABLE_137175 _let_2) BOUND_VARIABLE_137176) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_137175 _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) BOUND_VARIABLE_137175) (tptp.c_Set_Oinsert _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS BOUND_VARIABLE_137175 BOUND_VARIABLE_137175 _let_1) (tptp.c_Product__Type_OPair BOUND_VARIABLE_137175 BOUND_VARIABLE_137175)) (tptp.c_COMBI BOUND_VARIABLE_137175)))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))))))) (let ((_let_254 (ASSUME :args (_let_135)))) (let ((_let_255 (ASSUME :args (_let_134)))) (let ((_let_256 (ASSUME :args (_let_130)))) (let ((_let_257 (TRUST_SUBS_EQ (ASSUME :args (_let_129)) :args ((= tptp.c_Wellfounded_Omeasure (lambda ((BOUND_VARIABLE_137390 $$unsorted)) (tptp.hAPP (tptp.c_Relation_Oinv__image tptp.tc_Nat_Onat BOUND_VARIABLE_137390) tptp.c_Wellfounded_Oless__than))))))) (let ((_let_258 (TRUST_SUBS_EQ (ASSUME :args (_let_128)) :args ((= tptp.c_Wellfounded_Omlex__prod (lambda ((BOUND_VARIABLE_137397 $$unsorted) (BOUND_VARIABLE_137398 $$unsorted) (BOUND_VARIABLE_137399 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat BOUND_VARIABLE_137397))) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_1 BOUND_VARIABLE_137397) (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat BOUND_VARIABLE_137397 tptp.c_Wellfounded_Oless__than BOUND_VARIABLE_137399)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS BOUND_VARIABLE_137397 BOUND_VARIABLE_137397 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun BOUND_VARIABLE_137397 _let_1) BOUND_VARIABLE_137397) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat BOUND_VARIABLE_137397)) BOUND_VARIABLE_137398)) (tptp.c_COMBI BOUND_VARIABLE_137397)))))))))) (let ((_let_259 (ASSUME :args (_let_121)))) (let ((_let_260 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_118)) (MACRO_SR_EQ_INTRO :args (_let_118 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Wellfounded_Omax__extp (lambda ((BOUND_VARIABLE_137463 $$unsorted) (BOUND_VARIABLE_137464 $$unsorted) (BOUND_VARIABLE_137465 $$unsorted) (BOUND_VARIABLE_137466 $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite BOUND_VARIABLE_137463))) (and (tptp.hBOOL (tptp.hAPP _let_1 BOUND_VARIABLE_137465)) (tptp.hBOOL (tptp.hAPP _let_1 BOUND_VARIABLE_137466)) (not (= BOUND_VARIABLE_137466 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun BOUND_VARIABLE_137463 tptp.tc_HOL_Obool)))) (forall ((B_x $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_137463) B_x) BOUND_VARIABLE_137465))) (not (forall ((B_xa $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_137463) B_xa) BOUND_VARIABLE_137466))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP BOUND_VARIABLE_137464 B_x) B_xa)))))))))))))))) (let ((_let_261 (TRUST_SUBS_EQ (ASSUME :args (_let_117)) :args ((= tptp.c_Nitpick_Ozero__frac (lambda ((BOUND_VARIABLE_137501 $$unsorted)) (tptp.c_Nitpick_OAbs__Frac BOUND_VARIABLE_137501 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))) (let ((_let_262 (TRUST_SUBS_EQ (ASSUME :args (_let_116)) :args ((= tptp.c_Wellfounded_Owf (lambda ((BOUND_VARIABLE_137595 $$unsorted) (BOUND_VARIABLE_137596 $$unsorted)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_137595) B_x) (tptp.c_Wellfounded_Oacc BOUND_VARIABLE_137595 BOUND_VARIABLE_137596)))))))))) (let ((_let_263 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_115)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_115 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_FunDef_Oreduction__pair (lambda ((BOUND_VARIABLE_137742 $$unsorted) (BOUND_VARIABLE_137743 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod BOUND_VARIABLE_137742 BOUND_VARIABLE_137742) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) BOUND_VARIABLE_137743))) (and (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod BOUND_VARIABLE_137742 BOUND_VARIABLE_137742) tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_137742) B_x) (tptp.c_Wellfounded_Oacc BOUND_VARIABLE_137742 (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) BOUND_VARIABLE_137743)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp BOUND_VARIABLE_137742 BOUND_VARIABLE_137742 BOUND_VARIABLE_137742) _let_2) (tptp.hAPP (tptp.c_Product__Type_Osnd _let_1 _let_1) BOUND_VARIABLE_137743))) _let_2))))))))))) (let ((_let_264 (TRUST_SUBS_EQ (ASSUME :args (_let_114)) :args ((= tptp.c_Nitpick_Oone__frac (lambda ((BOUND_VARIABLE_137815 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (tptp.c_Nitpick_OAbs__Frac BOUND_VARIABLE_137815 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_1) _let_1))))))))) (let ((_let_265 (TRUST_SUBS_EQ (ASSUME :args (_let_113)) :args ((= tptp.c_Nitpick_Onumber__of__frac (lambda ((BOUND_VARIABLE_137821 $$unsorted) (BOUND_VARIABLE_137822 $$unsorted)) (tptp.c_Nitpick_OAbs__Frac BOUND_VARIABLE_137821 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) BOUND_VARIABLE_137822) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))) (let ((_let_266 (ASSUME :args (_let_112)))) (let ((_let_267 (TRUST_SUBS_EQ (ASSUME :args (_let_109)) :args ((= tptp.c_List_Olex (lambda ((BOUND_VARIABLE_137987 $$unsorted) (BOUND_VARIABLE_137988 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist BOUND_VARIABLE_137987))) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.c_List_Olexn BOUND_VARIABLE_137987 BOUND_VARIABLE_137988))))))))) (let ((_let_268 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_108)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_108 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Wellfounded_OwfP (lambda ((BOUND_VARIABLE_138014 $$unsorted) (BOUND_VARIABLE_138015 $$unsorted)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_138014) B_x) (tptp.c_Wellfounded_Oacc BOUND_VARIABLE_138014 (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod BOUND_VARIABLE_138014 BOUND_VARIABLE_138014)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_138014 BOUND_VARIABLE_138014 tptp.tc_HOL_Obool) BOUND_VARIABLE_138015)))))))))))) (let ((_let_269 (TRUST_SUBS_EQ (ASSUME :args (_let_107)) :args ((= tptp.c_Nitpick_Owf_H (lambda ((BOUND_VARIABLE_138470 $$unsorted) (BOUND_VARIABLE_138471 $$unsorted)) (and (tptp.c_Wellfounded_Oacyclic BOUND_VARIABLE_138470 BOUND_VARIABLE_138471) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod BOUND_VARIABLE_138470 BOUND_VARIABLE_138470)) BOUND_VARIABLE_138471)) (tptp.c_Nitpick_Ounknown tptp.tc_HOL_Obool))))))))) (let ((_let_270 (TRUST_SUBS_EQ (ASSUME :args (_let_106)) :args ((= tptp.c_Product__Type_Oapsnd (lambda ((BOUND_VARIABLE_138564 $$unsorted) (BOUND_VARIABLE_138565 $$unsorted) (BOUND_VARIABLE_138566 $$unsorted) (BOUND_VARIABLE_138567 $$unsorted)) (tptp.c_Product__Type_Omap__pair BOUND_VARIABLE_138566 BOUND_VARIABLE_138566 BOUND_VARIABLE_138564 BOUND_VARIABLE_138565 (tptp.c_Fun_Oid BOUND_VARIABLE_138566) BOUND_VARIABLE_138567))))))) (let ((_let_271 (TRUST_SUBS_EQ (ASSUME :args (_let_105)) :args ((= tptp.c_Product__Type_Oapfst (lambda ((BOUND_VARIABLE_138580 $$unsorted) (BOUND_VARIABLE_138581 $$unsorted) (BOUND_VARIABLE_138582 $$unsorted) (BOUND_VARIABLE_138583 $$unsorted)) (tptp.c_Product__Type_Omap__pair BOUND_VARIABLE_138580 BOUND_VARIABLE_138581 BOUND_VARIABLE_138582 BOUND_VARIABLE_138582 BOUND_VARIABLE_138583 (tptp.c_Fun_Oid BOUND_VARIABLE_138582)))))))) (let ((_let_272 (TRUST_SUBS_EQ (ASSUME :args (_let_104)) :args ((= tptp.c_Product__Type_Omap__pair (lambda ((BOUND_VARIABLE_138596 $$unsorted) (BOUND_VARIABLE_138597 $$unsorted) (BOUND_VARIABLE_138598 $$unsorted) (BOUND_VARIABLE_138599 $$unsorted) (BOUND_VARIABLE_138600 $$unsorted) (BOUND_VARIABLE_138601 $$unsorted)) (let ((_let_1 (tptp.tc_prod BOUND_VARIABLE_138597 BOUND_VARIABLE_138599))) (let ((_let_2 (tptp.tc_fun BOUND_VARIABLE_138599 _let_1))) (let ((_let_3 (tptp.tc_fun BOUND_VARIABLE_138598 _let_1))) (let ((_let_4 (tptp.tc_fun BOUND_VARIABLE_138598 BOUND_VARIABLE_138599))) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_138596 BOUND_VARIABLE_138598 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_138596 _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) BOUND_VARIABLE_138596) (tptp.c_COMBB BOUND_VARIABLE_138599 _let_1 BOUND_VARIABLE_138598)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB BOUND_VARIABLE_138597 _let_2 BOUND_VARIABLE_138596) (tptp.c_Product__Type_OPair BOUND_VARIABLE_138597 BOUND_VARIABLE_138599)) BOUND_VARIABLE_138600))) BOUND_VARIABLE_138601)))))))))))) (let ((_let_273 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_103)) (MACRO_SR_EQ_INTRO :args (_let_103 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Relation_Orefl__on (lambda ((BOUND_VARIABLE_138763 $$unsorted) (BOUND_VARIABLE_138764 $$unsorted) (BOUND_VARIABLE_138765 $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod BOUND_VARIABLE_138763 BOUND_VARIABLE_138763) tptp.tc_HOL_Obool)) BOUND_VARIABLE_138765) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma BOUND_VARIABLE_138763 BOUND_VARIABLE_138763) BOUND_VARIABLE_138764) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun BOUND_VARIABLE_138763 tptp.tc_HOL_Obool) BOUND_VARIABLE_138763) BOUND_VARIABLE_138764)))) (forall ((B_x $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_138763) B_x) BOUND_VARIABLE_138764))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod BOUND_VARIABLE_138763 BOUND_VARIABLE_138763)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair BOUND_VARIABLE_138763 BOUND_VARIABLE_138763) B_x) B_x)) BOUND_VARIABLE_138765))))))))))) (let ((_let_274 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_102)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_102 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Predicate_Oreflp (lambda ((BOUND_VARIABLE_139312 $$unsorted) (BOUND_VARIABLE_139313 $$unsorted)) (let ((_let_1 (tptp.tc_fun BOUND_VARIABLE_139312 tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (let ((_let_3 (tptp.tc_prod BOUND_VARIABLE_139312 BOUND_VARIABLE_139312))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect _let_3) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_139312 BOUND_VARIABLE_139312 tptp.tc_HOL_Obool) BOUND_VARIABLE_139313))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma BOUND_VARIABLE_139312 BOUND_VARIABLE_139312) _let_2) (tptp.hAPP (tptp.c_COMBK _let_1 BOUND_VARIABLE_139312) _let_2)))) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod BOUND_VARIABLE_139312 BOUND_VARIABLE_139312))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_139312) B_x) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun BOUND_VARIABLE_139312 tptp.tc_HOL_Obool))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair BOUND_VARIABLE_139312 BOUND_VARIABLE_139312) B_x) B_x)) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_139312 BOUND_VARIABLE_139312 tptp.tc_HOL_Obool) BOUND_VARIABLE_139313))))))))))))))))) (let ((_let_275 (TRUST_SUBS_EQ (ASSUME :args (_let_101)) :args ((= tptp.c_Set_OBall (lambda ((BOUND_VARIABLE_139480 $$unsorted)) (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI BOUND_VARIABLE_139480 tptp.tc_HOL_Obool))))))) (let ((_let_276 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_100)) (MACRO_SR_EQ_INTRO :args (_let_100 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Equiv__Relations_Ocongruent (lambda ((BOUND_VARIABLE_139486 $$unsorted) (BOUND_VARIABLE_139487 $$unsorted) (BOUND_VARIABLE_139488 $$unsorted) (BOUND_VARIABLE_139489 $$unsorted)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun BOUND_VARIABLE_139487 tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun BOUND_VARIABLE_139486 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun BOUND_VARIABLE_139486 BOUND_VARIABLE_139487))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod BOUND_VARIABLE_139486 BOUND_VARIABLE_139486)) B_x) BOUND_VARIABLE_139488))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_139486 BOUND_VARIABLE_139486 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_139486 _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_2) BOUND_VARIABLE_139486) (tptp.c_COMBB BOUND_VARIABLE_139487 tptp.tc_HOL_Obool BOUND_VARIABLE_139486)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB BOUND_VARIABLE_139487 _let_1 BOUND_VARIABLE_139486) tptp.c_fequal) BOUND_VARIABLE_139489))) BOUND_VARIABLE_139489)) B_x))))))))))))) (let ((_let_277 (TRUST_SUBS_EQ (EQ_RESOLVE (EQ_RESOLVE (ASSUME :args (_let_99)) (MACRO_SR_EQ_INTRO :args (_let_99 SB_DEFAULT SBA_FIXPOINT))) (MACRO_SR_EQ_INTRO (AND_INTRO _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args ((forall ((V_ts_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Hoare__Mirabelle_Ohoare__valids T_a V_Ga_2 V_ts_2) (forall ((B_n $$unsorted) (BOUND_VARIABLE_121000 $$unsorted)) (or (not (forall ((B_x $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple T_a)) B_x) V_Ga_2))) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a B_n B_x)))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple T_a)) BOUND_VARIABLE_121000) V_ts_2))) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a B_n BOUND_VARIABLE_121000))))) SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Hoare__Mirabelle_Ohoare__valids (lambda ((BOUND_VARIABLE_139755 $$unsorted) (BOUND_VARIABLE_139756 $$unsorted) (BOUND_VARIABLE_139757 $$unsorted)) (forall ((B_n $$unsorted) (BOUND_VARIABLE_121000 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Com_Ostate _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Com_Ostate) tptp.c_fimplies))) (let ((_let_5 (tptp.tc_fun tptp.tc_Com_Ostate _let_3))) (let ((_let_6 (tptp.tc_fun tptp.tc_Com_Ostate _let_5))) (let ((_let_7 (tptp.c_COMBS tptp.tc_Com_Ostate tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_1 _let_1))) (let ((_let_9 (tptp.tc_fun tptp.tc_Com_Ostate _let_8))) (let ((_let_10 (tptp.tc_fun _let_1 _let_2))) (let ((_let_11 (tptp.tc_fun BOUND_VARIABLE_139755 _let_2))) (let ((_let_12 (tptp.tc_fun BOUND_VARIABLE_139755 _let_1))) (let ((_let_13 (tptp.tc_fun _let_12 _let_11))) (let ((_let_14 (tptp.c_HOL_OAll tptp.tc_Com_Ostate))) (let ((_let_15 (tptp.tc_fun _let_12 _let_12))) (let ((_let_16 (tptp.tc_fun BOUND_VARIABLE_139755 _let_8))) (let ((_let_17 (tptp.tc_fun _let_15 _let_15))) (let ((_let_18 (tptp.tc_fun tptp.tc_Com_Ocom _let_15))) (let ((_let_19 (tptp.tc_fun BOUND_VARIABLE_139755 tptp.tc_HOL_Obool))) (let ((_let_20 (tptp.tc_fun _let_12 _let_19))) (let ((_let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_20))) (let ((_let_22 (tptp.tc_fun _let_12 tptp.tc_HOL_Obool))) (or (not (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Com_Ostate _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Com_Ostate) tptp.c_fimplies))) (let ((_let_5 (tptp.tc_fun tptp.tc_Com_Ostate _let_3))) (let ((_let_6 (tptp.tc_fun tptp.tc_Com_Ostate _let_5))) (let ((_let_7 (tptp.c_COMBS tptp.tc_Com_Ostate tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_1 _let_1))) (let ((_let_9 (tptp.tc_fun tptp.tc_Com_Ostate _let_8))) (let ((_let_10 (tptp.tc_fun _let_1 _let_2))) (let ((_let_11 (tptp.tc_fun BOUND_VARIABLE_139755 _let_2))) (let ((_let_12 (tptp.tc_fun BOUND_VARIABLE_139755 _let_1))) (let ((_let_13 (tptp.tc_fun _let_12 _let_11))) (let ((_let_14 (tptp.c_HOL_OAll tptp.tc_Com_Ostate))) (let ((_let_15 (tptp.tc_fun _let_12 _let_12))) (let ((_let_16 (tptp.tc_fun BOUND_VARIABLE_139755 _let_8))) (let ((_let_17 (tptp.tc_fun _let_15 _let_15))) (let ((_let_18 (tptp.tc_fun tptp.tc_Com_Ocom _let_15))) (let ((_let_19 (tptp.tc_fun BOUND_VARIABLE_139755 tptp.tc_HOL_Obool))) (let ((_let_20 (tptp.tc_fun _let_12 _let_19))) (let ((_let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_20))) (let ((_let_22 (tptp.tc_fun _let_12 tptp.tc_HOL_Obool))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple BOUND_VARIABLE_139755)) B_x) BOUND_VARIABLE_139756))) (tptp.hBOOL (tptp.c_Hoare__Mirabelle_Otriple_Otriple__case BOUND_VARIABLE_139755 tptp.tc_HOL_Obool (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_22) _let_12) (tptp.hAPP (tptp.c_COMBB _let_20 _let_22 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_19 tptp.tc_HOL_Obool _let_12) (tptp.c_HOL_OAll BOUND_VARIABLE_139755)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_18 _let_21 _let_12) (tptp.hAPP (tptp.c_COMBB _let_15 _let_20 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_12 _let_19 _let_12) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool BOUND_VARIABLE_139755) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_12 _let_18 _let_18) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_17 (tptp.tc_fun _let_18 _let_18) _let_12) (tptp.c_COMBB _let_15 _let_15 tptp.tc_Com_Ocom)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_15 _let_17 _let_12) (tptp.c_COMBB _let_12 _let_12 _let_12)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_16 _let_15 _let_12) (tptp.c_COMBS BOUND_VARIABLE_139755 _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun BOUND_VARIABLE_139755 _let_5) _let_16 _let_12) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 BOUND_VARIABLE_139755) _let_7)) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 BOUND_VARIABLE_139755) _let_4)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 _let_15 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 _let_12) (tptp.hAPP (tptp.c_COMBB _let_2 _let_1 BOUND_VARIABLE_139755) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool tptp.tc_Com_Ostate) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_13 tptp.tc_Com_Ocom) (tptp.c_COMBB _let_1 _let_2 BOUND_VARIABLE_139755)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_10 tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_9 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 tptp.tc_Com_Ostate) _let_7)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 tptp.tc_Com_Ostate) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Com_Ocom tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Com_Ostate (tptp.tc_fun tptp.tc_Nat_Onat _let_1)) (tptp.tc_fun tptp.tc_Nat_Onat _let_2) tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate tptp.tc_Nat_Onat _let_1)) tptp.c_Natural_Oevaln)) B_n))))))))) B_x))))))))))))))))))))))))))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple BOUND_VARIABLE_139755)) BOUND_VARIABLE_121000) BOUND_VARIABLE_139757))) (tptp.hBOOL (tptp.c_Hoare__Mirabelle_Otriple_Otriple__case BOUND_VARIABLE_139755 tptp.tc_HOL_Obool (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_21 (tptp.tc_fun tptp.tc_Com_Ocom _let_22) _let_12) (tptp.hAPP (tptp.c_COMBB _let_20 _let_22 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_19 tptp.tc_HOL_Obool _let_12) (tptp.c_HOL_OAll BOUND_VARIABLE_139755)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_18 _let_21 _let_12) (tptp.hAPP (tptp.c_COMBB _let_15 _let_20 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_12 _let_19 _let_12) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool BOUND_VARIABLE_139755) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_12 _let_18 _let_18) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_17 (tptp.tc_fun _let_18 _let_18) _let_12) (tptp.c_COMBB _let_15 _let_15 tptp.tc_Com_Ocom)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_15 _let_17 _let_12) (tptp.c_COMBB _let_12 _let_12 _let_12)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_16 _let_15 _let_12) (tptp.c_COMBS BOUND_VARIABLE_139755 _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun BOUND_VARIABLE_139755 _let_5) _let_16 _let_12) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 BOUND_VARIABLE_139755) _let_7)) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 BOUND_VARIABLE_139755) _let_4)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 _let_15 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 _let_12) (tptp.hAPP (tptp.c_COMBB _let_2 _let_1 BOUND_VARIABLE_139755) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool tptp.tc_Com_Ostate) _let_14)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_13 tptp.tc_Com_Ocom) (tptp.c_COMBB _let_1 _let_2 BOUND_VARIABLE_139755)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_10 tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_9 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 tptp.tc_Com_Ostate) _let_7)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 tptp.tc_Com_Ocom) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 tptp.tc_Com_Ostate) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Com_Ocom tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Com_Ostate (tptp.tc_fun tptp.tc_Nat_Onat _let_1)) (tptp.tc_fun tptp.tc_Nat_Onat _let_2) tptp.tc_Com_Ocom) (tptp.c_COMBC tptp.tc_Com_Ostate tptp.tc_Nat_Onat _let_1)) tptp.c_Natural_Oevaln)) B_n))))))))) BOUND_VARIABLE_121000)))))))))))))))))))))))))))))))) (let ((_let_278 (EQ_RESOLVE (ASSUME :args (_let_98)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_98 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_279 (TRUST_SUBS_EQ (ASSUME :args (_let_97)) :args ((= tptp.c_Wellfounded_Oacyclic (lambda ((BOUND_VARIABLE_139942 $$unsorted) (BOUND_VARIABLE_139943 $$unsorted)) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod BOUND_VARIABLE_139942 BOUND_VARIABLE_139942)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair BOUND_VARIABLE_139942 BOUND_VARIABLE_139942) B_x) B_x)) (tptp.c_Transitive__Closure_Otrancl BOUND_VARIABLE_139942 BOUND_VARIABLE_139943))))))))))) (let ((_let_280 (TRUST_SUBS_EQ (ASSUME :args (_let_96)) :args ((= tptp.c_Transitive__Closure_Otrancl (lambda ((BOUND_VARIABLE_139961 $$unsorted) (BOUND_VARIABLE_139962 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp BOUND_VARIABLE_139961 BOUND_VARIABLE_139961 BOUND_VARIABLE_139961) BOUND_VARIABLE_139962) (tptp.c_Transitive__Closure_Ortrancl BOUND_VARIABLE_139961 BOUND_VARIABLE_139962)))))))) (let ((_let_281 (TRUST_SUBS_EQ (ASSUME :args (_let_95)) :args ((= tptp.c_Relation_OId (lambda ((BOUND_VARIABLE_140107 $$unsorted)) (tptp.c_Transitive__Closure_Ortrancl BOUND_VARIABLE_140107 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod BOUND_VARIABLE_140107 BOUND_VARIABLE_140107) tptp.tc_HOL_Obool))))))))) (let ((_let_282 (EQ_RESOLVE (ASSUME :args (_let_94)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_94 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_283 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_92)) (MACRO_SR_EQ_INTRO :args (_let_92 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Relation_Osingle__valued (lambda ((BOUND_VARIABLE_140945 $$unsorted) (BOUND_VARIABLE_140946 $$unsorted) (BOUND_VARIABLE_140947 $$unsorted)) (forall ((B_x $$unsorted) (B_y $$unsorted) (BOUND_VARIABLE_121895 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OPair BOUND_VARIABLE_140945 BOUND_VARIABLE_140946) B_x))) (let ((_let_2 (tptp.c_member (tptp.tc_prod BOUND_VARIABLE_140945 BOUND_VARIABLE_140946)))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 B_y)) BOUND_VARIABLE_140947))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 BOUND_VARIABLE_121895)) BOUND_VARIABLE_140947))) (= B_y BOUND_VARIABLE_121895))))))))))) (let ((_let_284 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_91)) (MACRO_SR_EQ_INTRO :args (_let_91 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Relation_Ototal__on (lambda ((BOUND_VARIABLE_140972 $$unsorted) (BOUND_VARIABLE_140973 $$unsorted) (BOUND_VARIABLE_140974 $$unsorted)) (forall ((B_x $$unsorted) (BOUND_VARIABLE_121921 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair BOUND_VARIABLE_140972 BOUND_VARIABLE_140972))) (let ((_let_2 (tptp.c_member (tptp.tc_prod BOUND_VARIABLE_140972 BOUND_VARIABLE_140972)))) (let ((_let_3 (tptp.c_member BOUND_VARIABLE_140972))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 B_x) BOUND_VARIABLE_140973))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 BOUND_VARIABLE_121921) BOUND_VARIABLE_140973))) (= B_x BOUND_VARIABLE_121921) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) BOUND_VARIABLE_121921)) BOUND_VARIABLE_140974)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 BOUND_VARIABLE_121921) B_x)) BOUND_VARIABLE_140974))))))))))))) (let ((_let_285 (TRUST_SUBS_EQ (ASSUME :args (_let_90)) :args ((= tptp.c_Transitive__Closure_Ortrancl (lambda ((BOUND_VARIABLE_141208 $$unsorted) (BOUND_VARIABLE_141209 $$unsorted)) (let ((_let_1 (tptp.tc_prod BOUND_VARIABLE_141208 BOUND_VARIABLE_141208))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool)) BOUND_VARIABLE_141209)))))))))) (let ((_let_286 (TRUST_SUBS_EQ (ASSUME :args (_let_89)) :args ((= tptp.c_Nat_Ofunpow (lambda ((BOUND_VARIABLE_141225 $$unsorted)) (tptp.c_Nat_Ocompow BOUND_VARIABLE_141225 BOUND_VARIABLE_141225))))))) (let ((_let_287 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_88)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_88 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Nitpick_Oplus__frac (lambda ((BOUND_VARIABLE_141445 $$unsorted) (BOUND_VARIABLE_141446 $$unsorted) (BOUND_VARIABLE_141447 $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom BOUND_VARIABLE_141445))) (let ((_let_2 (tptp.hAPP _let_1 BOUND_VARIABLE_141447))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint)))) (let ((_let_4 (tptp.c_Nitpick_Onum BOUND_VARIABLE_141445))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_6 (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hAPP _let_1 BOUND_VARIABLE_141446))) (let ((_let_8 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (let ((_let_9 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_8 _let_2)))) (let ((_let_10 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_8 _let_7)))) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint BOUND_VARIABLE_141445) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_10) _let_9)) (tptp.c_Nitpick_Onat__gcd _let_10 _let_9)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint BOUND_VARIABLE_141445) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint BOUND_VARIABLE_141445) tptp.tc_Int_Oint) (tptp.c_Nitpick_Ofrac BOUND_VARIABLE_141445)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_4 BOUND_VARIABLE_141446))) (tptp.hAPP _let_3 _let_7)))) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_4 BOUND_VARIABLE_141447))) (tptp.hAPP _let_3 _let_2))))) (tptp.c_COMBI tptp.tc_Int_Oint))))))))))))))))))) (let ((_let_288 (TRUST_SUBS_EQ (ASSUME :args (_let_87)) :args ((= tptp.c_Nitpick_Otimes__frac (lambda ((BOUND_VARIABLE_141494 $$unsorted) (BOUND_VARIABLE_141495 $$unsorted) (BOUND_VARIABLE_141496 $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom BOUND_VARIABLE_141494))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Nitpick_Onum BOUND_VARIABLE_141494))) (tptp.hAPP (tptp.hAPP (tptp.c_Nitpick_Ofrac BOUND_VARIABLE_141494) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 BOUND_VARIABLE_141495)) (tptp.hAPP _let_3 BOUND_VARIABLE_141496))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 BOUND_VARIABLE_141495)) (tptp.hAPP _let_1 BOUND_VARIABLE_141496)))))))))))) (let ((_let_289 (TRUST_SUBS_EQ (ASSUME :args (_let_86)) :args ((= tptp.c_Nitpick_Oinverse__frac (lambda ((BOUND_VARIABLE_141518 $$unsorted) (BOUND_VARIABLE_141519 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Nitpick_Ofrac BOUND_VARIABLE_141518) (tptp.hAPP (tptp.c_Nitpick_Odenom BOUND_VARIABLE_141518) BOUND_VARIABLE_141519)) (tptp.hAPP (tptp.c_Nitpick_Onum BOUND_VARIABLE_141518) BOUND_VARIABLE_141519)))))))) (let ((_let_290 (TRUST_SUBS_EQ (ASSUME :args (_let_85)) :args ((= tptp.c_Nitpick_Ouminus__frac (lambda ((BOUND_VARIABLE_141533 $$unsorted) (BOUND_VARIABLE_141534 $$unsorted)) (tptp.c_Nitpick_OAbs__Frac BOUND_VARIABLE_141533 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nitpick_Onum BOUND_VARIABLE_141533) BOUND_VARIABLE_141534))) (tptp.hAPP (tptp.c_Nitpick_Odenom BOUND_VARIABLE_141533) BOUND_VARIABLE_141534))))))))) (let ((_let_291 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_84)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_84 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Nitpick_Oless__frac (lambda ((BOUND_VARIABLE_141608 $$unsorted) (BOUND_VARIABLE_141609 $$unsorted) (BOUND_VARIABLE_141610 $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom BOUND_VARIABLE_141608))) (let ((_let_2 (tptp.c_Nitpick_Onum BOUND_VARIABLE_141608))) (let ((_let_3 (tptp.c_Nitpick_OAbs__Frac BOUND_VARIABLE_141608 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_2 BOUND_VARIABLE_141610))) (tptp.hAPP _let_1 BOUND_VARIABLE_141610))))) (let ((_let_4 (tptp.hAPP _let_1 _let_3))) (let ((_let_5 (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint)))) (let ((_let_6 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_7 (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_8 (tptp.hAPP _let_1 BOUND_VARIABLE_141609))) (let ((_let_9 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (let ((_let_10 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_9 _let_4)))) (let ((_let_11 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_9 _let_8)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint BOUND_VARIABLE_141608) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_11) _let_10)) (tptp.c_Nitpick_Onat__gcd _let_11 _let_10)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint BOUND_VARIABLE_141608) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint BOUND_VARIABLE_141608) tptp.tc_Int_Oint) (tptp.c_Nitpick_Ofrac BOUND_VARIABLE_141608)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_2 BOUND_VARIABLE_141609))) (tptp.hAPP _let_5 _let_8)))) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_2 _let_3))) (tptp.hAPP _let_5 _let_4))))) (tptp.c_COMBI tptp.tc_Int_Oint))))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))))))))))))))))) (let ((_let_292 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_83)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_83 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Nitpick_Oless__eq__frac (lambda ((BOUND_VARIABLE_141726 $$unsorted) (BOUND_VARIABLE_141727 $$unsorted) (BOUND_VARIABLE_141728 $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom BOUND_VARIABLE_141726))) (let ((_let_2 (tptp.c_Nitpick_Onum BOUND_VARIABLE_141726))) (let ((_let_3 (tptp.c_Nitpick_OAbs__Frac BOUND_VARIABLE_141726 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_2 BOUND_VARIABLE_141728))) (tptp.hAPP _let_1 BOUND_VARIABLE_141728))))) (let ((_let_4 (tptp.hAPP _let_1 _let_3))) (let ((_let_5 (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint)))) (let ((_let_6 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_7 (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_8 (tptp.hAPP _let_1 BOUND_VARIABLE_141727))) (let ((_let_9 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (let ((_let_10 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_9 _let_4)))) (let ((_let_11 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_9 _let_8)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint BOUND_VARIABLE_141726) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_11) _let_10)) (tptp.c_Nitpick_Onat__gcd _let_11 _let_10)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint BOUND_VARIABLE_141726) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint BOUND_VARIABLE_141726) tptp.tc_Int_Oint) (tptp.c_Nitpick_Ofrac BOUND_VARIABLE_141726)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_2 BOUND_VARIABLE_141727))) (tptp.hAPP _let_5 _let_8)))) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP _let_2 _let_3))) (tptp.hAPP _let_5 _let_4))))) (tptp.c_COMBI tptp.tc_Int_Oint))))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))))))))))))))))) (let ((_let_293 (TRUST_SUBS_EQ (ASSUME :args (_let_82)) :args ((= tptp.c_Nitpick_Odenom (lambda ((BOUND_VARIABLE_141785 $$unsorted)) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint BOUND_VARIABLE_141785 (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.c_Nitpick_ORep__Frac BOUND_VARIABLE_141785)))))))) (let ((_let_294 (TRUST_SUBS_EQ (ASSUME :args (_let_81)) :args ((= tptp.c_Nitpick_Onum (lambda ((BOUND_VARIABLE_141793 $$unsorted)) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint BOUND_VARIABLE_141793 (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.c_Nitpick_ORep__Frac BOUND_VARIABLE_141793)))))))) (let ((_let_295 (TRUST_SUBS_EQ (ASSUME :args (_let_80)) :args ((= tptp.c_Product__Type_Ointernal__split (lambda ((BOUND_VARIABLE_141801 $$unsorted) (BOUND_VARIABLE_141802 $$unsorted) (BOUND_VARIABLE_141803 $$unsorted)) (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_141801 BOUND_VARIABLE_141802 BOUND_VARIABLE_141803))))))) (let ((_let_296 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_79)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_295 _let_294 _let_293 _let_292 _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_79 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Olenlex (lambda ((BOUND_VARIABLE_141853 $$unsorted) (BOUND_VARIABLE_141854 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist BOUND_VARIABLE_141853))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Orderings_Otop__class_Otop _let_3))) (let ((_let_5 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.hAPP (tptp.c_Set_OCollect _let_5) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_3 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.c_fequal)) tptp.c_Nat_OSuc))))) (let ((_let_7 (tptp.tc_fun _let_5 tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_2 _let_1) (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_7) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_7 _let_7) (tptp.c_Nat_Ocompow _let_5 tptp.tc_HOL_Obool)) _let_6))) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) _let_4) (tptp.c_List_Olexn BOUND_VARIABLE_141853 BOUND_VARIABLE_141854)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_1 _let_2) _let_1) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat _let_1)) (tptp.c_Nat_Osize__class_Osize _let_1))) (tptp.c_COMBI _let_1)))))))))))))))) (let ((_let_297 (TRUST_SUBS_EQ (ASSUME :args (_let_78)) :args ((= tptp.c_List_Oset__Cons (lambda ((BOUND_VARIABLE_141973 $$unsorted) (BOUND_VARIABLE_141974 $$unsorted) (BOUND_VARIABLE_141975 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist BOUND_VARIABLE_141973))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun BOUND_VARIABLE_141973 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun _let_2 _let_2))) (let ((_let_6 (tptp.tc_fun _let_1 _let_1))) (let ((_let_7 (tptp.tc_fun _let_6 _let_2))) (let ((_let_8 (tptp.tc_fun BOUND_VARIABLE_141973 _let_2))) (let ((_let_9 (tptp.tc_fun BOUND_VARIABLE_141973 _let_6))) (let ((_let_10 (tptp.tc_fun _let_1 _let_4))) (let ((_let_11 (tptp.tc_fun BOUND_VARIABLE_141973 _let_10))) (let ((_let_12 (tptp.tc_fun BOUND_VARIABLE_141973 _let_5))) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 tptp.tc_HOL_Obool _let_1) (tptp.c_HOL_OEx BOUND_VARIABLE_141973)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 _let_3 _let_1) (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool BOUND_VARIABLE_141973) (tptp.c_HOL_OEx _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_8 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_8 _let_8) _let_1) (tptp.c_COMBS BOUND_VARIABLE_141973 _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 _let_1) (tptp.hAPP (tptp.c_COMBB _let_10 _let_5 BOUND_VARIABLE_141973) (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 _let_11 _let_1) (tptp.hAPP (tptp.c_COMBB _let_2 _let_10 BOUND_VARIABLE_141973) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 _let_1) tptp.c_fconj))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_9 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_9 _let_8) _let_1) (tptp.c_COMBB _let_6 _let_2 BOUND_VARIABLE_141973)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_7 _let_1) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool _let_1)) tptp.c_fequal))) (tptp.c_List_Olist_OCons BOUND_VARIABLE_141973)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_141973 _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 BOUND_VARIABLE_141973) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 BOUND_VARIABLE_141973) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_141973 _let_3 tptp.tc_HOL_Obool) (tptp.c_member BOUND_VARIABLE_141973)) BOUND_VARIABLE_141974)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) BOUND_VARIABLE_141975)))))))))))))))))))))))) (let ((_let_298 (TRUST_SUBS_EQ (ASSUME :args (_let_77)) :args ((= tptp.c_List_Olists (lambda ((BOUND_VARIABLE_142517 $$unsorted) (BOUND_VARIABLE_142518 $$unsorted)) (let ((_let_1 (tptp.tc_fun BOUND_VARIABLE_142517 tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_List_Olist BOUND_VARIABLE_142517))) (tptp.hAPP (tptp.c_Set_OCollect _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) _let_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1)) (tptp.c_List_Oset BOUND_VARIABLE_142517))) BOUND_VARIABLE_142518)))))))))) (let ((_let_299 (EQ_RESOLVE (ASSUME :args (_let_76)) (MACRO_SR_EQ_INTRO :args (_let_76 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_300 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_75)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_299 _let_298 _let_297 _let_296 _let_295 _let_294 _let_293 _let_292 _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_75 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Onat__list (lambda ((BOUND_VARIABLE_142549 $$unsorted)) (forall ((B_x $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) BOUND_VARIABLE_142549)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)))))))))) (let ((_let_301 (TRUST_SUBS_EQ (ASSUME :args (_let_74)) :args ((= tptp.c_List_Obutlast (lambda ((BOUND_VARIABLE_142560 $$unsorted) (BOUND_VARIABLE_142561 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake BOUND_VARIABLE_142560) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_142560)) BOUND_VARIABLE_142561)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) BOUND_VARIABLE_142561))))))) (let ((_let_302 (TRUST_SUBS_EQ (ASSUME :args (_let_73)) :args ((= tptp.c_List_Orotate (lambda ((BOUND_VARIABLE_143098 $$unsorted) (BOUND_VARIABLE_143099 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist BOUND_VARIABLE_143098))) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 _let_1) BOUND_VARIABLE_143099) (tptp.c_List_Orotate1 BOUND_VARIABLE_143098))))))))) (let ((_let_303 (TRUST_SUBS_EQ (ASSUME :args (_let_72)) :args ((= tptp.c_Map_Omap__upds (lambda ((BOUND_VARIABLE_143191 $$unsorted) (BOUND_VARIABLE_143192 $$unsorted) (BOUND_VARIABLE_143193 $$unsorted) (BOUND_VARIABLE_143194 $$unsorted) (BOUND_VARIABLE_143195 $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption BOUND_VARIABLE_143192))) (let ((_let_2 (tptp.tc_fun BOUND_VARIABLE_143191 _let_1))) (let ((_let_3 (tptp.tc_fun BOUND_VARIABLE_143192 _let_2))) (let ((_let_4 (tptp.tc_fun BOUND_VARIABLE_143192 _let_1))) (let ((_let_5 (tptp.tc_fun _let_4 _let_3))) (let ((_let_6 (tptp.tc_fun _let_1 _let_2))) (let ((_let_7 (tptp.tc_fun BOUND_VARIABLE_143191 _let_5))) (let ((_let_8 (tptp.tc_fun BOUND_VARIABLE_143191 _let_3))) (let ((_let_9 (tptp.tc_prod BOUND_VARIABLE_143191 BOUND_VARIABLE_143192))) (tptp.hAPP (tptp.c_List_Ofoldl _let_2 _let_9 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_9 _let_2) _let_2) (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_143191 BOUND_VARIABLE_143192 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_4 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_4 _let_8) _let_2) (tptp.c_COMBC BOUND_VARIABLE_143191 _let_4 _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun BOUND_VARIABLE_143191 _let_6) _let_7 _let_2) (tptp.hAPP (tptp.c_COMBB _let_6 _let_5 BOUND_VARIABLE_143191) (tptp.c_COMBB _let_1 _let_2 BOUND_VARIABLE_143192))) (tptp.c_Fun_Ofun__upd BOUND_VARIABLE_143191 _let_1)))) (tptp.c_Option_Ooption_OSome BOUND_VARIABLE_143192))) BOUND_VARIABLE_143193) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip BOUND_VARIABLE_143191 BOUND_VARIABLE_143192) BOUND_VARIABLE_143194) BOUND_VARIABLE_143195))))))))))))))))) (let ((_let_304 (TRUST_SUBS_EQ (ASSUME :args (_let_71)) :args ((= tptp.c_Map_Oran (lambda ((BOUND_VARIABLE_143257 $$unsorted) (BOUND_VARIABLE_143258 $$unsorted) (BOUND_VARIABLE_143259 $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption BOUND_VARIABLE_143258))) (let ((_let_2 (tptp.tc_fun BOUND_VARIABLE_143257 tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.c_Set_OCollect BOUND_VARIABLE_143258) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool BOUND_VARIABLE_143258) (tptp.c_HOL_OEx BOUND_VARIABLE_143257)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 BOUND_VARIABLE_143258) (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_143257 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) BOUND_VARIABLE_143257) tptp.c_fequal) BOUND_VARIABLE_143259))) (tptp.c_Option_Ooption_OSome BOUND_VARIABLE_143258)))))))))))) (let ((_let_305 (ASSUME :args (_let_70)))) (let ((_let_306 (TRUST_SUBS_EQ (ASSUME :args (_let_68)) :args ((= tptp.c_Random_Oselect__weight (lambda ((BOUND_VARIABLE_143802 $$unsorted) (BOUND_VARIABLE_143803 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_2 (tptp.tc_prod BOUND_VARIABLE_143802 _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp _let_1 tptp.tc_Code__Numeral_Ocode__numeral _let_1 _let_2) (tptp.c_Random_Orange (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral BOUND_VARIABLE_143802) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_Ofst tptp.tc_Code__Numeral_Ocode__numeral BOUND_VARIABLE_143802)) BOUND_VARIABLE_143803)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB BOUND_VARIABLE_143802 (tptp.tc_fun _let_1 _let_2) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_OPair BOUND_VARIABLE_143802 _let_1)) (tptp.c_Random_Opick BOUND_VARIABLE_143802 BOUND_VARIABLE_143803))))))))))) (let ((_let_307 (TRUST_SUBS_EQ (ASSUME :args (_let_67)) :args ((= tptp.c_List_Olist_Olist__size (lambda ((BOUND_VARIABLE_143850 $$unsorted) (BOUND_VARIABLE_143851 $$unsorted) (BOUND_VARIABLE_143852 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap BOUND_VARIABLE_143850 tptp.tc_Nat_Onat) BOUND_VARIABLE_143851) BOUND_VARIABLE_143852))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_143850)) BOUND_VARIABLE_143852)))))))) (let ((_let_308 (ASSUME :args (_let_65)))) (let ((_let_309 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_61)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_308 _let_307 _let_306 _let_305 _let_304 _let_303 _let_302 _let_301 _let_300 _let_299 _let_298 _let_297 _let_296 _let_295 _let_294 _let_293 _let_292 _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_61 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Random_Oselect (lambda ((BOUND_VARIABLE_143957 $$unsorted) (BOUND_VARIABLE_143958 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_2 (tptp.tc_prod BOUND_VARIABLE_143957 _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp _let_1 tptp.tc_Code__Numeral_Ocode__numeral _let_1 _let_2) (tptp.c_Random_Orange (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_143957)) BOUND_VARIABLE_143958)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB BOUND_VARIABLE_143957 (tptp.tc_fun _let_1 _let_2) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_OPair BOUND_VARIABLE_143957 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat BOUND_VARIABLE_143957 tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.c_List_Onth BOUND_VARIABLE_143957) BOUND_VARIABLE_143958)) tptp.c_Code__Numeral_Onat__of))))))))))) (let ((_let_310 (ASSUME :args (_let_58)))) (let ((_let_311 (TRUST_SUBS_EQ (ASSUME :args (_let_57)) :args ((= tptp.c_Code__Numeral_Onat__of__aux (lambda ((BOUND_VARIABLE_144013 $$unsorted) (BOUND_VARIABLE_144014 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of BOUND_VARIABLE_144013)) BOUND_VARIABLE_144014))))))) (let ((_let_312 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_56)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_311 _let_310 _let_309 _let_308 _let_307 _let_306 _let_305 _let_304 _let_303 _let_302 _let_301 _let_300 _let_299 _let_298 _let_297 _let_296 _let_295 _let_294 _let_293 _let_292 _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_56 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_Code__Numeral_OSuc__code__numeral (lambda ((BOUND_VARIABLE_144028 $$unsorted)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Code__Numeral_Onat__of BOUND_VARIABLE_144028))))))))) (let ((_let_313 (SYMM (SYMM (ASSUME :args (_let_55)))))) (let ((_let_314 (TRUST_SUBS_EQ (ASSUME :args (_let_53)) :args ((= tptp.c_Map_Odom (lambda ((BOUND_VARIABLE_144092 $$unsorted) (BOUND_VARIABLE_144093 $$unsorted) (BOUND_VARIABLE_144094 $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption BOUND_VARIABLE_144093))) (tptp.hAPP (tptp.c_Set_OCollect BOUND_VARIABLE_144092) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool BOUND_VARIABLE_144092) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_144092 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) BOUND_VARIABLE_144092) tptp.c_fequal) BOUND_VARIABLE_144094)) (tptp.c_Option_Ooption_ONone BOUND_VARIABLE_144093))))))))))) (let ((_let_315 (TRUST_SUBS_EQ (ASSUME :args (_let_52)) :args ((= tptp.c_Smallcheck_Oorelse (lambda ((BOUND_VARIABLE_144446 $$unsorted) (BOUND_VARIABLE_144447 $$unsorted) (BOUND_VARIABLE_144448 $$unsorted)) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case (tptp.tc_Option_Ooption BOUND_VARIABLE_144446) BOUND_VARIABLE_144446 BOUND_VARIABLE_144448 (tptp.c_Option_Ooption_OSome BOUND_VARIABLE_144446)) BOUND_VARIABLE_144447))))))) (let ((_let_316 (TRUST_SUBS_EQ (ASSUME :args (_let_51)) :args ((= tptp.c_Enum_Oenum__the (lambda ((BOUND_VARIABLE_144461 $$unsorted) (BOUND_VARIABLE_144462 $$unsorted)) (tptp.c_HOL_OThe BOUND_VARIABLE_144461 BOUND_VARIABLE_144462))))))) (let ((_let_317 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_50)) (MACRO_SR_EQ_INTRO :args (_let_50 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Olist__all2 (lambda ((BOUND_VARIABLE_144482 $$unsorted) (BOUND_VARIABLE_144483 $$unsorted) (BOUND_VARIABLE_144484 $$unsorted) (BOUND_VARIABLE_144485 $$unsorted) (BOUND_VARIABLE_144486 $$unsorted)) (and (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_144482)) BOUND_VARIABLE_144485) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_144483)) BOUND_VARIABLE_144486)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod BOUND_VARIABLE_144482 BOUND_VARIABLE_144483))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip BOUND_VARIABLE_144482 BOUND_VARIABLE_144483) BOUND_VARIABLE_144485) BOUND_VARIABLE_144486))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_144482 BOUND_VARIABLE_144483 tptp.tc_HOL_Obool) BOUND_VARIABLE_144484) B_x)))))))))))) (let ((_let_318 (TRUST_SUBS_EQ (ASSUME :args (_let_49)) :args ((= tptp.c_List_Oconcat (lambda ((BOUND_VARIABLE_145369 $$unsorted) (BOUND_VARIABLE_145370 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist BOUND_VARIABLE_145369))) (tptp.hAPP (tptp.c_List_Ofoldl _let_1 _let_1 (tptp.c_List_Oappend BOUND_VARIABLE_145369) (tptp.c_List_Olist_ONil BOUND_VARIABLE_145369)) BOUND_VARIABLE_145370)))))))) (let ((_let_319 (TRUST_SUBS_EQ (ASSUME :args (_let_48)) :args ((= tptp.c_List_OremoveAll (lambda ((BOUND_VARIABLE_145536 $$unsorted) (BOUND_VARIABLE_145537 $$unsorted)) (tptp.c_List_Ofilter BOUND_VARIABLE_145536 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool BOUND_VARIABLE_145536) tptp.c_fNot) (tptp.hAPP tptp.c_fequal BOUND_VARIABLE_145537))))))))) (let ((_let_320 (TRUST_SUBS_EQ (ASSUME :args (_let_47)) :args ((= tptp.c_List_Opartition (lambda ((BOUND_VARIABLE_145549 $$unsorted) (BOUND_VARIABLE_145550 $$unsorted) (BOUND_VARIABLE_145551 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist BOUND_VARIABLE_145549))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.hAPP (tptp.c_List_Ofilter BOUND_VARIABLE_145549 BOUND_VARIABLE_145550) BOUND_VARIABLE_145551)) (tptp.hAPP (tptp.c_List_Ofilter BOUND_VARIABLE_145549 (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool BOUND_VARIABLE_145549 tptp.c_fNot) BOUND_VARIABLE_145550)) BOUND_VARIABLE_145551))))))))) (let ((_let_321 (TRUST_SUBS_EQ (ASSUME :args (_let_46)) :args ((= tptp.c_List_Omap__filter (lambda ((BOUND_VARIABLE_145605 $$unsorted) (BOUND_VARIABLE_145606 $$unsorted) (BOUND_VARIABLE_145607 $$unsorted) (BOUND_VARIABLE_145608 $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption BOUND_VARIABLE_145606))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap BOUND_VARIABLE_145605 BOUND_VARIABLE_145606) (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 BOUND_VARIABLE_145606 BOUND_VARIABLE_145605 (tptp.c_Option_Othe BOUND_VARIABLE_145606)) BOUND_VARIABLE_145607)) (tptp.hAPP (tptp.c_List_Ofilter BOUND_VARIABLE_145605 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool BOUND_VARIABLE_145605) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_145605 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) BOUND_VARIABLE_145605) tptp.c_fequal) BOUND_VARIABLE_145607)) (tptp.c_Option_Ooption_ONone BOUND_VARIABLE_145606)))) BOUND_VARIABLE_145608))))))))) (let ((_let_322 (TRUST_SUBS_EQ (ASSUME :args (_let_45)) :args ((= tptp.c_List_Osublist (lambda ((BOUND_VARIABLE_145754 $$unsorted) (BOUND_VARIABLE_145755 $$unsorted) (BOUND_VARIABLE_145756 $$unsorted)) (let ((_let_1 (tptp.tc_prod BOUND_VARIABLE_145754 tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 BOUND_VARIABLE_145754) (tptp.c_Product__Type_Ofst BOUND_VARIABLE_145754 tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_1) (tptp.c_member tptp.tc_Nat_Onat)) (tptp.c_Product__Type_Osnd BOUND_VARIABLE_145754 tptp.tc_Nat_Onat))) BOUND_VARIABLE_145756)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip BOUND_VARIABLE_145754 tptp.tc_Nat_Onat) BOUND_VARIABLE_145755) (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_145754)) BOUND_VARIABLE_145755))))))))))))) (let ((_let_323 (TRUST_SUBS_EQ (ASSUME :args (_let_44)) :args ((= tptp.c_List_Oupt (lambda ((BOUND_VARIABLE_145786 $$unsorted) (BOUND_VARIABLE_145787 $$unsorted)) (tptp.c_List_Olinorder__class_Osorted__list__of__set tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat BOUND_VARIABLE_145786) BOUND_VARIABLE_145787)))))))) (let ((_let_324 (TRUST_SUBS_EQ (ASSUME :args (_let_43)) :args ((= tptp.c_List_Ofoldr (lambda ((BOUND_VARIABLE_146256 $$unsorted) (BOUND_VARIABLE_146257 $$unsorted) (BOUND_VARIABLE_146258 $$unsorted) (BOUND_VARIABLE_146259 $$unsorted) (BOUND_VARIABLE_146260 $$unsorted)) (tptp.hAPP (tptp.c_List_Ofoldl BOUND_VARIABLE_146257 BOUND_VARIABLE_146256 (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_146256 BOUND_VARIABLE_146257 BOUND_VARIABLE_146257) BOUND_VARIABLE_146258) BOUND_VARIABLE_146260) (tptp.hAPP (tptp.c_List_Orev BOUND_VARIABLE_146256) BOUND_VARIABLE_146259)))))))) (let ((_let_325 (TRUST_SUBS_EQ (ASSUME :args (_let_42)) :args ((= tptp.c_List_Oreturn__list (lambda ((BOUND_VARIABLE_146319 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Int_Oint tptp.tc_Nat_Onat) tptp.c_Int_Onat) BOUND_VARIABLE_146319))))))) (let ((_let_326 (TRUST_SUBS_EQ (ASSUME :args (_let_41)) :args ((= tptp.c_List_Oembed__list (lambda ((BOUND_VARIABLE_146368 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat tptp.tc_Int_Oint) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) BOUND_VARIABLE_146368))))))) (let ((_let_327 (TRUST_SUBS_EQ (ASSUME :args (_let_40)) :args ((= tptp.c_List_OdropWhile (lambda ((BOUND_VARIABLE_146374 $$unsorted) (BOUND_VARIABLE_146375 $$unsorted) (BOUND_VARIABLE_146376 $$unsorted)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop BOUND_VARIABLE_146374) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_146374)) (tptp.c_List_OtakeWhile BOUND_VARIABLE_146374 BOUND_VARIABLE_146375 BOUND_VARIABLE_146376))) BOUND_VARIABLE_146376))))))) (let ((_let_328 (TRUST_SUBS_EQ (ASSUME :args (_let_39)) :args ((= tptp.c_Map_Omap__comp (lambda ((BOUND_VARIABLE_146481 $$unsorted) (BOUND_VARIABLE_146482 $$unsorted) (BOUND_VARIABLE_146483 $$unsorted) (BOUND_VARIABLE_146484 $$unsorted) (BOUND_VARIABLE_146485 $$unsorted) (BOUND_VARIABLE_146486 $$unsorted)) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case (tptp.tc_Option_Ooption BOUND_VARIABLE_146482) BOUND_VARIABLE_146481 (tptp.c_Option_Ooption_ONone BOUND_VARIABLE_146482) BOUND_VARIABLE_146484) (tptp.hAPP BOUND_VARIABLE_146485 BOUND_VARIABLE_146486)))))))) (let ((_let_329 (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_38)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_328 _let_327 _let_326 _let_325 _let_324 _let_323 _let_322 _let_321 _let_320 _let_319 _let_318 _let_317 _let_316 _let_315 _let_314 _let_313 _let_312 _let_311 _let_310 _let_309 _let_308 _let_307 _let_306 _let_305 _let_304 _let_303 _let_302 _let_301 _let_300 _let_299 _let_298 _let_297 _let_296 _let_295 _let_294 _let_293 _let_292 _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_38 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Omaps (lambda ((BOUND_VARIABLE_146871 $$unsorted) (BOUND_VARIABLE_146872 $$unsorted) (BOUND_VARIABLE_146873 $$unsorted) (BOUND_VARIABLE_146874 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist BOUND_VARIABLE_146872))) (tptp.hAPP (tptp.c_List_Ofoldl _let_1 _let_1 (tptp.c_List_Oappend BOUND_VARIABLE_146872) (tptp.c_List_Olist_ONil BOUND_VARIABLE_146872)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap BOUND_VARIABLE_146871 _let_1) BOUND_VARIABLE_146873) BOUND_VARIABLE_146874))))))))) (let ((_let_330 (AND_INTRO (TRUST_SUBS_EQ (ASSUME :args (_let_18)) :args ((= tptp.c_Lazy__Sequence_Oproduct (lambda ((BOUND_VARIABLE_148130 $$unsorted) (BOUND_VARIABLE_148131 $$unsorted) (BOUND_VARIABLE_148132 $$unsorted) (BOUND_VARIABLE_148133 $$unsorted)) (let ((_let_1 (tptp.tc_prod BOUND_VARIABLE_148130 BOUND_VARIABLE_148131))) (let ((_let_2 (tptp.tc_Lazy__Sequence_Olazy__sequence _let_1))) (let ((_let_3 (tptp.tc_fun BOUND_VARIABLE_148131 _let_2))) (tptp.hAPP (tptp.c_Lazy__Sequence_Obind BOUND_VARIABLE_148130 _let_1 BOUND_VARIABLE_148132) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 BOUND_VARIABLE_148130) (tptp.c_Lazy__Sequence_Obind BOUND_VARIABLE_148131 _let_1 BOUND_VARIABLE_148133)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun BOUND_VARIABLE_148131 _let_1) _let_3 BOUND_VARIABLE_148130) (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 BOUND_VARIABLE_148131) (tptp.c_Lazy__Sequence_Osingle _let_1))) (tptp.c_Product__Type_OPair BOUND_VARIABLE_148130 BOUND_VARIABLE_148131))))))))))) (TRUST_SUBS_EQ (ASSUME :args (_let_19)) :args ((= tptp.c_New__Random__Sequence_Opos__map (lambda ((BOUND_VARIABLE_148042 $$unsorted) (BOUND_VARIABLE_148043 $$unsorted) (BOUND_VARIABLE_148044 $$unsorted) (BOUND_VARIABLE_148045 $$unsorted)) (tptp.c_New__Random__Sequence_Opos__bind BOUND_VARIABLE_148042 BOUND_VARIABLE_148043 BOUND_VARIABLE_148045 (tptp.hAPP (tptp.c_Fun_Ocomp BOUND_VARIABLE_148043 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence BOUND_VARIABLE_148043))))) BOUND_VARIABLE_148042 (tptp.c_New__Random__Sequence_Opos__single BOUND_VARIABLE_148043)) BOUND_VARIABLE_148044)))))) (TRUST_SUBS_EQ (ASSUME :args (_let_20)) :args ((= tptp.c_New__Random__Sequence_Oneg__map (lambda ((BOUND_VARIABLE_148003 $$unsorted) (BOUND_VARIABLE_148004 $$unsorted) (BOUND_VARIABLE_148005 $$unsorted) (BOUND_VARIABLE_148006 $$unsorted)) (tptp.c_New__Random__Sequence_Oneg__bind BOUND_VARIABLE_148003 BOUND_VARIABLE_148004 BOUND_VARIABLE_148006 (tptp.hAPP (tptp.c_Fun_Ocomp BOUND_VARIABLE_148004 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption BOUND_VARIABLE_148004)))))) BOUND_VARIABLE_148003 (tptp.c_New__Random__Sequence_Oneg__single BOUND_VARIABLE_148004)) BOUND_VARIABLE_148005)))))) (TRUST_SUBS_EQ (ASSUME :args (_let_21)) :args ((= tptp.c_New__Random__Sequence_Opos__empty (lambda ((BOUND_VARIABLE_147988 $$unsorted) (BOUND_VARIABLE_147989 $$unsorted) (BOUND_VARIABLE_147990 $$unsorted) (BOUND_VARIABLE_147991 $$unsorted)) (tptp.c_New__DSequence_Opos__empty BOUND_VARIABLE_147988))))) (TRUST_SUBS_EQ (ASSUME :args (_let_22)) :args ((= tptp.c_New__Random__Sequence_Oneg__decr__bind (lambda ((BOUND_VARIABLE_147942 $$unsorted) (BOUND_VARIABLE_147943 $$unsorted) (BOUND_VARIABLE_147944 $$unsorted) (BOUND_VARIABLE_147945 $$unsorted) (BOUND_VARIABLE_147946 $$unsorted) (BOUND_VARIABLE_147947 $$unsorted) (BOUND_VARIABLE_147948 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption BOUND_VARIABLE_147943))))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (tptp.c_New__DSequence_Oneg__decr__bind BOUND_VARIABLE_147942 BOUND_VARIABLE_147943 (tptp.hAPP (tptp.hAPP (tptp.hAPP BOUND_VARIABLE_147944 BOUND_VARIABLE_147946) BOUND_VARIABLE_147947) BOUND_VARIABLE_147948) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_147942 _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_147942 tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_147942 tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) BOUND_VARIABLE_147945) BOUND_VARIABLE_147946)) BOUND_VARIABLE_147947)) BOUND_VARIABLE_147948))))))))) (TRUST_SUBS_EQ (ASSUME :args (_let_23)) :args ((= tptp.c_New__Random__Sequence_Opos__decr__bind (lambda ((BOUND_VARIABLE_147902 $$unsorted) (BOUND_VARIABLE_147903 $$unsorted) (BOUND_VARIABLE_147904 $$unsorted) (BOUND_VARIABLE_147905 $$unsorted) (BOUND_VARIABLE_147906 $$unsorted) (BOUND_VARIABLE_147907 $$unsorted) (BOUND_VARIABLE_147908 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence BOUND_VARIABLE_147903)))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (tptp.c_New__DSequence_Opos__decr__bind BOUND_VARIABLE_147902 BOUND_VARIABLE_147903 (tptp.hAPP (tptp.hAPP (tptp.hAPP BOUND_VARIABLE_147904 BOUND_VARIABLE_147906) BOUND_VARIABLE_147907) BOUND_VARIABLE_147908) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_147902 _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_147902 tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_147902 tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) BOUND_VARIABLE_147905) BOUND_VARIABLE_147906)) BOUND_VARIABLE_147907)) BOUND_VARIABLE_147908))))))))) (TRUST_SUBS_EQ (ASSUME :args (_let_24)) :args ((= tptp.c_Lazy__Sequence_Oyieldn (lambda ((BOUND_VARIABLE_147869 $$unsorted)) (tptp.c_Lazy__Sequence_Oanamorph (tptp.tc_Lazy__Sequence_Olazy__sequence BOUND_VARIABLE_147869) BOUND_VARIABLE_147869 (tptp.c_Lazy__Sequence_Oyield BOUND_VARIABLE_147869)))))) (TRUST_SUBS_EQ (ASSUME :args (_let_25)) :args ((= tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case (lambda ((BOUND_VARIABLE_147844 $$unsorted) (BOUND_VARIABLE_147845 $$unsorted) (BOUND_VARIABLE_147846 $$unsorted) (BOUND_VARIABLE_147847 $$unsorted) (BOUND_VARIABLE_147848 $$unsorted)) (let ((_let_1 (tptp.tc_Lazy__Sequence_Olazy__sequence BOUND_VARIABLE_147845))) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case BOUND_VARIABLE_147844 (tptp.tc_prod BOUND_VARIABLE_147845 _let_1) BOUND_VARIABLE_147846 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case BOUND_VARIABLE_147845 _let_1 BOUND_VARIABLE_147844) BOUND_VARIABLE_147847)) (tptp.hAPP (tptp.c_Lazy__Sequence_Oyield BOUND_VARIABLE_147845) BOUND_VARIABLE_147848))))))) (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_30)) (MACRO_SR_EQ_INTRO :args (_let_30 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Olist__ex1 (lambda ((BOUND_VARIABLE_147611 $$unsorted) (BOUND_VARIABLE_147612 $$unsorted) (BOUND_VARIABLE_147613 $$unsorted)) (not (forall ((B_x $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_147611) B_x) (tptp.hAPP (tptp.c_List_Oset BOUND_VARIABLE_147611) BOUND_VARIABLE_147613)))) (not (tptp.hBOOL (tptp.hAPP BOUND_VARIABLE_147612 B_x))) (not (forall ((B_y $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member BOUND_VARIABLE_147611) B_y) (tptp.hAPP (tptp.c_List_Oset BOUND_VARIABLE_147611) BOUND_VARIABLE_147613)))) (not (tptp.hBOOL (tptp.hAPP BOUND_VARIABLE_147612 B_y))) (= B_x B_y))))))))))) (TRUST_SUBS_EQ (ASSUME :args (_let_33)) :args ((= tptp.c_FunDef_Ois__measure (lambda ((BOUND_VARIABLE_147477 $$unsorted) (BOUND_VARIABLE_147478 $$unsorted)) true)))) (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_34)) (MACRO_SR_EQ_INTRO :args (_let_34 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Olist__all (lambda ((BOUND_VARIABLE_147349 $$unsorted) (BOUND_VARIABLE_147350 $$unsorted) (BOUND_VARIABLE_147351 $$unsorted)) (forall ((B_n $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_147349)) BOUND_VARIABLE_147351)))) (tptp.hBOOL (tptp.hAPP BOUND_VARIABLE_147350 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth BOUND_VARIABLE_147349) BOUND_VARIABLE_147351) B_n))))))))) (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_35)) (MACRO_SR_EQ_INTRO :args (_let_35 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Olist__ex (lambda ((BOUND_VARIABLE_147159 $$unsorted) (BOUND_VARIABLE_147160 $$unsorted) (BOUND_VARIABLE_147161 $$unsorted)) (not (forall ((B_n $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist BOUND_VARIABLE_147159)) BOUND_VARIABLE_147161)))) (not (tptp.hBOOL (tptp.hAPP BOUND_VARIABLE_147160 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth BOUND_VARIABLE_147159) BOUND_VARIABLE_147161) B_n))))))))))) (TRUST_SUBS_EQ (ASSUME :args (_let_36)) :args ((= tptp.c_New__Random__Sequence_Opos__not__random__dseq (lambda ((BOUND_VARIABLE_147059 $$unsorted) (BOUND_VARIABLE_147060 $$unsorted) (BOUND_VARIABLE_147061 $$unsorted) (BOUND_VARIABLE_147062 $$unsorted)) (tptp.c_New__DSequence_Opos__not__seq (tptp.hAPP (tptp.hAPP (tptp.hAPP BOUND_VARIABLE_147059 BOUND_VARIABLE_147060) BOUND_VARIABLE_147061) BOUND_VARIABLE_147062)))))) (TRUST_SUBS_EQ (EQ_RESOLVE (ASSUME :args (_let_37)) (MACRO_SR_EQ_INTRO (AND_INTRO _let_329 _let_328 _let_327 _let_326 _let_325 _let_324 _let_323 _let_322 _let_321 _let_320 _let_319 _let_318 _let_317 _let_316 _let_315 _let_314 _let_313 _let_312 _let_311 _let_310 _let_309 _let_308 _let_307 _let_306 _let_305 _let_304 _let_303 _let_302 _let_301 _let_300 _let_299 _let_298 _let_297 _let_296 _let_295 _let_294 _let_293 _let_292 _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219) :args (_let_37 SB_DEFAULT SBA_FIXPOINT))) :args ((= tptp.c_List_Omeasures (lambda ((BOUND_VARIABLE_147008 $$unsorted) (BOUND_VARIABLE_147009 $$unsorted)) (let ((_let_1 (tptp.tc_fun BOUND_VARIABLE_147008 tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_List_Olist _let_1))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.hAPP (tptp.c_Set_OCollect _let_5) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.c_fequal)) tptp.c_Nat_OSuc))))) (let ((_let_7 (tptp.tc_fun _let_5 tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.c_Orderings_Otop__class_Otop _let_4))) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_2 BOUND_VARIABLE_147008) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat (tptp.tc_fun (tptp.tc_prod _let_2 _let_2) tptp.tc_HOL_Obool)) _let_8) (tptp.c_List_Olexn tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_7) _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_7 _let_7) (tptp.c_Nat_Ocompow _let_5 tptp.tc_HOL_Obool)) _let_6)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC BOUND_VARIABLE_147008 _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_Nat_Onat) (tptp.tc_fun _let_3 _let_2) BOUND_VARIABLE_147008) (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_COMBC _let_1 BOUND_VARIABLE_147008 tptp.tc_Nat_Onat) (tptp.c_COMBI _let_1)))) BOUND_VARIABLE_147009)))))))))))))) _let_329 _let_328 _let_327 _let_326 _let_325 _let_324 _let_323 _let_322 _let_321 _let_320 _let_319 _let_318 _let_317 _let_316 _let_315 _let_314 _let_313 _let_312 _let_311 _let_310 _let_309 _let_308 _let_307 _let_306 _let_305 _let_304 _let_303 _let_302 _let_301 _let_300 _let_299 _let_298 _let_297 _let_296 _let_295 _let_294 _let_293 _let_292 _let_291 _let_290 _let_289 _let_288 _let_287 _let_286 _let_285 _let_284 _let_283 _let_282 _let_281 _let_280 _let_279 _let_278 _let_277 _let_276 _let_275 _let_274 _let_273 _let_272 _let_271 _let_270 _let_269 _let_268 _let_267 _let_266 _let_265 _let_264 _let_263 _let_262 _let_261 _let_260 _let_259 _let_258 _let_257 _let_256 _let_255 _let_254 _let_253 _let_252 _let_251 _let_250 _let_249 _let_248 _let_247 _let_246 _let_245 _let_244 _let_243 _let_242 _let_241 _let_240 _let_239 _let_238 _let_237 _let_236 _let_235 _let_234 _let_233 _let_232 _let_231 _let_230 _let_229 _let_228 _let_227 _let_226 _let_225 _let_224 _let_223 _let_222 _let_221 _let_220 _let_219))) (let ((_let_331 (EQ_RESOLVE (ASSUME :args (_let_214)) (TRANS (MACRO_SR_EQ_INTRO :args (_let_214 SB_DEFAULT SBA_FIXPOINT)) (MACRO_SR_EQ_INTRO _let_330 :args ((forall ((V_s1 $$unsorted) (V_s0 $$unsorted) (V_pn $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_pn))) V_s0) V_s1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn)) V_s0) V_s1)))) SB_DEFAULT SBA_FIXPOINT)))))) (let ((_let_332 (forall ((B_Z $$unsorted) (B_s $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of tptp.tc_Com_Opname tptp.tc_Com_Ocom) tptp.c_Com_Obodies) tptp.v_pn))) B_Z) B_s))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP tptp.c_Com_Ocom_OBODY tptp.v_pn)) B_Z) B_s)))))) (let ((_let_333 (not _let_218))) (let ((_let_334 (tptp.hAPP _let_7 _let_216))) (let ((_let_335 (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_14 (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP _let_334 _let_4)) _let_3)))) (let ((_let_336 (not _let_332))) (let ((_let_337 (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_14 (tptp.hAPP (tptp.hAPP _let_9 (tptp.hAPP _let_334 _let_217)) _let_3)))) (let ((_let_338 (not _let_337))) (let ((_let_339 (or _let_338 _let_336 _let_335))) (let ((_let_340 (forall ((V_Q_2 $$unsorted) (V_Q_H_2 $$unsorted) (V_c_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2) V_c_2))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (or (not (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_Q_H_2)) _let_2))) (not (forall ((B_Z $$unsorted) (B_s $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_H_2 B_Z) B_s))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 B_Z) B_s))))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_Q_2)) _let_2)))))))))) (let ((_let_341 (EQ_RESOLVE (ASSUME :args (_let_212)) (MACRO_SR_EQ_INTRO :args (_let_212 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_342 (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2) V_c_2))) (let ((_let_343 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_344 (_let_336))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_331 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_203 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_202 tptp.v_pn QUANTIFIERS_INST_CBQI_CONFLICT)) :args (_let_215))) (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE (ASSUME :args _let_344)) :args _let_344)) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_336) _let_332))) (REFL :args (_let_333)) :args (or))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_339)) :args ((or _let_335 _let_338 _let_336 (not _let_339)))) (EQ_RESOLVE (ASSUME :args (_let_15)) (MACRO_SR_EQ_INTRO _let_330 :args (_let_15 SB_DEFAULT SBA_FIXPOINT))) (EQ_RESOLVE (ASSUME :args (_let_16)) (MACRO_SR_EQ_INTRO _let_330 :args (_let_16 SB_DEFAULT SBA_FIXPOINT))) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_341 :args (_let_4 _let_217 _let_216 tptp.c_fequal _let_14 tptp.tc_Com_Ostate QUANTIFIERS_INST_E_MATCHING ((not (= (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_343) (tptp.hAPP _let_342 V_Q_H_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_343 tptp.tc_HOL_Obool)))) false)) (tptp.hAPP _let_342 V_Q_2)))) :args (_let_340))) _let_341 :args (_let_339 false _let_340)) :args (_let_336 true _let_335 false _let_337 false _let_339)) :args (_let_333 true _let_332)) _let_331 :args (false true _let_218 false _let_215)) :args ((forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))) (= V_f_2 V_g_2))) (forall ((V_Ga_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_Hoare__Mirabelle_Otriple T_a) tptp.tc_HOL_Obool)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_Hoare__Mirabelle_Otriple T_a) tptp.tc_HOL_Obool)))) (forall ((V_fun2_H_2 $$unsorted) (V_com_H_2 $$unsorted) (V_fun1_H_2 $$unsorted) (V_fun2_2 $$unsorted) (V_com_2 $$unsorted) (V_fun1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 V_fun1_2) V_com_2) V_fun2_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 V_fun1_H_2) V_com_H_2) V_fun2_H_2)) (and (= V_fun1_2 V_fun1_H_2) (= V_com_2 V_com_H_2) (= V_fun2_2 V_fun2_H_2))))) (forall ((V_Ga_2 $$unsorted) (V_ts_2 $$unsorted) (V_G_H_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_G_H_2 V_ts_2) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_G_H_2) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_2)))) (forall ((V_ts_2 $$unsorted) (V_t_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_t_2))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_2) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP _let_2 V_ts_2))))))) (forall ((V_ts_2 $$unsorted) (V_t_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_t_2))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP _let_2 V_ts_2)) (and (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_2)))))) (forall ((V_Q_2 $$unsorted) (V_pn_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_pn_2))) V_Q_2)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn_2)) V_Q_2)) _let_2)))))))) (forall ((V_Procsa_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.tc_fun T_a (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBS tptp.tc_Com_Opname tptp.tc_Com_Ocom _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun tptp.tc_Com_Ocom _let_3) tptp.tc_Com_Opname) (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a)) V_P_2)))) (let ((_let_5 (tptp.c_COMBS tptp.tc_Com_Opname _let_2 _let_1))) (let ((_let_6 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 tptp.c_Com_Ocom_OBODY)) V_Q_2)) V_Procsa_2))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_Ga_2) _let_6) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_Option_Ooption tptp.tc_Com_Ocom) tptp.tc_Com_Ocom tptp.tc_Com_Opname) (tptp.c_Option_Othe tptp.tc_Com_Ocom)) tptp.c_Com_Obody))) V_Q_2)) V_Procsa_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 _let_6))))))))) (forall ((V_Ga_2 $$unsorted) (V_Q_2 $$unsorted) (V_pn_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (let ((_let_5 (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn_2)) V_Q_2)))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a (tptp.hAPP _let_5 V_Ga_2) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_pn_2))) V_Q_2)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP _let_5 _let_2))))))))) (forall ((V_s1 $$unsorted) (V_s $$unsorted) (V_P $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP tptp.c_Com_Ocom_OBODY V_P)) V_s) V_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_P))) V_s) V_s1)))) _let_214 (forall ((V_Procsa_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.tc_fun T_a (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBS tptp.tc_Com_Opname tptp.tc_Com_Ocom _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun tptp.tc_Com_Ocom _let_3) tptp.tc_Com_Opname) (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a)) V_P_2)))) (let ((_let_5 (tptp.c_COMBS tptp.tc_Com_Opname _let_2 _let_1))) (let ((_let_6 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 tptp.c_Com_Ocom_OBODY)) V_Q_2)) V_Procsa_2))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__valids T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_Ga_2) _let_6) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_Option_Ooption tptp.tc_Com_Ocom) tptp.tc_Com_Ocom tptp.tc_Com_Opname) (tptp.c_Option_Othe tptp.tc_Com_Ocom)) tptp.c_Com_Obody))) V_Q_2)) V_Procsa_2)) (tptp.c_Hoare__Mirabelle_Ohoare__valids T_a V_Ga_2 _let_6))))))))) (forall ((V_a3_2 $$unsorted) (V_a2_2 $$unsorted) (V_a1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP tptp.c_Com_Ocom_OBODY V_a1_2)) V_a2_2) V_a3_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_a1_2))) V_a2_2) V_a3_2)))) (forall ((V_Ga_2 $$unsorted) (V_pn_2 $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert _let_1))) (let ((_let_4 (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_OMGT (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn_2))))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate (tptp.hAPP _let_4 V_Ga_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_OMGT (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_pn_2)))) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate V_Ga_2 (tptp.hAPP _let_4 _let_2)))))))) (forall ((V_pn_2 $$unsorted) (V_Procsa_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a))) (let ((_let_4 (tptp.tc_fun T_a (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool)))) (let ((_let_5 (tptp.tc_fun _let_4 _let_1))) (let ((_let_6 (tptp.hAPP (tptp.c_COMBS tptp.tc_Com_Opname tptp.tc_Com_Ocom _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun tptp.tc_Com_Ocom _let_5) tptp.tc_Com_Opname) _let_3) V_P_2)))) (let ((_let_7 (tptp.c_COMBS tptp.tc_Com_Opname _let_4 _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_Ga_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 tptp.c_Com_Ocom_OBODY)) V_Q_2)) V_Procsa_2)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_Option_Ooption tptp.tc_Com_Ocom) tptp.tc_Com_Ocom tptp.tc_Com_Opname) (tptp.c_Option_Othe tptp.tc_Com_Ocom)) tptp.c_Com_Obody))) V_Q_2)) V_Procsa_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Com_Opname) V_pn_2) V_Procsa_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP V_P_2 V_pn_2)) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn_2)) (tptp.hAPP V_Q_2 V_pn_2))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))))))))) (forall ((V_c_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b (tptp.hAPP (tptp.c_COMBK T_b T_a) V_c_2)) V_A_2))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_c_2) _let_1)))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_A_2))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)) (=> (not (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))) (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2))))) (forall ((T_a $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (=> (not (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2))) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b T_a)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Obot T_a) (= (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b T_a)) V_x_2) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))) (=> (= V_A_2 V_B_2) (=> (=> _let_3 (not _let_2)) (not (=> (not _let_3) _let_2)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_a_2))) (=> (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))) (= V_a_2 V_b_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) V_A_2))) (=> (not (= V_a_2 V_b_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((T_a $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP tptp.c_fequal V_y_2))) (= V_x_2 V_y_2))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (= (= (tptp.hAPP _let_1 V_R_2) (tptp.hAPP _let_1 V_S_2)) (= V_R_2 V_S_2)))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (= V_A_2 V_B_2) (= (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (=> (= V_x_2 V_y_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (=> (= V_a_2 V_b_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_A_2)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_c_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2))) (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2) V_A_2))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 V_A_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 V_B_2))) (= (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2)) (= V_A_2 V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_a_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) V_A_2))) (or (= V_a_2 V_b_2) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2)))) (forall ((T_b $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (=> (= V_b_2 (tptp.hAPP V_f_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_z_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= V_z_2 (tptp.hAPP V_f_2 B_x)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) _let_1))))) (forall ((V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.c_member T_a))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) _let_3))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_4 V_R_2)) (tptp.hAPP _let_4 V_S_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_x_2) (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2))))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (= V_b_2 V_a_2))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (= V_b_2 V_a_2))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) (tptp.hAPP V_f_2 V_x_2)) _let_1) _let_1)))) (forall ((V_c_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_b (tptp.hAPP (tptp.c_COMBK T_b T_a) V_c_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_c_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_ts_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_2) (tptp.c_Hoare__Mirabelle_Ohoare__valids T_a V_Ga_2 V_ts_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_y_2) V_A_2) V_x_2)) (or (= V_y_2 V_x_2) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_y_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (= (tptp.hAPP _let_1 _let_2) _let_2)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_c)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_x) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_a) V_a))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)))) (forall ((V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (and (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_B_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))))) (forall ((V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (or (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_B_2)) (tptp.hBOOL (tptp.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 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) V_C_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.hAPP _let_1 V_B_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_C_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.hAPP _let_1 V_B_2))) (= (tptp.hAPP _let_1 _let_2) _let_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_A_2) V_A_2)) (forall ((V_u $$unsorted) (V_t $$unsorted) (V_s $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c) V_s))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_t)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_u)) (= V_u V_t))))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_g_2) V_A_2)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_a T_c) V_f_2) V_g_2)) V_A_2))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_c_2 $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert _let_1))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple tptp.tc_Com_Ostate) V_P_2) V_c_2) V_Q_2)) _let_2))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_OMGT V_c_2)) _let_2)) (=> (tptp.c_Hoare__Mirabelle_Ohoare__valids tptp.tc_Com_Ostate _let_2 _let_4) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_2 _let_4)))))))) _let_213 (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Obot T_a) (= (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b T_a)) V_x_2) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_a))) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (= V_a_2 V_b_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_a))) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_a_2) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_c_2) (tptp.hAPP (tptp.hAPP _let_2 V_d_2) _let_1))) (or (and (= V_a_2 V_c_2) (= V_b_2 V_d_2)) (and (= V_a_2 V_d_2) (= V_b_2 V_c_2))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot T_a))) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) (tptp.c_Orderings_Obot__class_Obot T_a)) V_x))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.c_Orderings_Obot__class_Obot T_a)) V_x) V_x))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2)) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) _let_2) (and (= V_A_2 _let_2) (= V_B_2 _let_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_B_2) V_B_2))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) V_C_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2)))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b T_a)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Set_Othe__elem T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2)) (forall ((V_c_2 $$unsorted) (V_Ga_2 $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Com_Ostate _let_3))) (let ((_let_5 (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_fun tptp.tc_Com_Ostate _let_5))) (let ((_let_7 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate V_Ga_2 (tptp.hAPP (tptp.hAPP _let_7 (tptp.c_Hoare__Mirabelle_OMGT V_c_2)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate V_Ga_2 (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple tptp.tc_Com_Ostate) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_5 tptp.tc_Com_Ostate) (tptp.hAPP (tptp.c_COMBB _let_5 tptp.tc_HOL_Obool tptp.tc_Com_Ostate) (tptp.c_HOL_OAll tptp.tc_Com_Ostate))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_6 tptp.tc_Com_Ostate) (tptp.hAPP (tptp.c_COMBC tptp.tc_Com_Ostate _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_5 _let_5) tptp.tc_Com_Ostate) (tptp.c_COMBS tptp.tc_Com_Ostate tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_4 tptp.tc_Com_Ostate) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Com_Ostate) tptp.c_fimplies)) (tptp.hAPP tptp.c_Natural_Oevalc V_c_2))))) tptp.c_fequal))) V_c_2) tptp.c_fequal)) _let_2))))))))))) (forall ((V_fun2_2 $$unsorted) (V_com_2 $$unsorted) (V_fun1_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Hoare__Mirabelle_Otriple_Otriple__case T_b T_a V_f1_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_b) V_fun1_2) V_com_2) V_fun2_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP V_f1_2 V_fun1_2) V_com_2) V_fun2_2))) (forall ((V_fun2_2 $$unsorted) (V_com_2 $$unsorted) (V_fun1_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Hoare__Mirabelle_Otriple_Otriple__rec T_b T_a V_f1_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_b) V_fun1_2) V_com_2) V_fun2_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP V_f1_2 V_fun1_2) V_com_2) V_fun2_2))) (forall ((V_Q_2 $$unsorted) (V_c_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (=> (forall ((B_Z $$unsorted) (B_s $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_COMBK (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool) T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_Z) B_s)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Com_Ostate tptp.tc_Com_Ostate tptp.tc_HOL_Obool) tptp.c_fequal) B_s))) V_c_2) (tptp.hAPP _let_2 (tptp.hAPP V_Q_2 B_Z)))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2) V_c_2) V_Q_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((V_c_2 $$unsorted) (V_Ga_2 $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert _let_1))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun tptp.tc_Com_Ostate _let_4))) (let ((_let_6 (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun tptp.tc_Com_Ostate _let_6))) (=> tptp.c_Hoare__Mirabelle_Ostate__not__singleton (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate V_Ga_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple tptp.tc_Com_Ostate) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 _let_6 tptp.tc_Com_Ostate) (tptp.hAPP (tptp.c_COMBB _let_6 tptp.tc_HOL_Obool tptp.tc_Com_Ostate) (tptp.c_HOL_OAll tptp.tc_Com_Ostate))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_7 tptp.tc_Com_Ostate) (tptp.hAPP (tptp.c_COMBC tptp.tc_Com_Ostate _let_6 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_6 _let_6) tptp.tc_Com_Ostate) (tptp.c_COMBS tptp.tc_Com_Ostate tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_5 tptp.tc_Com_Ostate) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 tptp.tc_Com_Ostate) tptp.c_fimplies)) (tptp.hAPP tptp.c_Natural_Oevalc V_c_2))))) tptp.c_fequal))) V_c_2) tptp.c_fequal)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate V_Ga_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_OMGT V_c_2)) _let_2)))))))))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_a1_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_a2_2) tptp.c_Com_Ocom_OSKIP) V_a2_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (forall ((V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2) tptp.c_Com_Ocom_OSKIP) V_P_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (forall ((V_P_2 $$unsorted) (V_Q_2 $$unsorted) (V_c_2 $$unsorted) (V_P_H_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_P_H_2) V_c_2) V_Q_2)) _let_2)) (=> (forall ((B_Z $$unsorted) (B_s $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_Z) B_s)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_H_2 B_Z) B_s)))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_P_2) V_c_2) V_Q_2)) _let_2))))))))) _let_212 (forall ((V_R_2 $$unsorted) (V_d_2 $$unsorted) (V_Q_2 $$unsorted) (V_c_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a))) (let ((_let_4 (tptp.hAPP _let_3 V_P_2))) (let ((_let_5 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_c_2) V_Q_2)) _let_2)) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_Q_2) V_d_2) V_R_2)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Com_Ocom_OSemi V_c_2 V_d_2)) V_R_2)) _let_2)))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_Com_Ostate _let_2))) (let ((_let_4 (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_4) T_a) (tptp.c_COMBS tptp.tc_Com_Ostate tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_3 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 tptp.tc_Com_Ostate) tptp.c_fconj)) V_P_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Com_Ostate) tptp.c_fNot) V_b_2))) (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2)) V_P_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))))) (forall ((V_c_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2)) V_s_2) V_s_2)))) (forall ((V_s2_2 $$unsorted) (V_s1_2 $$unsorted) (V_c_2 $$unsorted) (V_s0_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2)))) (=> (tptp.hBOOL (tptp.hAPP V_b_2 V_s0_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c_2) V_s0_2) V_s1_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_s1_2) V_s2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_s0_2) V_s2_2))))))) (forall ((V_s2 $$unsorted) (V_c1 $$unsorted) (V_s1 $$unsorted) (V_s0 $$unsorted) (V_c0 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c0) V_s0) V_s1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c1) V_s1) V_s2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OSemi V_c0 V_c1)) V_s0) V_s2))))) (forall ((V_s $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc tptp.c_Com_Ocom_OSKIP) V_s) V_s))) (forall ((V_t $$unsorted) (V_s $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc tptp.c_Com_Ocom_OSKIP) V_s) V_t)) (= V_t V_s))) (forall ((V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) V_a2_2) V_a2_2)) (=> (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)) (exists ((B_s1 $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a3_2) V_a2_2) B_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) B_s1) V_a2_2))))))) (forall ((V_a5_2 $$unsorted) (V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) V_a2_2) V_a5_2)) (or (and (= V_a5_2 V_a2_2) (not _let_1)) (and _let_1 (exists ((B_s1 $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a3_2) V_a2_2) B_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) B_s1) V_a5_2))))))))) (forall ((V_a5_2 $$unsorted) (V_a2_2 $$unsorted) (V_a4_2 $$unsorted) (V_a1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OSemi V_a1_2 V_a4_2)) V_a2_2) V_a5_2)) (exists ((B_s1 $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a1_2) V_a2_2) B_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a4_2) B_s1) V_a5_2)))))) (forall ((V_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc tptp.c_Com_Ocom_OSKIP) V_a) V_a))) (forall ((V_com2_H $$unsorted) (V_com1_H $$unsorted)) (not (= (tptp.c_Com_Ocom_OSemi V_com1_H V_com2_H) tptp.c_Com_Ocom_OSKIP))) (forall ((V_com2_H $$unsorted) (V_com1_H $$unsorted)) (not (= tptp.c_Com_Ocom_OSKIP (tptp.c_Com_Ocom_OSemi V_com1_H V_com2_H)))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2) tptp.c_Com_Ocom_OSKIP))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= tptp.c_Com_Ocom_OSKIP (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2)))) (forall ((V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2) (tptp.c_Com_Ocom_OSemi V_com1_2 V_com2_2)))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_com2_2 $$unsorted) (V_com1_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OSemi V_com1_2 V_com2_2) (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2)))) (forall ((V_pname_H $$unsorted)) (not (= tptp.c_Com_Ocom_OSKIP (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H)))) (forall ((V_pname_H $$unsorted)) (not (= (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H) tptp.c_Com_Ocom_OSKIP))) (forall ((V_com2 $$unsorted) (V_com1 $$unsorted) (V_pname_H $$unsorted)) (not (= (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H) (tptp.c_Com_Ocom_OSemi V_com1 V_com2)))) (forall ((V_pname_H $$unsorted) (V_com2 $$unsorted) (V_com1 $$unsorted)) (not (= (tptp.c_Com_Ocom_OSemi V_com1 V_com2) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H)))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_pname_H_2 $$unsorted)) (not (= (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H_2) (tptp.c_Com_Ocom_OWhile V_fun_2 V_com_2)))) (forall ((V_pname_H_2 $$unsorted) (V_pname_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_2) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H_2)) (= V_pname_2 V_pname_H_2))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_com_2 $$unsorted) (V_fun_2 $$unsorted)) (= (= (tptp.c_Com_Ocom_OWhile V_fun_2 V_com_2) (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2)) (and (= V_fun_2 V_fun_H_2) (= V_com_2 V_com_H_2)))) (forall ((V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_com2_2 $$unsorted) (V_com1_2 $$unsorted)) (= (= (tptp.c_Com_Ocom_OSemi V_com1_2 V_com2_2) (tptp.c_Com_Ocom_OSemi V_com1_H_2 V_com2_H_2)) (and (= V_com1_2 V_com1_H_2) (= V_com2_2 V_com2_H_2)))) (forall ((V_pname_H_2 $$unsorted) (V_com_2 $$unsorted) (V_fun_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OWhile V_fun_2 V_com_2) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H_2)))) _let_211 (forall ((V_c_2 $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (=> tptp.c_Hoare__Mirabelle_Ostate__not__singleton (=> tptp.c_Com_OWT__bodies (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_Hoare__Mirabelle_OMGT V_c_2)) _let_2)))))))) (forall ((V_Y_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a (tptp.c_COMBI T_a)) V_Y_2) V_Y_2)) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2))) (not (forall ((B_x $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 B_x)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)))))))) (forall ((V_t $$unsorted) (V_s $$unsorted) (V_c2 $$unsorted) (V_c1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OSemi V_c1 V_c2)) V_s) V_t)) (not (forall ((B_s1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c1) V_s) B_s1)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c2) B_s1) V_t)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__valids T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_Opeek__and T_a V_P_2 V_b_2)) V_c_2) V_P_2)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__valids T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_P_2) (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2)) (tptp.c_Hoare__Mirabelle_Opeek__and T_a V_P_2 (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Com_Ostate tptp.c_fNot) V_b_2)))) _let_2)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_Opeek__and T_a V_P_2 V_b_2)) V_c_2) V_P_2)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_P_2) (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2)) (tptp.c_Hoare__Mirabelle_Opeek__and T_a V_P_2 (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Com_Ostate tptp.c_fNot) V_b_2)))) _let_2)))))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_Q_H_2 $$unsorted) (V_c0_2 $$unsorted) (V_P_H_2 $$unsorted) (V_Ga_2 $$unsorted) (T_b $$unsorted) (V_U_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_b))) (let ((_let_2 (tptp.tc_fun T_b (tptp.tc_fun tptp.tc_Com_Ostate tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun tptp.tc_Com_Ocom _let_3) T_a) (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_b)))) (let ((_let_5 (tptp.c_COMBS T_a tptp.tc_Com_Ocom _let_3))) (let ((_let_6 (tptp.c_COMBS T_a _let_2 _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_U_2)) (=> (forall ((B_p $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_b))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.hAPP V_c0_2 B_p))) (let ((_let_4 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_b))) (let ((_let_5 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_b V_Ga_2 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP V_P_H_2 B_p)) _let_3) (tptp.hAPP V_Q_H_2 B_p))) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_b V_Ga_2 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP V_P_2 B_p)) _let_3) (tptp.hAPP V_Q_2 B_p))) _let_2))))))))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_b V_Ga_2 (tptp.hAPP (tptp.c_Set_Oimage T_a _let_1 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_P_H_2)) V_c0_2)) V_Q_H_2)) V_U_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_b V_Ga_2 (tptp.hAPP (tptp.c_Set_Oimage T_a _let_1 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_P_2)) V_c0_2)) V_Q_2)) V_U_2)))))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_x) V_x))) (forall ((V_c_2 $$unsorted) (V_Ga_2 $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (=> tptp.c_Hoare__Mirabelle_Ostate__not__singleton (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple tptp.tc_Com_Ostate))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Com_Opname) B_x) (tptp.c_Map_Odom tptp.tc_Com_Opname tptp.tc_Com_Ocom tptp.c_Com_Obody))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_Hoare__Mirabelle_OMGT (tptp.hAPP tptp.c_Com_Ocom_OBODY B_x))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate V_Ga_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_Hoare__Mirabelle_OMGT V_c_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (tptp.hAPP _let_1 V_B_2)))))) (forall ((T_b $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted)) (=> (= (tptp.hAPP V_f_2 V_a_2) V_b_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a V_f_2) V_B_2)))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2))) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c_2)))) (forall ((V_c2 $$unsorted) (V_c1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OSemi V_c1 V_c2))) (not (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c1)) (not (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) V_A_2) V_x_2)) (tptp.hBOOL (tptp.hAPP V_A_2 (tptp.hAPP V_f_2 V_x_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (T_d $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_a_2) V_b_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_b T_a V_c_2) V_d_2)) (forall ((B_v $$unsorted)) (= (tptp.hAPP V_a_2 (tptp.hAPP V_b_2 B_v)) (tptp.hAPP V_c_2 (tptp.hAPP V_d_2 B_v)))))) (forall ((V_v_2 $$unsorted) (V_d_2 $$unsorted) (V_c_2 $$unsorted) (T_d $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_a_2) V_b_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_b T_a V_c_2) V_d_2)) (= (tptp.hAPP V_a_2 (tptp.hAPP V_b_2 V_v_2)) (tptp.hAPP V_c_2 (tptp.hAPP V_d_2 V_v_2))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c V_f_2) V_g_2) V_x_2) (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_g_2) V_f_2)) V_x_2) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_c V_f_2) (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_g_2) V_x_2)))) (forall ((V_Y_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_a (tptp.c_COMBI T_a)) V_Y_2) V_Y_2)) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_a V_g_2) V_h_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_b T_a (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_d V_f_2) V_g_2)) V_h_2))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c V_f_2) V_g_2) V_x_2) (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)))) (forall ((V_r_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_a T_b V_f_2) V_g_2)) V_r_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_a V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_g_2) V_r_2)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Com_Opname) (tptp.c_Map_Odom tptp.tc_Com_Opname tptp.tc_Com_Ocom tptp.c_Com_Obody))) (forall ((T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP V_f_2 V_a_2)) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a V_f_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_a_2)) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_a_2)) V_B_2)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_s_2 $$unsorted) (V_Z_2 $$unsorted) (V_p_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Opeek__and T_a V_P_2 V_p_2) V_Z_2) V_s_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_Z_2) V_s_2)) (tptp.hBOOL (tptp.hAPP V_p_2 V_s_2))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c_2)) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2))))) (forall ((V_a1_2 $$unsorted) (V_a2_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OWhile V_a2_2 V_a1_2))) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_a1_2)))) (forall ((V_c1 $$unsorted) (V_c0 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c0)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c1)) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OSemi V_c0 V_c1)))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OSemi V_a1_2 V_a2_2))) (and (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_a1_2)) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_a2_2))))) _let_210 _let_210 (forall ((V_b_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (= (tptp.hAPP V_f_2 V_a_2) V_b_2))) (forall ((V_h_2 $$unsorted) (T_b $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_F_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_h_2) V_F_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)))))) (forall ((T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_d_2 $$unsorted) (V_Q_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_P_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Hoare__Mirabelle_Otriple T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_Opeek__and T_a V_P_2 V_b_2)) V_c_2) V_Q_2)) _let_2)) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Hoare__Mirabelle_Opeek__and T_a V_P_2 (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Com_Ostate tptp.c_fNot) V_b_2))) V_d_2) V_Q_2)) _let_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_P_2) (tptp.c_Com_Ocom_OCond V_b_2 V_c_2 V_d_2)) V_Q_2)) _let_2))))))))) (forall ((V_Ga_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_F_2) V_Ga_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_F_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_Ga_2)))))) (forall ((V_Ga_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_F_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_Ga_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_F_2) V_Ga_2))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))))) (forall ((T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image _let_1 T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_Set_Oinsert T_b)) V_f_2)) _let_2)) _let_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_HOL_OThe T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2)) V_a_2)) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_c2_2 $$unsorted) (V_c1_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OCond V_b_2 V_c1_2 V_c2_2)) V_s_2) V_t_2)) (=> (=> _let_1 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c1_2) V_s_2) V_t_2)))) (not (=> (not _let_1) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c2_2) V_s_2) V_t_2))))))))) (forall ((V_c1_2 $$unsorted) (V_s1_2 $$unsorted) (V_c0_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c0_2) V_s_2) V_s1_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OCond V_b_2 V_c0_2 V_c1_2)) V_s_2) V_s1_2))))) (forall ((V_c0_2 $$unsorted) (V_s1_2 $$unsorted) (V_c1_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c1_2) V_s_2) V_s1_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OCond V_b_2 V_c0_2 V_c1_2)) V_s_2) V_s1_2))))) (forall ((V_c2_2 $$unsorted) (V_c1_2 $$unsorted) (V_b_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OCond V_b_2 V_c1_2 V_c2_2))) (not (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c1_2)) (not (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c2_2))))))) (forall ((V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_fun_2 $$unsorted)) (= (= (tptp.c_Com_Ocom_OCond V_fun_2 V_com1_2 V_com2_2) (tptp.c_Com_Ocom_OCond V_fun_H_2 V_com1_H_2 V_com2_H_2)) (and (= V_fun_2 V_fun_H_2) (= V_com1_2 V_com1_H_2) (= V_com2_2 V_com2_H_2)))) (forall ((V_z_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_a T_b V_f_2) V_g_2) V_z_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_z_2)) (forall ((V_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a5_2 $$unsorted) (V_a1_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OCond V_a1_2 V_a5_2 V_a3_2)) V_a2_2) V_a4_2)) (or (and _let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a5_2) V_a2_2) V_a4_2))) (and (not _let_1) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a3_2) V_a2_2) V_a4_2))))))) (forall ((V_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a5_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc (tptp.c_Com_Ocom_OCond V_a1_2 V_a3_2 V_a5_2)) V_a2_2) V_a4_2)) (or (and _let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a3_2) V_a2_2) V_a4_2))) (and (not _let_1) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_a5_2) V_a2_2) V_a4_2))))))) (forall ((V_b_2 $$unsorted) (V_c1_2 $$unsorted) (V_c0_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c0_2)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c1_2)) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OCond V_b_2 V_c0_2 V_c1_2)))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_a3_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OCond V_a3_2 V_a1_2 V_a2_2))) (and (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_a1_2)) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_a2_2))))) (forall ((V_pname_H_2 $$unsorted) (V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_fun_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OCond V_fun_2 V_com1_2 V_com2_2) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H_2)))) (forall ((V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_fun_2 $$unsorted) (V_pname_H_2 $$unsorted)) (not (= (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H_2) (tptp.c_Com_Ocom_OCond V_fun_2 V_com1_2 V_com2_2)))) (forall ((V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_fun_2 $$unsorted) (V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2) (tptp.c_Com_Ocom_OCond V_fun_2 V_com1_2 V_com2_2)))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_fun_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OCond V_fun_2 V_com1_2 V_com2_2) (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2)))) (forall ((V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_com2_2 $$unsorted) (V_com1_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OSemi V_com1_2 V_com2_2) (tptp.c_Com_Ocom_OCond V_fun_H_2 V_com1_H_2 V_com2_H_2)))) (forall ((V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OCond V_fun_H_2 V_com1_H_2 V_com2_H_2) (tptp.c_Com_Ocom_OSemi V_com1_2 V_com2_2)))) (forall ((V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= tptp.c_Com_Ocom_OSKIP (tptp.c_Com_Ocom_OCond V_fun_H_2 V_com1_H_2 V_com2_H_2)))) (forall ((V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OCond V_fun_H_2 V_com1_H_2 V_com2_H_2) tptp.c_Com_Ocom_OSKIP))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) _let_1) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_B_2)))))))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_HOL_OThe T_a (tptp.hAPP tptp.c_fequal V_x_2)) V_x_2)) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) V_A_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (=> (tptp.c_Big__Operators_Ocomm__monoid__big T_a T_b V_f_2 V_z_2 V_F_2) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_a T_b V_f_2) V_g_2) V_z_2) V_A_2))) (=> (not _let_2) (= _let_1 V_z_2))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_F_2 V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding__idem T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__idem T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_f_2 V_x_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__idem T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) (tptp.hAPP V_f_2 V_x_2)) _let_1)))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_N_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_F_2 V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_f_2 V_x_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a _let_3)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (not (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ominus T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b T_a)) V_A_2) V_B_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP V_A_2 V_x_2)) (tptp.hAPP V_B_2 V_x_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2))) (= (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_B_2) _let_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ominus T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b T_a)) V_A_2) V_B_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP V_A_2 V_x_2)) (tptp.hAPP V_B_2 V_x_2))))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_F_2) (= (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_x_2) V_x_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) _let_2) V_A_2) _let_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_B_2) V_A_2)) (tptp.hAPP _let_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_B_2) V_A_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 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 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_c $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_b T_b T_c (tptp.hAPP V_f_2 V_y_2)))) (let ((_let_2 (tptp.c_Fun_Ocomp T_b T_b T_c (tptp.hAPP V_f_2 V_x_2)))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_g_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_g_2))))))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_y_2))) (let ((_let_2 (tptp.hAPP V_f_2 V_x_2))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_2) _let_1)))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (T_f $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_e $$unsorted) (V_F_2 $$unsorted) (V_fa_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_fa_2 V_F_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_d T_c V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_e T_c V_g_2) V_h_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_d T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_d T_f V_f_2) V_g_2)) V_h_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.hAPP V_F_2 (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_F_2 _let_4)))))))))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_A_2)) V_B_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_B_2)))) (and (=> _let_5 (= _let_4 _let_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 _let_2)))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_4 (tptp.hAPP _let_3 V_A_2))) (= (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_4 V_B_2)) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_4 (tptp.hAPP _let_3 V_A_2))) (= (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_4 (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP _let_1 (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP _let_1 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (tptp.hAPP V_f_2 V_x_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2)))) (let ((_let_4 (tptp.hAPP V_F_2 V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_F_2 _let_3)))))))))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_b T_b T_c (tptp.hAPP V_f_2 V_x_2)))) (let ((_let_2 (tptp.hAPP _let_1 V_g_2))) (=> (tptp.c_Finite__Set_Ofolding__idem T_a T_b V_f_2 V_F_2) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (=> (tptp.c_Finite__Set_Ofolding__idem T_a T_b V_f_2 V_F_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) _let_1) _let_1)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (tptp.hAPP V_f_2 V_x_2)))))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_A_2))))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_b T_b T_c (tptp.hAPP V_F_2 V_B_2)))) (let ((_let_2 (tptp.c_Fun_Ocomp T_b T_b T_c (tptp.hAPP V_F_2 V_A_2)))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_g_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_g_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_B_2))) (let ((_let_2 (tptp.hAPP V_F_2 V_A_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_2) _let_1))))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_b T_b T_c (tptp.hAPP V_f_2 V_x_2)))) (let ((_let_2 (tptp.c_Fun_Ocomp T_b T_b T_c (tptp.hAPP V_F_2 V_A_2)))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_g_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_g_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_2) _let_1))))))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) _let_1) _let_1)))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Big__Operators_Ocomm__monoid__big T_a T_b V_f_2 V_z_2 V_F_2) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))) (= (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) V_A_2) V_z_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a _let_4)))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP V_f_2 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP V_F_2 V_A_2)) V_A_2))))))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_x_2) V_y_2)) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)))) (let ((_let_4 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_y_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_4) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_4))))))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 V_A_2))))))))) (forall ((V_P_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_A_2)) (=> (forall ((B_a $$unsorted) (B_A $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) B_A)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_a) B_A)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_A)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) B_A) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) B_a) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_F_2) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP V_f_2 B_x) B_y)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.hAPP V_F_2 V_N_2)) (tptp.hAPP V_F_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2)))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Finite__Set_Ofun__left__comm__idem T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_1) (tptp.c_Groups_Ominus__class_Ominus _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_F_2 V_A_2))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Rings_Oring T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP V_f_2 V_a_2))))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))) (=> (tptp.class_Groups_Oab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) _let_2) (tptp.hAPP V_f_2 V_a_2)))) (=> (not _let_5) (= _let_4 _let_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)) (not (=> (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)) (not (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2))))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_x_2) (tptp.hAPP V_f_2 V_x_2)) V_f_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (not (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_Ga_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (or (tptp.hBOOL (tptp.hAPP _let_1 V_F_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_Ga_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_F_2) V_Ga_2)))))) (forall ((T_a $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (=> (= (tptp.hAPP V_f_2 V_x_2) V_y_2) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b T_a) V_f_2) V_x_2) V_y_2) V_f_2))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted)) (=> (not (= V_z_2 V_x_2)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_x_2) V_y_2) V_z_2) (tptp.hAPP V_f_2 V_z_2)))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ofun__upd T_a T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_m_2))) (=> (not (= V_a_2 V_c_2)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_b_2)) V_c_2) V_d_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_c_2) V_d_2)) V_a_2) V_b_2)))))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_x_2) V_y_2) V_z_2))) (let ((_let_2 (= V_z_2 V_x_2))) (and (=> _let_2 (= _let_1 V_y_2)) (=> (not _let_2) (= _let_1 (tptp.hAPP V_f_2 V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b T_a) V_f_2) V_x_2) V_y_2) V_x_2) V_y_2)) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ofun__upd T_a T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_x_2))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y_2)) V_x_2) V_z_2) (tptp.hAPP _let_2 V_z_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_x_2) V_y_2) V_f_2) (= (tptp.hAPP V_f_2 V_x_2) V_y_2))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_x) V_x))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)) (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b T_a)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) V_C_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_c)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.hAPP _let_1 V_B_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_C_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.hAPP _let_1 V_B_2))) (= (tptp.hAPP _let_1 _let_2) _let_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b T_a)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_x) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_a) V_a))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_A_2) V_A_2)) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.hAPP _let_1 V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_b V_f_2) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_a))) (let ((_let_4 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_a) (tptp.tc_fun _let_4 T_a) T_b) _let_3) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_a) (tptp.tc_fun _let_2 T_a) T_c) _let_1) (tptp.hAPP (tptp.c_COMBC T_b T_c T_a) V_f_2))) V_A_2)) V_B_2)))))))) (forall ((V_b_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_a_2) V_b_2) V_x_2))) (let ((_let_2 (= V_x_2 V_a_2))) (and (=> _let_2 (= _let_1 V_b_2)) (=> (not _let_2) (= _let_1 (tptp.hAPP V_f_2 V_x_2))))))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_b V_f_2) (tptp.c_Finite__Set_Ofun__left__comm__idem T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b T_b) T_c) V_f_2) V_g_2)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot T_a))) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) _let_1) _let_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot T_a))) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) _let_1) V_x) _let_1)))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) V_x) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) V_x) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) V_x))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y)) V_x))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_B_2)) (not (= B_x B_xa))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) _let_2) _let_2)))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) _let_2) V_B_2) _let_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _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 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2))) (tptp.hAPP (tptp.hAPP _let_3 V_C_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2))) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_C_2)))) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2)) (tptp.hAPP _let_3 V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)) V_C_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_C_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))) (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2))))))))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_2) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.c_member T_a))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) _let_3))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_4 V_R_2)) (tptp.hAPP _let_4 V_S_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_x_2) (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_c T_b T_a V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_c) V_g_2) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) (tptp.hAPP _let_1 V_g_2)) V_x_2) (tptp.hAPP V_f_2 V_y_2))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_2) (tptp.hAPP V_f_2 V_a_2)))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))))) (forall ((T_a $$unsorted)) (tptp.c_Finite__Set_Ofun__left__comm__idem T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Set_Oinsert T_a))) (forall ((T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_b V_f_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) _let_1) _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.hAPP _let_1 V_B_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_B_2)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP _let_3 _let_2))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_B_2)) V_C_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_C_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP _let_3 _let_2))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_C_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_C_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) V_C_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_B_2) V_A_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2) V_A_2)))) (forall ((V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (= (tptp.hAPP V_F_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_z_2))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_B_2) V_C_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 V_C_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2) V_C_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 V_C_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)) V_A_2))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)) V_a_2))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_a T_b V_f_2) V_g_2) V_z_2) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))) (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) _let_2) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))) (tptp.hAPP V_F_2 _let_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) _let_2) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_B_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a _let_3)))))))))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) _let_2) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_B_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a _let_4)))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (= (tptp.hAPP V_g_2 B_x) V_z_2))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a)) V_x_2) V_A_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Oab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))))))))))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (= (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_x_2) V_x_2))) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_a V_f_2) V_z_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_z_2)) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_2) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)) V_a_2) V_A_2))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_a T_b V_f_2) V_g_2) (tptp.c_Finite__Set_Ofold T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) V_f_2) V_g_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)) V_x_2) V_A_2)))))) (forall ((V_s_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_F_2 V_A_2) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_s_2) V_A_2))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) _let_2) V_g_2)) V_h_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_h_2) V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a _let_1 (tptp.c_Set_Oinsert T_a)) V_B_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)))))))))) (forall ((V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold T_a T_b V_f_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_z_2)) V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a V_f_2) V_a_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) V_a_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) _let_1) _let_1)))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_F_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_F_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_F_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP _let_1 V_F_2)))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a)) V_a_2) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_1) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_B_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a V_f_2) V_x_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_B_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) _let_1) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_F_2 V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) V_A_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) V_b) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b)) V_b) V_a))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_N_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2)))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a T_b) V_A_2) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b)) V_f_2)) (tptp.c_Orderings_Obot__class_Obot T_b)) V_A_2))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold T_a T_b V_f_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_z_2)) V_A_2))))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a)) (tptp.c_Orderings_Obot__class_Obot T_a)) V_A_2))))) (forall ((T_b $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2) V_c_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.hAPP V_f_2 V_y_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z_2))))))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (tptp.c_Finite__Set_Ofun__left__comm T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b T_b) T_c) V_f_2) V_g_2)))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_B_2 B_x)))))))) (forall ((V_f_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c _let_1 T_b V_g_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_g_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_A_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.hAPP V_f_2 V_y_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_2) _let_1)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Oab__semigroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (= (= (tptp.hAPP _let_1 V_b_2) (tptp.hAPP _let_1 V_c_2)) (= V_b_2 V_c_2))))) (forall ((V_c_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_a_2)) (= V_b_2 V_c_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_c)) (= V_b V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Groups_Ocancel__ab__semigroup__add T_a) (=> (= (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_c)) (= V_b V_c))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a)) (= V_b V_c))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2) (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)) (= (= V_a_2 V_b_2) (= V_c_2 V_d_2)))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofold T_a T_b V_f_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z_2)) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_B_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_B_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_A_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a T_b) V_A_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) _let_1) V_f_2)) V_B_2) V_A_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a_2) V_b_2))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_2 V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_3) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_Set_Oinsert T_a)) V_f_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_B_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_b_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP V_B_2 V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_A_2))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_a_2))) (forall ((T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((V_M_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK T_b T_a) V_M_2)) V_M_2)))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_X_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_X_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_X_2)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_C_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP V_B_2 V_a_2)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_B_2) V_A_2))))) (forall ((V_S_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_1) V_S_2) (tptp.c_COMBI _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 (tptp.tc_fun T_b _let_2) _let_2) _let_3) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)) V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_2))) (let ((_let_4 (tptp.tc_fun T_c _let_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_4 _let_2) T_b) _let_3) V_B_2)) V_C_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2)) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_3 V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_3 _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 (tptp.tc_fun T_b _let_1) _let_1) _let_2) V_B_2))))))) (forall ((V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) (tptp.hAPP V_B_2 B_x))) (tptp.hBOOL (tptp.hAPP V_P_2 B_xa)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (let ((_let_4 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_4) V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_1) T_c) _let_2) V_B_2)) V_C_2)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (let ((_let_4 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_4) V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_1) T_c) _let_2) V_B_2)) V_C_2)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2) V_c_2)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oinsert _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_a_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_a_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP V_B_2 B_x))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.c_Orderings_Obot__class_Obot _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) _let_2)) _let_2)))) (forall ((V_c_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_c_2)))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_3) (= _let_2 V_c_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_1 T_b) V_B_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_1 T_c) V_B_2) V_f_2))))) (forall ((V_S_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_I_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_M_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) V_M_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_M_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_M_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_2) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_A_2) V_C_2))) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_J_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.tc_fun _let_1 _let_1))) (let ((_let_4 (tptp.tc_fun T_c _let_1))) (let ((_let_5 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_J_2))) (let ((_let_6 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_I_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_6 V_A_2)) (tptp.hAPP _let_5 V_B_2)) (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_1 T_b) _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_4) T_b) (tptp.c_COMBB _let_1 _let_1 T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) _let_2) V_A_2))) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) V_B_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_2) V_A_2)) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_B_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_I_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_1) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_1) _let_2) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_1) V_B_2) _let_2))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c T_a)) V_A_2) V_f_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_a) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_a) V_f_2) V_x_2))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_a) V_A_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c T_a) V_B_2))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_a) T_a T_b) _let_2) V_f_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_a) T_a T_c) _let_1) (tptp.hAPP (tptp.c_COMBC T_b T_c T_a) V_f_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) V_B_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_2) V_A_2)) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) _let_1))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_C_2)) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) V_C_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) B_x) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_u_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_u_2) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2)))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (V_I_2 $$unsorted) (V_k_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_I_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_k_2) V_I_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP V_A_2 V_k_2)) _let_2) _let_2))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_3) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_1))) V_A_2))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_2) V_C_2))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.hAPP _let_1 (tptp.c_Orderings_Obot__class_Obot _let_2)))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_a) _let_1) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_3) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_C_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) _let_3) V_A_2)) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_C_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 V_A_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_C_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) V_B_2))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 V_B_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) _let_2) V_A_2)) V_B_2))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_A_2)) (=> (forall ((B_M $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) B_M) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) B_M)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2))))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_k)) (tptp.hAPP (tptp.hAPP _let_2 V_n) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_i))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_j)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_k)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)) V_n) V_m)) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) V_n) V_m)) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_B_2 B_a))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_F_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_x_2) V_F_2))) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_x_2) V_F_2))) (forall ((B_xa $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_xa) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_x_2) V_F_2))) (=> (not (= B_x B_xa)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) B_x) B_xa) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_x_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_F_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a T_b) V_A_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) _let_1) V_f_2)) V_B_2) V_A_2)))))) (forall ((V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_A_2) (tptp.c_Complete__Lattice_OSup__class_OSup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_C_2))) (not (forall ((B_X $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_A_2) B_X)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) B_X) V_C_2)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_b_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP V_B_2 V_a_2))))))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_b_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP V_B_2 V_a_2)))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2) V_c_2))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2) V_c_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (V_I_2 $$unsorted) (T_a $$unsorted)) (=> (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_I_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_A_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_I_2) V_A_2))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c T_a)) V_A_2) V_f_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_a) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_a) V_f_2) V_x_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_3 V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI _let_3 _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 (tptp.tc_fun T_b _let_1) _let_1) _let_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI _let_1 _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 (tptp.tc_fun T_b _let_2) _let_2) _let_3) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)) V_B_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_a) V_A_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c T_a) V_B_2))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_a) T_a T_b) _let_2) V_f_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_a) T_a T_c) _let_1) (tptp.hAPP (tptp.c_COMBC T_b T_c T_a) V_f_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP V_B_2 B_x))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_2) V_C_2))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) _let_1) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_1 T_c) V_B_2) V_f_2))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_1 T_b) V_B_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_J_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.tc_fun _let_1 _let_1))) (let ((_let_4 (tptp.tc_fun T_c _let_1))) (let ((_let_5 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_J_2))) (let ((_let_6 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_I_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_6 V_A_2)) (tptp.hAPP _let_5 V_B_2)) (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_1 T_b) _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_4) T_b) (tptp.c_COMBB _let_1 _let_1 T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) _let_2) V_A_2))) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) V_B_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_2) V_A_2)) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_I_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_I_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (let ((_let_4 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_4) V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_1) T_c) _let_2) V_B_2)) V_C_2)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2))) (let ((_let_4 (tptp.tc_fun T_c _let_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_4 _let_2) T_b) _let_3) V_B_2)) V_C_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2)) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_u_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_u_2) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2)))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (V_I_2 $$unsorted) (V_k_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_I_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_k_2) V_I_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP V_A_2 V_k_2)) _let_2) _let_2))))) (forall ((V_M_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK T_b T_a) V_M_2)) V_M_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) V_B_2))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 V_B_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) _let_2) V_A_2)) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 V_A_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP V_B_2 V_a_2)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)))))) (forall ((V_M_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) V_M_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_M_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_M_2)))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_C_2) V_B_2)))) (let ((_let_4 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_4 (= _let_3 V_A_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_2) V_B_2)))))))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2)) (= V_m_2 V_n_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2)))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_n)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_i))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_j)) V_k) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_k)) V_j))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a T_b) V_A_2) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) V_f_2)) (tptp.c_Orderings_Otop__class_Otop T_b)) V_A_2))))) (forall ((V_P_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_F_2)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_F $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) B_F)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) B_F))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_F)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) B_x) B_F))))))) (tptp.hBOOL (tptp.hAPP V_P_2 V_F_2)))))) (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 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (=> (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) V_y_2) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 V_A_2 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_y_2)))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_c_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c (tptp.hAPP (tptp.c_COMBK T_a T_b) V_c_2)) V_f_2) V_x_2) V_c_2)) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_y_2))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (not (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2) V_c_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_3 V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) T_a) (tptp.c_If T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_B_2))) V_f_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))) V_A_2)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_A_2)))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_M_2 $$unsorted) (T_a $$unsorted)) (=> (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) B_x) V_M_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) B_x)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Complete__Lattice_OInf__class_OInf (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_M_2))))) (forall ((V_X_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OInf__class_OInf _let_2 V_C_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) V_X_2) V_C_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_X_2))))))) (forall ((V_X_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OInf__class_OInf _let_2 V_C_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) V_X_2) V_C_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_X_2))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2)))))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2) V_c_2)))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Ozero T_a) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Otop T_a) (= (tptp.hAPP (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b T_a)) V_x_2) (tptp.c_Orderings_Otop__class_Otop T_a)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) _let_1)) (=> (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP _let_2 _let_1)) (= V_A_2 _let_1)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_sum T_a T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.c_Orderings_Otop__class_Otop _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Otop__class_Otop _let_1)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((V_S_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI _let_1 _let_1) V_S_2) (tptp.c_COMBI _let_1))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop T_a)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop T_a)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (or (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_a1_2 V_a2_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_a2_2)) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_z_2)) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (= (tptp.c_Orderings_Otop__class_Otop _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2)) (= V_a_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) V_a))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) _let_2) V_B_2) _let_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) _let_2) _let_2)))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_Orderings_Otop__class_Otop _let_1)) V_B_2) V_B_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) (tptp.c_Orderings_Otop__class_Otop _let_1)) V_A_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_a)) (= V_a_2 V_b_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (= V_a_2 V_b_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n) V_n)) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n) V_m) (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n) _let_1) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_n) V_m) _let_1) (= V_m V_n)))))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_z_2))) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_z_2))) (forall ((V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_x_2)) (= V_x_2 V_z_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (let ((_let_3 (tptp.c_member T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) _let_3)) V_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_x_2) (tptp.hAPP _let_2 V_r_2)))))))) (forall ((V_c_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_c_2)))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_3) (= _let_2 V_c_2))))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_B_2) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c T_a T_b) V_f_2) V_g_2)) _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_c T_a V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_g_2) _let_1))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) (tptp.hAPP (tptp.c_COMBK T_a T_b) _let_1)) V_A_2) _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI _let_1 _let_1) V_B_2) _let_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (= V_y_2 V_x_2)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop T_a))) (=> (tptp.class_Lattices_Obounded__lattice__top T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) _let_1) V_x) _let_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop T_a))) (=> (tptp.class_Lattices_Obounded__lattice__top T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) _let_1) _let_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice__top T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.c_Orderings_Otop__class_Otop T_a)) V_x) V_x))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice__top T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) (tptp.c_Orderings_Otop__class_Otop T_a)) V_x))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop T_a))) (=> (tptp.class_Lattices_Obounded__lattice__top T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_B_2) V_A_2))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))))) (forall ((T_b $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)) (tptp.c_Orderings_Otop__class_Otop T_a)) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.c_Orderings_Otop__class_Otop _let_1)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) _let_2) _let_3) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_c V_g_2) _let_3) _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_c T_b V_g_2) V_f_2)) _let_2) _let_1))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_g_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) V_g_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (=> (= (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a V_f_2) V_A_2)) V_A_2)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_y_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_y_2)))))) (forall ((V_f_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_F_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2) V_F_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_F_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (let ((_let_3 (tptp.c_member T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) _let_3)) V_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_x_2) (tptp.hAPP _let_2 V_r_2)))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_g_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_g_2) _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_c T_b) V_f_2) V_g_2)) _let_1)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_3) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) _let_3) V_A_2)) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) _let_3) V_A_2)) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) V_B_2))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Orderings_Otop__class_Otop _let_1)) V_B_2))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) _let_2) V_A_2)) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_a_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_a_2) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2)) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_B_2))))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_z_2) V_A_2) V_y_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_2) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_A_2) V_C_2))) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 V_B_2) V_C_2))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_a_2))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_A_2))))) (forall ((V_F_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_h_2) V_F_2))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_h_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2))))) (forall ((V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_a V_f_2) V_z_2) V_A_2) (tptp.c_HOL_OThe T_a (tptp.c_Finite__Set_Ofold__graph T_b T_a V_f_2 V_z_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))))) (forall ((V_v_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_a_2) V_b_2) V_c_2) (= (tptp.hAPP V_a_2 (tptp.hAPP V_b_2 V_v_2)) (tptp.hAPP V_c_2 V_v_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) _let_2) V_B_2)))) (let ((_let_4 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_4 (= _let_3 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_C_2) V_B_2)))))))))) (forall ((T_b $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a (tptp.hAPP (tptp.c_COMBK T_a T_b) V_c_2)) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_c_2) V_A_2)))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_3) (= _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) T_a) (tptp.c_If T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_b_2)) _let_1)) V_S_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (and (=> _let_4 (= _let_3 (tptp.hAPP V_b_2 V_a_2))) (=> (not _let_4) (= _let_3 _let_1)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) T_a) (tptp.c_If T_b)) (tptp.hAPP tptp.c_fequal V_a_2))) V_b_2)) _let_1)) V_S_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (and (=> _let_4 (= _let_3 (tptp.hAPP V_b_2 V_a_2))) (=> (not _let_4) (= _let_3 _let_1)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))) (let ((_let_3 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) _let_2)) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 _let_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oinsert _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))))) (forall ((V_z_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_z_2) V_A_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a_2) V_b_2))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a (tptp.c_Groups_Oplus__class_Oplus T_b)) V_g_2) _let_1) V_A_2))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a (tptp.c_Groups_Oplus__class_Oplus T_b)) V_g_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_B_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_B_2) V_A_2)))))) (forall ((V_fun2_2 $$unsorted) (V_com_2 $$unsorted) (V_fun1_2 $$unsorted) (V_fa_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Hoare__Mirabelle_Otriple_Otriple__size T_a V_fa_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_fun1_2) V_com_2) V_fun2_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_a V_f_2 V_a_2 V_A_2) V_x_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) V_x_2))))) (forall ((V_a_2 $$unsorted) (V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (exists ((B_y_H $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (and (= V_y_2 (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) B_y_H)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) B_y_H))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (= (tptp.hAPP V_g_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_f_2 $$unsorted) (V_T_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_S_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_1 V_S_2)) (tptp.hAPP _let_1 V_T_2)))))))))) (forall ((T_b $$unsorted) (V_h_2 $$unsorted) (V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_S_2)) (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 B_x))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_y) V_S_2)) (=> (not (= B_x B_y)) (=> (= _let_1 (tptp.hAPP V_f_2 B_y)) (= (tptp.hAPP V_h_2 _let_1) (tptp.c_Groups_Ozero__class_Ozero T_c))))))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_h_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_c) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_h_2) V_f_2)) V_S_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (=> (not _let_5) (= _let_4 _let_2)))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_x_2))) (= (tptp.c_Orderings_Obot__class_Obot tptp.tc_Nat_Onat) _let_31) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Product__Type_Ounit) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Product__Type_Ounit tptp.tc_HOL_Obool))) _let_62) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Groups_Oone T_a) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_a V_f_2 V_A_2) V_x_2)) (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_a V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_x_2)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2) V_A_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_A_2)) (=> (not (= B_x B_xa)) (= (tptp.hAPP V_f_2 B_xa) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBK tptp.tc_Nat_Onat T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_A_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image tptp.tc_Nat_Onat T_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_COMBK tptp.tc_Nat_Onat T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) _let_1) V_A_2))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image tptp.tc_Nat_Onat T_a (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_COMBK tptp.tc_Nat_Onat T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_A_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_b_2)) (= V_a_2 V_b_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_member T_a) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP _let_3 V_B_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_B_2))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_3 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_fun2_2 $$unsorted) (V_com_2 $$unsorted) (V_fun1_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_Hoare__Mirabelle_Otriple T_a)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_fun1_2) V_com_2) V_fun2_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))) (= (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP V_a3_2 V_a1_2) V_a5_2))) (= (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_a3_2 V_a4_2 V_a2_2 _let_1) (or (and (= V_a2_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= _let_1 V_a4_2)) (exists ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (and (= (tptp.hAPP (tptp.hAPP V_a3_2 V_a1_2) V_a5_2) (tptp.hAPP (tptp.hAPP V_a3_2 B_x) B_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_a2_2)) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_a3_2 V_a4_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_a2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) B_x) (tptp.c_Orderings_Obot__class_Obot _let_1))) B_y)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Ozero__neq__one T_a) (not (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Ozero__neq__one T_a) (not (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (= (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_b_2) V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_c)) V_b)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_c)) V_d)))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (let ((_let_3 (tptp.hAPP _let_1 V_c))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_d)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_d)))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_sum T_a T_b)) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)))) (let ((_let_2 (and (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2))))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_B_2)))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a _let_1) V_g_2) (tptp.c_Groups_Oone__class_Oone T_b)))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_3)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Set_Oinsert T_a))) (let ((_let_5 (tptp.c_Groups_Ominus__class_Ominus _let_2))) (let ((_let_6 (tptp.c_member T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_6 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_6 V_y_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_5 V_A_2) (tptp.hAPP (tptp.hAPP _let_4 V_x_2) _let_3))) (tptp.hAPP (tptp.hAPP _let_4 V_y_2) _let_3)))) (tptp.hAPP _let_1 V_A_2))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_2))))) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_2))))) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_Q_2 $$unsorted) (V_pn_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn_2)) V_Q_2))) (forall ((V_x_2 $$unsorted) (V_X_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_X_2)) V_x_2)) (not (forall ((B_a $$unsorted) (B_A $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_X_2) (tptp.hAPP (tptp.hAPP _let_1 B_a) B_A)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_a V_f_2 B_a B_A) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_a) B_A))))))))) (forall ((V_a3_2 $$unsorted) (V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a1_2) V_a2_2)) V_a3_2)) (exists ((B_a $$unsorted) (B_A $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (and (= (tptp.hAPP (tptp.hAPP _let_1 V_a1_2) V_a2_2) (tptp.hAPP (tptp.hAPP _let_1 B_a) B_A)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_a V_f_2 B_a B_A) V_a3_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_a) B_A)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Oring__char__0 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.c_Int_Oring__1__class_OInts T_a))) (not (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone T_a)) V_a_2)) V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n)))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__comm__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_a_2)) (tptp.hAPP _let_3 V_b_2))) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_c_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)))))))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP _let_3 V_b_2) V_c_2))) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_c_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__ring T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 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(tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_lx) V_ly)))) (let ((_let_3 (tptp.hAPP _let_1 V_rx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_ry)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ry)))))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_rx) V_ry))) (let ((_let_3 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_ly)) _let_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_ly) _let_2)))))))) (forall ((V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ly)) V_rx) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_rx)) V_ly)))))) (forall ((V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ly)) V_rx) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ly) V_rx))))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_rx) V_ry)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_rx)) V_ry)))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (let ((_let_3 (tptp.hAPP _let_1 V_rx))) (=> (tptp.class_Rings_Ocomm__semiring__1 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(forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))) (=> (not (= V_x V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_z))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (tptp.hBOOL (tptp.hAPP _let_2 V_x)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)) (tptp.hBOOL (tptp.hAPP _let_2 V_z)))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a))) (=> (tptp.class_Orderings_Oord T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP _let_1 V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_c))) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 V_a))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y)) (not (= V_y V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y)) (not (= V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y)) (not (= V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (= V_x V_y)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2))) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2))) (= V_x_2 V_y_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (= V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_x) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a) V_a))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (= V_x_2 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (= V_x_2 V_y_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_x))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) (tptp.hAPP _let_1 V_n))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)) (exists ((B_k $$unsorted)) (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) B_k)))))) (forall ((V_m $$unsorted) (V_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_i))))) (forall ((V_m $$unsorted) (V_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_m))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_n_2) V_m_2)))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_n_2)) _let_1) (=> (=> (= V_m_2 V_n_2) _let_1) (=> (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_m_2)) _let_1) _let_1)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (= (tptp.hAPP tptp.c_Nat_OSuc V_x) (tptp.hAPP tptp.c_Nat_OSuc V_y)) (= V_x V_y))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_n))) (= V_m V_n))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_k)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (=> (not (= _let_1 V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_n))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)))))) (forall ((V_t $$unsorted) (V_s $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_s) V_t)) (not (= V_s V_t)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_m)) (not (= V_m V_n)))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_n)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (not (= V_x V_y)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m))) (= V_m V_n))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_m_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m_2))) (= V_n_2 V_m_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))) (forall ((V_nat_H_2 $$unsorted) (V_nat_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Nat_OSuc V_nat_2) (tptp.hAPP tptp.c_Nat_OSuc V_nat_H_2)) (= V_nat_2 V_nat_H_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (not (= V_m_2 V_n_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_m_2)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2)))))) (forall ((V_n $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_n) V_n))) (forall ((V_n $$unsorted)) (not (= V_n (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_n)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_m)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) V_m)))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_n)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_m)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j)))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)))))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_k_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (or (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (exists ((B_j $$unsorted)) (and (= V_m_2 (tptp.hAPP tptp.c_Nat_OSuc B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) V_n_2))))))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= V_n_2 _let_1)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_m_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_n_2)))))) (forall ((V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2)) (exists ((B_m $$unsorted)) (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc B_m))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_b_2) V_a_2)) V_e_2)) V_d_2))))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2)) V_e_2)) V_c_2)) V_d_2)))))))) (forall ((V_i $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP tptp.c_Nat_OSuc V_i))) V_n))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_n)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP tptp.c_Nat_OSuc _let_1))) V_n)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Omult__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_a) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_a) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Omult__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oring__no__zero__divisors T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_b_2) _let_1) (or (= V_a_2 _let_1) (= V_b_2 _let_1)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ono__zero__divisors T_a) (=> (not (= V_a _let_1)) (=> (not (= V_b _let_1)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ono__zero__divisors T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) _let_1) (or (= V_a _let_1) (= V_b _let_1)))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_w_2))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (= (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_y_2)) (tptp.hAPP _let_2 V_z_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_z_2)) (tptp.hAPP _let_2 V_y_2))) (or (= V_w_2 V_x_2) (= V_y_2 V_z_2))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_e $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Osemiring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_e)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_e)) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_e)) V_c)))))) (forall ((V_b $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_m)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_m)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_m)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b_2))) (let ((_let_3 (tptp.hAPP _let_1 V_a_2))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (= (and (not (= V_a_2 V_b_2)) (not (= V_c_2 V_d_2))) (not (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) (tptp.hAPP _let_2 V_d_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_d_2)) (tptp.hAPP _let_2 V_c_2))))))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c_2))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b)))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_m $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_nat_H $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_nat_H)))) (forall ((V_m $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_nat_H_1 $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_nat_H_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_m $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_b_2) (tptp.hAPP (tptp.hAPP _let_2 V_c_2) V_d_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP tptp.c_Nat_OSuc V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_n))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b))))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) _let_1) _let_1))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (or (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (or (= V_m_2 V_n_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2)) (or (= V_m_2 V_n_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_a $$unsorted) (V_x $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_b)) V_x)))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_b)) V_x)))))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (not (= V_n_2 _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_n_2))))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not (= V_n _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)) (tptp.hAPP tptp.c_Nat_OSuc V_k)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) V_k)))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) V_k) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_j)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l)) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_m) V_l) (tptp.hAPP (tptp.hAPP _let_2 V_k) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k) V_l)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_l)))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_j)))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_m)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2)))))) (forall ((V_i $$unsorted) (V_j $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_i)) V_i)))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_j)) V_i)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) V_k) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_j $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j) V_n)) V_k))))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_l))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP _let_2 V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_n)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_l)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 V_m))))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2)) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n) V_n)) (forall ((V_S_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_S_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2) B_x))))))) (forall ((V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_S_2)) V_a_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) B_x) V_a_2))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (tptp.c_Finite__Set_Ofun__left__comm T_a T_a (tptp.c_Groups_Otimes__class_Otimes T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_a (tptp.c_Groups_Otimes__class_Otimes T_a)))) (forall ((V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n))) (forall ((V_B_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.tc_fun T_a T_a))) (let ((_let_3 (tptp.tc_fun T_c T_a))) (let ((_let_4 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_a))) (let ((_let_5 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (=> (tptp.class_Rings_Osemiring__0 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_6 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_4 V_g_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_5 T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_5 T_a) T_b) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_3 _let_3) T_b) (tptp.c_COMBB T_a T_a T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a _let_2 T_b) _let_1) V_f_2))) V_g_2))) V_B_2)) V_A_2)))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (=> (tptp.class_Rings_Osemiring__0 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2)) V_r_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_1) V_f_2)) V_r_2)) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_r_2))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (=> (tptp.class_Rings_Osemiring__0 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a T_b) _let_1) V_f_2)) V_A_2)))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_a) V_A_2) V_f_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2) (tptp.hAPP V_f_2 B_x)))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_a) V_A_2) V_f_2)) V_a_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP V_f_2 B_x)) V_a_2))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (V_r $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_r))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (=> (not (= V_r (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (= V_a V_b) (not (= V_c V_d))) (not (= (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) (tptp.hAPP _let_1 V_d)))))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) _let_1)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2)) (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Oring T_a) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2)) (= V_c_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_b_2) V_a_2)) V_e_2)) V_d_2))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Oring T_a) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2)) V_e_2)) V_c_2) V_d_2)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 V_m) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_1)) V_m)))))) (forall ((V_a $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP (tptp.hAPP _let_2 V_a) V_m)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.c_Groups_Oone__class_Oone T_a))) V_m)))))) (forall ((V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_m)) V_m) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.c_Groups_Oone__class_Oone T_a))) V_m)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2) _let_2) (or (and (= V_m_2 _let_2) (= V_n_2 _let_1)) (and (= V_m_2 _let_1) (= V_n_2 _let_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2)) (or (and (= V_m_2 _let_2) (= V_n_2 _let_1)) (and (= V_m_2 _let_1) (= V_n_2 _let_2))))))) (= _let_62 _let_174) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_m_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_n_2)))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n_2) V_m_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_n)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_m)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_m))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (= V_m (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (or (= V_n (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (= V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (forall ((V_k_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i_2) V_k_2)) V_j_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_m))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.c_Int_Oring__1__class_OInts T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Oone__class_Oone T_a)) V_a_2)) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Int_Oring__1__class_OInts T_a))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_a))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_b_2)) _let_1)))))))) (forall ((V_Q_2 $$unsorted) (V_pn_2 $$unsorted) (V_P_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2))) (= (tptp.c_Hoare__Mirabelle_Otriple__valid T_a V_n_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_pn_2))) V_Q_2)) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a (tptp.hAPP tptp.c_Nat_OSuc V_n_2) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn_2)) V_Q_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_a))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2)) _let_1)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Int_Oring__1__class_OInts T_a))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_1) _let_1)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_a_2) V_b_2))) (and (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((B_d $$unsorted)) (=> (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_b_2) B_d)) (tptp.hBOOL (tptp.hAPP V_P_2 B_d))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_a_2) V_b_2))) (not (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_a_2) V_b_2)) (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (exists ((B_d $$unsorted)) (and (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_b_2) B_d)) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_d))))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_a _let_1 V_b_2 V_A_2) V_y_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_a _let_1 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_z_2) V_y_2)))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2) V_A_2) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_A_2)) (=> (not (= B_x B_xa)) (= (tptp.hAPP V_f_2 B_xa) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.c_Finite__Set_Ofold__image T_b T_a _let_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) _let_2) V_g_2)) V_h_2)) _let_1) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_g_2) _let_1) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_h_2) _let_1) V_A_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP tptp.c_Nat_OSuc _let_2)))))))))) (forall ((V_z_2 $$unsorted) (V_g_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a _let_1) V_g_2) V_z_2))) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_g_2 V_a_2)) (tptp.hAPP _let_2 V_A_2))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))) (and (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) _let_1)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Int_Oring__char__0 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.c_Int_Oring__1__class_OInts T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2) _let_1) (= V_a_2 _let_1)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b)) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b)) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b)) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_a T_b)) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_3 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_sum T_a T_b)) (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_B_2)))))) (forall ((V_a_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oinsert T_a))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_4 (tptp.c_member T_a))) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_a _let_3 V_b_2 V_A_2) V_x_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_a_2) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_b_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_a _let_3 V_a_2 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))) V_x_2))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a _let_2) V_g_2) (tptp.c_Groups_Oone__class_Oone T_b)))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)))))))))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (=> (not (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))) (let ((_let_3 (= V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_3 (= _let_2 V_n)) (=> (not _let_3) (= _let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n))))))))) (forall ((V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= V_n (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Sum__Type_OPlus T_a T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun (tptp.tc_sum T_a T_b) tptp.tc_HOL_Obool)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_4 (= V_m _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n)))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_1 V_x) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_y))) (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2)))) (or (not (= V_x_2 _let_1)) (not (= V_y_2 _let_1)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (= (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_n))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_n)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)))) (forall ((V_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Sum__Type_OPlus T_a T_b V_A_2 V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_sum T_a T_b) tptp.tc_HOL_Obool))) (and (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_B_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2))))) (forall ((V_f_2 $$unsorted) (V_T_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a _let_1) V_f_2) (tptp.c_Groups_Oone__class_Oone T_b)))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_S_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_b)))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_S_2)) (tptp.hAPP _let_2 V_T_2))))))))))) (forall ((V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (exists ((B_b $$unsorted) (B_B $$unsorted)) (and (= V_A_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) B_b) B_B)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_b) B_B))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) B_B) V_k_2) (=> (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= B_B (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_u $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_i) V_u)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_u)) V_k)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_j)) V_u)) V_k))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (or (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_m_2 V_n_2))))) (forall ((V_y $$unsorted) (V_z $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_C_2))))))) (forall ((T_e $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (V_x_H_2 $$unsorted) (V_g_H_2 $$unsorted) (V_f_H_2 $$unsorted) (V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted)) (=> (= (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_f_H_2 (tptp.hAPP V_g_H_2 V_x_H_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_a T_e V_f_H_2) V_g_H_2) V_x_H_2)))) (forall ((V_com2 $$unsorted) (V_com1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.c_Com_Ocom_Ocom__size (tptp.c_Com_Ocom_OSemi V_com1 V_com2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Com_Ocom_Ocom__size V_com1)) (tptp.c_Com_Ocom_Ocom__size V_com2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_fun_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.c_Com_Ocom_Ocom__size (tptp.c_Com_Ocom_OCond V_fun_2 V_com1_2 V_com2_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Com_Ocom_Ocom__size V_com1_2)) (tptp.c_Com_Ocom_Ocom__size V_com2_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted)) (= (tptp.c_Com_Ocom_Ocom__size (tptp.c_Com_Ocom_OWhile V_fun_2 V_com_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Com_Ocom_Ocom__size V_com_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_com2 $$unsorted) (V_com1 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize tptp.tc_Com_Ocom))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.c_Com_Ocom_OSemi V_com1 V_com2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_com1)) (tptp.hAPP _let_1 V_com2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_com2_2 $$unsorted) (V_com1_2 $$unsorted) (V_fun_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize tptp.tc_Com_Ocom))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.c_Com_Ocom_OCond V_fun_2 V_com1_2 V_com2_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_com1_2)) (tptp.hAPP _let_1 V_com2_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize tptp.tc_Com_Ocom))) (= (tptp.hAPP _let_1 (tptp.c_Com_Ocom_OWhile V_fun_2 V_com_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_com_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2))))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m_2))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_COMBI tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_n_2))))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nat_Onat_Onat__case T_a V_f1_2 V_f2_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_f1_2)) (forall ((V_nat_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nat_Onat_Onat__case T_a V_f1_2 V_f2_2 (tptp.hAPP tptp.c_Nat_OSuc V_nat_2)) (tptp.hAPP V_f2_2 V_nat_2))) (forall ((V_pname $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_pname $$unsorted)) (= (tptp.c_Com_Ocom_Ocom__size (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Com_Ocom) tptp.c_Com_Ocom_OSKIP) _let_31) (= (tptp.c_Com_Ocom_Ocom__size tptp.c_Com_Ocom_OSKIP) _let_31) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_a_2) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_a _let_1))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Orderings_Obot__class_Obot _let_3))) (let ((_let_5 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (=> (tptp.hBOOL (tptp.hAPP _let_5 V_A_2)) (=> (not (= V_A_2 _let_4)) (=> (tptp.hBOOL (tptp.hAPP _let_5 V_B_2)) (=> (not (= V_B_2 _let_4)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2) _let_4) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))))))))))) (forall ((V_s1 $$unsorted) (V_n $$unsorted) (V_s0 $$unsorted) (V_pn $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_pn))) V_s0) V_n) V_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn)) V_s0) (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_s1)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_a _let_1))) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_2 V_A_2)))))))))) (forall ((V_v_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) V_v_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (not (forall ((B_y $$unsorted)) (=> (= V_v_2 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) B_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_a T_b V_f_2 V_z_2 V_A_2) B_y)))))))))) (forall ((V_n_2 $$unsorted) (V_d_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_a_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_d_2))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_n_2))) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) (tptp.hAPP _let_2 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_d_2)))))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)))) (let ((_let_4 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless _let_1))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)))) (let ((_let_7 (tptp.hAPP _let_5 V_A_2))) (= (tptp.hBOOL (tptp.hAPP _let_7 (tptp.hAPP _let_4 V_B_2))) (and (=> _let_6 (tptp.hBOOL (tptp.hAPP _let_7 V_B_2))) (=> (not _let_6) (and (=> _let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_4 (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_B_2))) (=> (not _let_3) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))))))))))))) (forall ((V_n_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2) V_A_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP V_f_2 B_x))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_x)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)) (= V_A_2 V_B_2))))) (forall ((V_k_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_k_2)))) (forall ((V_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2)))) (forall ((V_s2_2 $$unsorted) (V_s1_2 $$unsorted) (V_n_2 $$unsorted) (V_c_2 $$unsorted) (V_s0_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2)))) (=> (tptp.hBOOL (tptp.hAPP V_b_2 V_s0_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c_2) V_s0_2) V_n_2) V_s1_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 V_s1_2) V_n_2) V_s2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 V_s0_2) V_n_2) V_s2_2))))))) (forall ((V_n_2 $$unsorted) (V_c_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OWhile V_b_2 V_c_2)) V_s_2) V_n_2) V_s_2)))) (forall ((V_t_2 $$unsorted) (V_n_2 $$unsorted) (V_s_2 $$unsorted) (V_c2_2 $$unsorted) (V_c1_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OCond V_b_2 V_c1_2 V_c2_2)) V_s_2) V_n_2) V_t_2)) (=> (=> _let_1 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c1_2) V_s_2) V_n_2) V_t_2)))) (not (=> (not _let_1) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c2_2) V_s_2) V_n_2) V_t_2))))))))) (forall ((V_c1_2 $$unsorted) (V_s1_2 $$unsorted) (V_n_2 $$unsorted) (V_c0_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c0_2) V_s_2) V_n_2) V_s1_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OCond V_b_2 V_c0_2 V_c1_2)) V_s_2) V_n_2) V_s1_2))))) (forall ((V_c0_2 $$unsorted) (V_s1_2 $$unsorted) (V_n_2 $$unsorted) (V_c1_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_b_2 V_s_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c1_2) V_s_2) V_n_2) V_s1_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OCond V_b_2 V_c0_2 V_c1_2)) V_s_2) V_n_2) V_s1_2))))) (forall ((V_s2 $$unsorted) (V_c1 $$unsorted) (V_s1 $$unsorted) (V_n $$unsorted) (V_s0 $$unsorted) (V_c0 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c0) V_s0) V_n) V_s1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c1) V_s1) V_n) V_s2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OSemi V_c0 V_c1)) V_s0) V_n) V_s2))))) (forall ((V_t $$unsorted) (V_n $$unsorted) (V_s $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln tptp.c_Com_Ocom_OSKIP) V_s) V_n) V_t)) (= V_t V_s))) (forall ((V_n $$unsorted) (V_s $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln tptp.c_Com_Ocom_OSKIP) V_s) V_n) V_s))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP _let_1 V_k_2))))) (= (tptp.hAPP _let_200 _let_209) _let_60) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_f_2)) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_f_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b)) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2)))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (= V_A_2 V_B_2) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_z))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (tptp.hBOOL (tptp.hAPP _let_2 V_x)))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (= V_x V_y)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)) (tptp.hBOOL (tptp.hAPP _let_2 V_z)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (= V_x V_y)))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a))) (=> (tptp.class_Orderings_Oord T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP _let_1 V_c))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_c))) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 V_a))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))))) (forall ((V_x_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_P_2) V_Q_2)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (tptp.hBOOL (tptp.hAPP V_Q_2 V_x_2))))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (= V_x_2 V_y_2)))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b)) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (= V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_B_2) V_A_2)))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))) (forall ((V_Q_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_P_2) V_Q_2)) (tptp.hBOOL (tptp.hAPP V_Q_2 V_x_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oorder T_a) (= (= V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2))))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (= V_A_2 V_B_2) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b)) V_f_2) V_g_2)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_A_2))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_f_2)) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_f_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_B_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2)) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_B_2)))))) (forall ((V_u_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_u_2)) V_u_2)) (= (tptp.hAPP _let_209 _let_31) _let_201) (forall ((V_s_H $$unsorted) (V_n $$unsorted) (V_s $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c) V_s))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) V_s_H)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_s_H))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_z))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 V_x))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_z) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_z)) (tptp.hBOOL (tptp.hAPP _let_1 V_z))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b) V_a)) (=> (not (= V_a V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_b) V_a)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) V_b)) (=> (not (= V_a V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) V_b)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y)) (= V_x V_y))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) V_y_2))) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (not (= V_a V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_b) V_a)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (not (= V_a V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) V_b)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) V_y_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Opreorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) V_y_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) V_y_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (not (= V_x_2 V_y_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_y) V_x))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_y_2) V_x_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y_2) V_x_2))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c_2))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b)))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Obot T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Orderings_Obot__class_Obot T_a)) V_x)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_b_2) (tptp.hAPP (tptp.hAPP _let_2 V_c_2) V_d_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) _let_2)) (= V_A_2 _let_2))))) (forall ((V_x $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b)) V_x)) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x))))))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d))))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_y) V_z)) V_x))))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b)) V_x))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y) V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y) V_y)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) V_y_2)) V_z_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) V_y_2) V_y_2)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_b))) (not (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (not (tptp.hBOOL (tptp.hAPP _let_1 V_b))))))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d))))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_y) V_z)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_b)))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y) V_x)))) (forall ((V_a $$unsorted) (V_x $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_b)) V_x)))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_b)) V_x)))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_y_2) V_z_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) V_y_2) V_x_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) V_x)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Otop T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.c_Orderings_Otop__class_Otop T_a))))) (forall ((V_a_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_D_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_C_2)) (tptp.hAPP _let_1 V_D_2))))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) V_B_2)))))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.c_Orderings_Otop__class_Otop _let_1))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2))) (exists ((B_AA $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_AA) V_A_2)) (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) B_AA)))))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_D_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_D_2))))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) V_C_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_D_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_D_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_C_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) 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 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (= (tptp.hAPP (tptp.hAPP _let_2 V_B_2) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2)) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_D_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_D_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) V_A_2)))) (forall ((V_t $$unsorted) (V_n $$unsorted) (V_s $$unsorted) (V_c $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c) V_s) V_n) V_t)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c) V_s) V_t)))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_c_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevalc V_c_2) V_s_2) V_t_2)) (exists ((B_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c_2) V_s_2) B_n) V_t_2))))) (forall ((V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_A_2)) V_b_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) B_x) V_b_2))))))) (forall ((V_a6_2 $$unsorted) (V_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) V_a2_2) V_a4_2) V_a6_2)) (or (and (= V_a6_2 V_a2_2) (not _let_1)) (and _let_1 (exists ((B_s1 $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a3_2) V_a2_2) V_a4_2) B_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) B_s1) V_a4_2) V_a6_2))))))))) (forall ((V_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) V_a2_2) V_a4_2) V_a2_2)) (=> (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)) (exists ((B_s1 $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a3_2) V_a2_2) V_a4_2) B_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OWhile V_a1_2 V_a3_2)) B_s1) V_a4_2) V_a2_2))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_f_2) V_g_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_g_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_g_2) V_f_2))))))))) (forall ((V_Ga_2 $$unsorted) (V_ts_2 $$unsorted) (V_G_H_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_G_H_2 V_ts_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_Hoare__Mirabelle_Otriple T_a) tptp.tc_HOL_Obool)) V_G_H_2) V_Ga_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_2)))) (forall ((V_ts_2 $$unsorted) (V_ts_H_2 $$unsorted) (V_Ga_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_H_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_Hoare__Mirabelle_Otriple T_a) tptp.tc_HOL_Obool)) V_ts_2) V_ts_H_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_2)))) (forall ((V_Ga_2 $$unsorted) (V_ts_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_Hoare__Mirabelle_Otriple T_a) tptp.tc_HOL_Obool)) V_ts_2) V_Ga_2)) (tptp.c_Hoare__Mirabelle_Ohoare__derivs T_a V_Ga_2 V_ts_2))) (forall ((V_a5_2 $$unsorted) (V_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a6_2 $$unsorted) (V_a1_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OCond V_a1_2 V_a6_2 V_a3_2)) V_a2_2) V_a4_2) V_a5_2)) (or (and _let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a6_2) V_a2_2) V_a4_2) V_a5_2))) (and (not _let_1) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a3_2) V_a2_2) V_a4_2) V_a5_2))))))) (forall ((V_a5_2 $$unsorted) (V_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a6_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_a1_2 V_a2_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OCond V_a1_2 V_a3_2 V_a6_2)) V_a2_2) V_a4_2) V_a5_2)) (or (and _let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a3_2) V_a2_2) V_a4_2) V_a5_2))) (and (not _let_1) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a6_2) V_a2_2) V_a4_2) V_a5_2))))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_A_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b_2) B_x))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_a V_f_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_a6_2 $$unsorted) (V_a3_2 $$unsorted) (V_a2_2 $$unsorted) (V_a5_2 $$unsorted) (V_a1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.c_Com_Ocom_OSemi V_a1_2 V_a5_2)) V_a2_2) V_a3_2) V_a6_2)) (exists ((B_s1 $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a1_2) V_a2_2) V_a3_2) B_s1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_a5_2) B_s1) V_a3_2) V_a6_2)))))) (forall ((V_a2 $$unsorted) (V_a1 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln tptp.c_Com_Ocom_OSKIP) V_a1) V_a2) V_a1))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_C_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)) (= V_A_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (not (= V_A_2 V_B_2)))))) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc) (tptp.hAPP _let_1 V_n_2)))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_R_2)) (tptp.hAPP _let_3 V_S_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)) V_A_2)) V_a_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)) V_A_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) B_x))))))))) (forall ((V_u_2 $$unsorted) (V_M_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_a) V_A_2) V_M_2)) V_u_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP V_M_2 B_x)) V_u_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_I_2) V_A_2)) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_I_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP V_A_2 B_x)) V_B_2))))))) (forall ((V_M_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_u_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_u_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_a) V_A_2) V_M_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_u_2) (tptp.hAPP V_M_2 B_x)))))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_I_2) V_A_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_I_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP V_A_2 B_x)))))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_i_2) V_k_2))))) (forall ((V_n_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n_2))) (tptp.hAPP V_f_2 V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_m_2) V_n_2)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) _let_1))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_x_2) V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_a)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (tptp.hBOOL (tptp.hAPP _let_3 V_b)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Rings_Oordered__ring T_a) (=> (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a)) (tptp.hBOOL (tptp.hAPP _let_3 V_b))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)))) (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_b)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)))))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)))))))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__comm__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Oordered__cancel__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_b_2)) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_b_2))) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__ring T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_c) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2)) (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) V_B_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x))) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) (tptp.hAPP _let_2 V_z))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) _let_3)) (or (= V_A_2 _let_2) (= V_A_2 _let_3))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_A_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)) V_C_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_C_2) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) 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 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) V_C_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_B_2) V_A_2)) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_A_2)) V_x_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a V_f_2) V_A_2))) V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_B_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (V_g_2 $$unsorted)) (=> (= V_g_2 (tptp.c_Finite__Set_Ofold1 T_a V_f_2)) (= (tptp.hAPP V_g_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_a_2))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_a_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2)) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_B_2))) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_B_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2)) V_B_2))))) (forall ((V_M_2 $$unsorted) (V_A_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_M_2 V_i_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a T_b) V_A_2) V_M_2)))))) (forall ((V_M_2 $$unsorted) (V_A_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a T_b) V_A_2) V_M_2)) (tptp.hAPP V_M_2 V_i_2)))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP V_B_2 V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_A_2) V_B_2)))))) (forall ((V_Q_2 $$unsorted) (V_c_2 $$unsorted) (V_P_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Hoare__Mirabelle_Otriple__valid T_a V_n_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Hoare__Mirabelle_Otriple_Otriple T_a) V_P_2) V_c_2) V_Q_2)) (forall ((B_Z $$unsorted) (B_s $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_Z) B_s)) (forall ((B_s_H $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c_2) B_s) V_n_2) B_s_H)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 B_Z) B_s_H)))))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_1) V_A_2) V_B_2)) (tptp.hAPP V_B_2 V_a_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_a T_b V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_B_2)) _let_1) _let_1)))))) (forall ((V_k_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_k_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) _let_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_k_2)) _let_2))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a V_f_2) V_A_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) _let_1) V_b)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_b)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_b))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_c)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Rings_Olinordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__semiring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_3) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_3) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)))))))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__semiring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) _let_3) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_1 V_x) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_y))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__ring__strict T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2))) _let_1)) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_y) V_x)) V_x))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_y)) V_x))))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_b_2) V_a_2)) V_e_2)) V_d_2))))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Oordered__ring T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_e_2)) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_e_2)) V_d_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a_2) V_b_2)) V_e_2)) V_c_2)) V_d_2)))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop _let_3)) (tptp.c_Orderings_Otop__class_Otop _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a V_f_2) V_B_2)) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) (tptp.hAPP _let_1 V_A_2))))))))) (forall ((V_N_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) V_N_2)) (exists ((B_m $$unsorted)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_x) B_m))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a V_f_2) V_A_2) (tptp.c_HOL_OThe T_a (tptp.c_Finite__Set_Ofold1Set T_a V_f_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_B_2)) _let_1) _let_1)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a)) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__idem T_a T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b _let_1) (tptp.hAPP V_F_2 V_B_2)) _let_1)))))) (forall ((V_v $$unsorted) (V_u $$unsorted) (V_y $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_4 (tptp.hAPP _let_3 (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_x) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_y) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_u)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_v)) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_u) V_v) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_u) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_v) V_y))) V_a))))))))))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP _let_1 (tptp.c_Orderings_Obot__class_Obot _let_2)))) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) (tptp.hAPP _let_1 V_B_2))))))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (let ((_let_3 (tptp.hAPP _let_2 V_A_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)))) (let ((_let_5 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP _let_5 V_B_2))) (and (=> _let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_5 (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_B_2))) (=> (not _let_4) (tptp.hBOOL (tptp.hAPP _let_3 V_B_2))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2) V_A_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Oab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_B_2)) _let_1) _let_1)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_B_2)) _let_1) _let_1)))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)))))))) _let_208 (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_a _let_1))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_2 V_A_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_a _let_1))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Orderings_Obot__class_Obot _let_3))) (let ((_let_5 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (=> (tptp.hBOOL (tptp.hAPP _let_5 V_A_2)) (=> (not (= V_A_2 _let_4)) (=> (tptp.hBOOL (tptp.hAPP _let_5 V_B_2)) (=> (not (= V_B_2 _let_4)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)))))))))))))) (forall ((V_a4_2 $$unsorted) (V_a3_2 $$unsorted) (V_a2_2 $$unsorted) (V_a1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.hAPP tptp.c_Com_Ocom_OBODY V_a1_2)) V_a2_2) (tptp.hAPP tptp.c_Nat_OSuc V_a3_2)) V_a4_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_a1_2))) V_a2_2) V_a3_2) V_a4_2)))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_a_2) V_A_2)))))) (forall ((V_v $$unsorted) (V_u $$unsorted) (V_y $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_4 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semiring__1__strict T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_x) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_y) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_u)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_v)) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_u) V_v) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_u) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_v) V_y))) V_a))))))))))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Big__Operators_Osemilattice__big T_a V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a V_f_2) V_A_2))))) (forall ((V_s1 $$unsorted) (V_n $$unsorted) (V_s $$unsorted) (V_P $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.hAPP tptp.c_Com_Ocom_OBODY V_P)) V_s) V_n) V_s1)) (not (forall ((B_n $$unsorted)) (=> (= V_n (tptp.hAPP tptp.c_Nat_OSuc B_n)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln (tptp.hAPP (tptp.c_Option_Othe tptp.tc_Com_Ocom) (tptp.hAPP tptp.c_Com_Obody V_P))) V_s) B_n) V_s1)))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (forall ((B_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_B_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP V_f_2 B_b))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))) (forall ((V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (let ((_let_3 (tptp.c_Partial__Function_Oflat__lub T_a V_b_2 V_A_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) _let_2)))) (and (=> _let_4 (= _let_3 V_b_2)) (=> (not _let_4) (= _let_3 (tptp.c_HOL_OThe T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) _let_2))))))))))) (forall ((V_n_2 $$unsorted) (V_d_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_4 (tptp.hAPP _let_3 V_a_2))) (let ((_let_5 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 _let_5) _let_5)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a tptp.tc_Nat_Onat) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) tptp.tc_Nat_Onat) _let_2) _let_1)) V_d_2))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_n_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n_2)) (tptp.hAPP _let_4 (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) V_d_2)))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__semigroup__mult T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Groups_Otimes__class_Otimes T_a)) V_A_2)) V_A_2))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Oordered__ab__semigroup__add T_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_B_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (forall ((V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)))))) (forall ((V_N_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) V_N_2)) (exists ((B_m $$unsorted)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_x) B_m))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Nat_Osemiring__char__0 T_a) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2))))) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) _let_60))) (forall ((V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))))))) (forall ((V_n_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= _let_206 _let_126) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) V_z_2)) (exists ((B_n $$unsorted)) (= V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_w_2) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Nat_OSuc B_n))))))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) (tptp.hAPP _let_1 V_m))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2))))) (forall ((V_z $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n))) V_z))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))))) (= _let_207 _let_127) (forall ((V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n_2) _let_1)) (= V_n_2 _let_1)))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_n)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_m_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2))) (= (tptp.hBOOL (tptp.hAPP _let_2 _let_1)) (or (tptp.hBOOL (tptp.hAPP _let_2 V_n_2)) (= V_m_2 _let_1)))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP tptp.c_Nat_OSuc V_m_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_m_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_2 _let_1)) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 V_n))) (= V_m _let_1)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)) (not (= V_m_2 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)) (= V_m_2 V_n_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)) (=> (not (= V_m V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (= V_m V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_x) V_y)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)) (exists ((B_k $$unsorted)) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) B_k))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2)))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_m)))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_j)))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))))))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k) V_l)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_l)))))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_k)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k) V_n))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_k)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_k)) V_n)) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k) V_n)))))))) (forall ((V_y $$unsorted) (V_z $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP _let_1 V_k_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_m_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_n_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_n)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)) (tptp.hAPP _let_2 V_n)))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_m_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_n_2)) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2)) (= V_m_2 V_n_2))))))) (forall ((V_n $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_n)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_i)) V_i)))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_l)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_l))))))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_l))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 V_m))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_m))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k) V_l)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_l)))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))))))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_m)))))) (forall ((V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_m)))) (forall ((V_m $$unsorted) (V_t $$unsorted) (V_n $$unsorted) (V_s $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP tptp.c_Natural_Oevaln V_c) V_s))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) V_t)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_t)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) V_n_2)) (tptp.hBOOL (tptp.c_Nat_Onat_Onat__case tptp.tc_HOL_Obool tptp.c_fFalse (tptp.hAPP _let_1 V_m_2) V_n_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_a) _let_1) V_f_2)) V_A_2)))) (= (tptp.hAPP _let_29 _let_174) _let_127) (forall ((V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_k)))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j)))))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_m))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_m)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_m_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) V_n_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) (tptp.hAPP tptp.c_Nat_OSuc V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2))) (= V_n_2 V_m_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m_2) V_n_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_m_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_n_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_c) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)))))))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_i)) V_k) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)) V_i)))))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_k)) V_i) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j) V_i)) V_k)))))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_j)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))))))) (forall ((V_k_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i_2) V_j_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_i_2) V_k_2) (= V_j_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2) V_i_2))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_n) V_m))) (forall ((V_i_2 $$unsorted) (V_j_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_j_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i_2) V_k_2)) V_j_2)))))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_k)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_j)) V_k)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_m))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)) V_k)))))) (forall ((V_i_2 $$unsorted) (V_k_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_k_2)) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_j_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i_2) V_k_2)))))) (forall ((V_i $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 V_i) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_k)) V_j))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_n_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n_2)) (tptp.c_Int_Oring__1__class_OInts T_a))))) (forall ((V_n_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a T_b) _let_1) V_f_2)) V_A_2))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 V_m)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_m_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_n_2)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_n_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_B_2)) (tptp.hAPP _let_1 V_A_2))) (= V_A_2 V_B_2)))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_k)) (tptp.hAPP tptp.c_Nat_OSuc V_j)))))) (forall ((V_m $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))) V_m) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k) V_m)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i_2) V_j_2)) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_n_2)) (= V_m_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_i_2)) V_u_2)) V_n_2))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_u $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_i) V_u)) V_m)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_u)) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j) V_i)) V_u)) V_n)))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_i_2) V_j_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_i_2)) V_u_2)) V_n_2))))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j_2) V_i_2)) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_n_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_i_2) V_j_2)) V_u_2)) V_m_2) V_n_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_u $$unsorted) (V_i $$unsorted) (V_j $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j) V_i)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_i) V_u)) V_m)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_u)) V_n)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_j)) V_u)) V_m)) V_n))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_j_2) V_i_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_i_2) V_j_2)) V_u_2)) V_m_2)) V_n_2)))))))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) (tptp.hAPP (tptp.c_COMBK T_a T_b) V_y_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))) V_y_2)))) (forall ((V_n_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j_2) V_i_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_i_2) V_j_2)) V_u_2)) V_m_2)) V_n_2)))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i_2) V_j_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_i_2)) V_u_2)) V_n_2))))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_2))))) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_n_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a T_a) (tptp.c_Groups_Oplus__class_Oplus T_a)) (tptp.c_Groups_Oone__class_Oone T_a)) V_n_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_K_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Groups_Oordered__ab__semigroup__add T_b) (tptp.class_Rings_Osemiring__1 T_b)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_i) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 B_i)) V_K_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))) V_K_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OgreaterThanAtMost T_a V_less__eq_2 V_less_2 V_l_2 V_u_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_i_2) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OatLeastLessThan T_a V_less__eq_2 V_less_2 V_l_2 V_u_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_i_2) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OgreaterThanLessThan T_a V_less_2 V_l_2 V_u_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_i_2) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OatLeastAtMost T_a V_less__eq_2 V_l_2 V_u_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_i_2) V_u_2))))) (forall ((V_k_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OgreaterThan T_a V_less_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_k_2) V_i_2)))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n)))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j))))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_w_2) V_z_2)) (exists ((B_n $$unsorted)) (= V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_w_2) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_n)))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)) (= V_z V_w))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)) (tptp.hBOOL (tptp.hAPP _let_2 V_k))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z))))) (forall ((V_w $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_w) V_w))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)) (= V_m V_n))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k)) (tptp.hBOOL (tptp.hAPP _let_2 V_k))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (= V_m V_n) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_n))) (forall ((V_w_2 $$unsorted) (V_z_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_z_2) V_w_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z_2) V_w_2)) (not (= V_z_2 V_w_2))))) (forall ((V_z $$unsorted) (V_z_H $$unsorted) (V_w $$unsorted) (V_w_H $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_w_H) V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z_H) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_w_H) V_z_H)) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))))))) (forall ((V_z $$unsorted) (V_w $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_w) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_z)))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_w_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) V_z_2)))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_w_2) V_z_2)))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_w_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) V_z_2)))) (forall ((V_z_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2)))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)))))) (forall ((V_z_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z_2)) V_z_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_z_2) _let_1))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j)))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_m_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_m_2) V_n_2) _let_1) (and (= V_m_2 _let_1) (= V_n_2 _let_1)))))) (tptp.hBOOL (tptp.hAPP _let_161 _let_127)) (forall ((V_k $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_k)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k_2) V_l_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_z_2)) (= V_w_2 V_z_2))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (= V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2)))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z) V_z)) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)) (forall ((V_w $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) V_w) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z1) V_w)) (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_w)))))) (forall ((V_z2 $$unsorted) (V_z1 $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z1)) (tptp.hAPP _let_1 V_z2)))))) (forall ((V_z3 $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_z1))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z2)) V_z3) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_z3)))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))) (forall ((V_z2 $$unsorted) (V_z1 $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z1)) (tptp.hAPP _let_1 V_z2)))))) (forall ((V_w $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) V_w) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z1) V_w)) (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_w)))))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z) V_z)) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (not (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (not (= _let_126 _let_127)) (forall ((V_z3 $$unsorted) (V_z2 $$unsorted) (V_z1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_z1))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z2)) V_z3) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z2) V_z3)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z))))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))) (forall ((V_i_2 $$unsorted) (V_inc_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_a V_inc_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_i_2) V_i_2))) (forall ((V_i_2 $$unsorted) (V_n_2 $$unsorted) (V_inc_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_a V_inc_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2) V_i_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_a V_inc_2 V_n_2 (tptp.hAPP V_inc_2 V_i_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_x_2) V_y_2)))) (and (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_y_2) V_x_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))))) (forall ((V_q $$unsorted) (V_r $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_2) V_a)) (=> (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_r) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a) V_q))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_r)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_q) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))) (forall ((V_q $$unsorted) (V_r $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_a)) (=> (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_r) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a) V_q))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_r) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_q))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_5 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_b_H) V_q_H)) V_r_H))) (=> (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_b) V_q)) V_r) _let_6) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_6) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_r_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_H) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_q_H) V_q))))))))))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_r_H $$unsorted) (V_q_H $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_q_H)) V_r_H)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_q)) V_r))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_r) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_r)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_r_H)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_q) V_q_H))))))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hAPP _let_1 _let_2))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_6 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 V_b_H) V_q_H)) V_r_H))) (=> (= (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 V_b) V_q)) V_r) _let_7) (=> (tptp.hBOOL (tptp.hAPP _let_4 _let_7)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r_H) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_r)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_H) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_q) V_q_H)))))))))))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b_H) V_q_H)) V_r_H))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r_H) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_b_H)) (tptp.hBOOL (tptp.hAPP _let_2 V_q_H))))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b_H) V_q_H)) V_r_H)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_b_H)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_q_H) _let_1))))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_r_H $$unsorted) (V_q_H $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_q_H)) V_r_H)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_q)) V_r))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_r_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r_H) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_q_H) V_q))))))))))) (= _let_127 _let_207) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_x) V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_x) V_y))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_x_2) V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x_2) V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_1) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)))) (= (= V_A_2 V_B_2) (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2)))) (tptp.hBOOL (tptp.hAPP _let_181 _let_126)) (= _let_197 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_29) _let_201)) (forall ((V_A_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_A_2)))) (forall ((V_A_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_A_2))))) (= _let_126 _let_206) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_z $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_z)))) (forall ((V_P_2 $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_x)))))) (forall ((V_P_2 $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_x)))))) (tptp.hBOOL (tptp.hAPP _let_181 _let_127)) (forall ((V_k_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OatMost T_a V_less__eq_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_i_2) V_k_2)))) (forall ((V_k_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OatLeast T_a V_less__eq_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_k_2) V_i_2)))) (forall ((V_k_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.c_SetInterval_Oord_OlessThan T_a V_less_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_i_2) V_k_2)))) (forall ((V_x $$unsorted) (V_y $$unsorted)) (let ((_let_1 (tptp.c_Nat__Transfer_Otsub V_x V_y))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_y) V_x)))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x) V_y))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_c_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_2) V_c_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_x_2) V_y_2)))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_c_2) V_y_2)))) (and (=> _let_6 (= _let_3 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_c_2)))) (=> (not _let_6) (and (=> _let_5 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) _let_1))) (=> (not _let_5) (= _let_3 _let_1)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) (tptp.c_Predicate_OPowp T_a V_A_2)) (tptp.c_Predicate_OPowp T_a V_B_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_l_2) V_u_2)))) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.c_Orderings_Otop__class_Otop _let_158)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2) V_d_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)) (= V_b_2 V_d_2))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2) V_d_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)) (= V_a_2 V_c_2))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)) (= (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2) V_d_2)) (and (= V_a_2 V_c_2) (= V_b_2 V_d_2)))))))) (forall ((V_k_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j_2) V_k_2))) (let ((_let_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_i_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i_2) V_j_2)) (= (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.hAPP _let_2 V_j_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_j_2) _let_1))))))) (forall ((V_m_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (forall ((V_n_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_n_2))) (forall ((V_k_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_k_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) V_A_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) V_A_2) _let_1)) (= V_A_2 _let_1)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_l_2) V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_u_2) V_l_2))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_i_2) V_j_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_i_2)) (tptp.hAPP tptp.c_Nat_OSuc V_j_2)))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b_2) V_a_2)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2) V_b_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2) V_b_2)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2) V_d_2))) (or (tptp.hBOOL (tptp.hAPP _let_2 V_a_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_a_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_d_2))))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Nat_Onat) V_f_2) tptp.c_Nat_OSuc)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) V_n_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_m_2) V_u_2)) (tptp.hAPP _let_1 V_u_2)))))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_2 V_f_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Nat_Onat) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_k_2))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) V_n_2))))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_i_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_i_2) V_n_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_n_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_n_2) V_m_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_m_2) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_i_2) V_j_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_k_2))))) (forall ((V_f_2 $$unsorted) (V_p_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m_2) V_n_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_n_2) V_p_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_n_2) V_p_2))) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_p_2)))))))))) (forall ((V_f_2 $$unsorted) (V_p_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (let ((_let_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.class_Groups_Oab__group__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m_2) V_n_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_n_2) V_p_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p_2))) (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_n_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_n_2) V_p_2)))))))))) (forall ((V_m_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_m_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2))) (let ((_let_4 (tptp.hAPP _let_3 (tptp.c_Orderings_Otop__class_Otop _let_1)))) (let ((_let_5 (tptp.tc_fun tptp.tc_Nat_Onat _let_2))) (= (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_5 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_5 _let_2) tptp.tc_Nat_Onat) _let_3) (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) V_A_2)) (tptp.hAPP _let_4 V_A_2)))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_u_2)) (tptp.hAPP _let_1 V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (= (tptp.hAPP V_f_2 _let_1) (tptp.c_Groups_Ozero__class_Ozero T_a)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc _let_1)) V_k_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat _let_1) V_k_2)))))))) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP V_f_2 V_m_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) V_n_2)))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP _let_1 V_n_2)))) (=> (not _let_3) (= _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))))))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) V_m_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (and (=> _let_4 (= _let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n_2))) (tptp.hAPP V_f_2 V_n_2))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Nat__Transfer_Otsub V_x V_y))))))) (forall ((V_x $$unsorted) (V_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_y) V_x)) (= (tptp.c_Nat__Transfer_Otsub V_x V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Nat__Transfer_Otsub (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_x) V_y))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (forall ((B_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_m) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (exists ((B_m $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_m) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_B_2 $$unsorted) (V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (=> (forall ((B_n $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_1 B_n)) V_A_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_n) V_k_2))) V_B_2))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (=> (forall ((B_n $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 B_n)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_n) V_k_2))) V_B_2))))) (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (forall ((B_n $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) B_n)) V_A_2)) V_C_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) V_A_2)) V_C_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Nat_Onat V_l_2 V_u_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint V_l_2) V_u_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Int_Oint V_l_2 V_u_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_l_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_u_2) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Int_Oint V_l_2 V_u_2))) (forall ((V_u_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_u_2)))) (forall ((V_k_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_k_2)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_l_2)) V_u_2) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Nat_Onat V_l_2 V_u_2))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) V_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_u_2) V_l_2))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint V_l_2) V_u_2))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_m_2) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Nat_Onat V_l_2 V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_u_2) (tptp.hAPP tptp.c_Nat_OSuc V_l_2)))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_m_2) V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_u_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2)))))) (forall ((V_i_2 $$unsorted) (V_f_2 $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_s_2)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_i) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP V_f_2 B_i))))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2) V_s_2) _let_1) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) V_s_2)) (= (tptp.hAPP V_f_2 V_i_2) _let_1)))))))) (forall ((V_i_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_s_2)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_i) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP V_f_2 B_i))))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2) V_s_2) V_B_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 V_i_2)) V_B_2)))))))) (= (tptp.hAPP _let_205 _let_31) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc) _let_60)) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted) (V_Q_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_n_2))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP V_Q_2 B_x)) (tptp.hAPP V_P_2 B_x)))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_P_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_3 V_Q_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_Nat_Onat) _let_2) V_P_2)) V_Q_2)) _let_1))))))) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) tptp.tc_Nat_Onat) _let_3) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Nat_Onat) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_2) _let_1)))) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)))) (=> (tptp.class_Groups_Oab__group__add T_a) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP V_f_2 V_m_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP _let_2 V_n_2) _let_1))))) (=> (not _let_5) (= _let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))))))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_l_2 V_u_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Int_Oint V_l_2 V_u_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_l_2 V_u_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OgreaterThan T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_l_2) V_u_2) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_l_2 V_u_2))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OgreaterThan T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2)) (tptp.hAPP _let_1 V_u_2)) (tptp.hAPP _let_1 V_l_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_P_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)) (= (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP V_P_2 V_m_2)) (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)))))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (forall ((B_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_m) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted)) (= (exists ((B_m $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_m) V_n_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_m)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_l_2) (tptp.hAPP tptp.c_Nat_OSuc V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_l_2 V_u_2))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a_2) V_b_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a_2) V_b_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_b_2) V_a_2)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 V_d_2))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_d_2)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_c_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (and (= V_a_2 V_b_2) (= V_b_2 V_c_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a_2 V_b_2) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_i_2 V_j_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_i_2) (tptp.hAPP tptp.c_Nat_OSuc V_j_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Nat_Onat) V_f_2) tptp.c_Nat_OSuc)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)))))) (forall ((V_n_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_2 V_f_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Nat_Onat) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_k_2))) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2))))))) (forall ((V_k_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_k_2)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_k_2 V_l_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_k_2) V_l_2)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_k_2) V_l_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_k_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_i_2 V_j_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_k_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_l_2 V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_u_2) V_l_2))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_k_2) V_i_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OgreaterThan T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y_2) V_x_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_l_2 V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_u_2)) V_l_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) _let_1)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_f_2))) (=> (= (tptp.hAPP V_f_2 _let_1) _let_1) (= (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc _let_1) V_k_2)) (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_k_2))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_c_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 V_d_2))) (and (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP _let_2 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_d_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) V_d_2))))) (tptp.hBOOL (tptp.hAPP _let_2 V_d_2))))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_l_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI tptp.tc_Nat_Onat _let_59) _let_60) _let_205) _let_201) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)) (= (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP V_f_2 V_m_2)) (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) V_n_2)))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_k_2))) (let ((_let_3 (tptp.c_SetInterval_Oord__class_OgreaterThan tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_3 V_k_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) _let_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 _let_1)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_m_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (and (=> _let_4 (= _let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2))) (tptp.hAPP V_f_2 _let_1))))))))))) (forall ((V_p_2 $$unsorted) (V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n_2))) (let ((_let_2 (tptp.hAPP _let_1 V_p_2))) (let ((_let_3 (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (let ((_let_4 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) _let_3)) (= (tptp.hAPP _let_4 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_4 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2))) (tptp.hAPP _let_4 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat _let_3 _let_2))))))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_u_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_m_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_u_2)))))) (forall ((V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_n_2))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 _let_1) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) _let_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) V_n_2))) (tptp.hAPP (tptp.hAPP _let_5 _let_3) (tptp.hAPP (tptp.hAPP _let_4 _let_3) _let_1)))))))))) (forall ((V_M_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_n_2)) V_M_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) V_n_2)) V_M_2)) (tptp.hAPP V_M_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_u_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_u_2)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint _let_1) V_u_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_u_2))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_l_2))))) (forall ((V_k_2 $$unsorted) (V_P_2 $$unsorted) (V_d_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_d_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_d_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_k_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) V_d_2))))))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Groups_Otimes__class_Otimes T_a)))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.hAPP _let_1 V_N_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2))))))))) (forall ((V_k_2 $$unsorted) (V_P_2 $$unsorted) (V_d_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_d_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_d_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_k_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) V_d_2))))))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_l_2 V_u_2)))) (forall ((V_k_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_k_2)))) (forall ((V_n_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeast T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatMost T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_n_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n)) V_n)) (= _let_31 _let_204) (= _let_204 _let_31) (forall ((V_P_2 $$unsorted)) (= (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_Onat B_x))))))) (forall ((V_P_2 $$unsorted)) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_Onat B_x))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (= (= (tptp.hAPP tptp.c_Int_Onat V_x_2) (tptp.hAPP tptp.c_Int_Onat V_y_2)) (= V_x_2 V_y_2)))))) (forall ((V_z_H_2 $$unsorted) (V_z_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z_H_2)) (= (= (tptp.hAPP tptp.c_Int_Onat V_z_2) (tptp.hAPP tptp.c_Int_Onat V_z_H_2)) (= V_z_2 V_z_H_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_l_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_u_2) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Int_Oint V_l_2 V_u_2))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_l_2 V_u_2)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_u_2) V_l_2)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (= _let_62 _let_203) (forall ((V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_n_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_n_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_A_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_A_2)) V_A_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Int_Oint V_l_2 V_u_2)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_u_2) V_l_2)))) (forall ((V_n_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_n_2))) (= _let_201 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) _let_197)) (forall ((V_k_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_k_2))) (forall ((V_u_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_u_2)) (tptp.hAPP tptp.c_Nat_OSuc V_u_2))) (forall ((V_i_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Int_Onat V_i_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP tptp.c_Int_Onat V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_w_2)) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_w_2) V_z_2)))))) (forall ((V_w_2 $$unsorted) (V_z_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_w_2)) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_w_2) V_z_2)))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_k_2))) (= (tptp.hAPP _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) _let_2) (tptp.hAPP _let_1 V_k_2)))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_k_2) V_i_2))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_i_2) V_k_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x_2)) (tptp.hAPP tptp.c_Int_Onat V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))))))) (= _let_203 _let_174) (forall ((V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)) V_z))) (forall ((V_z_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2) V_z_2) (and (= V_m_2 (tptp.hAPP tptp.c_Int_Onat V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2))))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_z)))) (and (=> _let_3 (= _let_2 V_z)) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_z_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2)) V_z_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatMost T_a))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeast T_a))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y_2) V_x_2)))))) (= (tptp.hAPP _let_199 _let_31) _let_60) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint V_l_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_u_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_l_2 V_u_2))) (forall ((V_k_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan tptp.tc_Nat_Onat) V_k_2))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_a) (tptp.class_Groups_Ocomm__monoid__mult T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Int_Oint) V_f_2) tptp.c_Int_Onat)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_A_2))))) (forall ((V_u_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_u_2)) (tptp.hAPP tptp.c_Int_Onat V_u_2))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint V_l_2) V_u_2)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_u_2) V_l_2)))) (forall ((V_z_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x_2)) (tptp.hAPP tptp.c_Int_Onat V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_x_2) V_y_2))))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_w_2)) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) V_z_2))))) (forall ((V_w_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)))) (= (= V_m_2 (tptp.hAPP tptp.c_Int_Onat V_w_2)) (and (=> _let_1 (= V_w_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2))) (=> (not _let_1) (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_m_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)))) (= (= (tptp.hAPP tptp.c_Int_Onat V_w_2) V_m_2) (and (=> _let_1 (= V_w_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2))) (=> (not _let_1) (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_2) V_w_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_z_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_w_2)) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_w_2) V_z_2))))))) (forall ((V_i_2 $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_Onat V_i_2))) (and (forall ((B_n $$unsorted)) (=> (= V_i_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_n)) (tptp.hBOOL (tptp.hAPP V_P_2 B_n)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_z_H $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z_H)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP tptp.c_Int_Onat V_z_H))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_z_H $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP tptp.c_Int_Onat V_z_H))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_z $$unsorted) (V_z_H $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z_H) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP tptp.c_Int_Onat V_z_H))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) V_f_2) V_A_2) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_a) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_f_2)) V_A_2)))) (= (tptp.hAPP _let_200 _let_202) _let_60) (forall ((V_n_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (let ((_let_2 (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_n_2))) (tptp.hAPP V_f_2 _let_1)))))))) (= (tptp.hAPP _let_202 _let_31) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) _let_31) _let_201)) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.c_Nat__Transfer_Otsub V_x V_y))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeast T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_u_2)) (tptp.hAPP _let_1 V_u_2)) (tptp.hAPP _let_1 V_l_2)))))) (forall ((V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_i_2 V_j_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_j_2) V_i_2)))) (and (=> _let_2 (= _let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Int_Oint) V_i_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_j_2)))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_l_2) V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (= (tptp.hAPP _let_200 _let_199) _let_60) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatMost T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2)) (tptp.hAPP _let_1 V_u_2)))))) (forall ((V_k_2 $$unsorted) (V_M_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2))) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_2 tptp.tc_Nat_Onat) V_M_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_k_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_k_2 (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_k_2))) V_M_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_l_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_SetInterval_Oord__class_OatLeast tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_2 V_k_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_k_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))) (forall ((V_z_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z_2)))) (forall ((V_m_2 $$unsorted) (V_w_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_w_2)) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2)))))) (forall ((V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Int_Oint V_l_2 V_u_2)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_u_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_l_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))) (forall ((V_u_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_u_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_l_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_u_2))))) (forall ((V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_l_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_l_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_u_2))))) _let_198 (tptp.c_Nat__Transfer_Otransfer__morphism tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat _let_181) (forall ((V_A_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_t_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_A_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_A_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_D_2)) V_t_2))))))))) (forall ((V_B_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_t_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_B_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_t_2))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_B_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_D_2)))))))))) (forall ((V_A_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_t_2) V_A_2)) (forall ((B_x $$unsorted)) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_A_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (not (= B_x V_t_2)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_D_2) V_t_2)))))))) (forall ((V_B_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_t_2) V_B_2)) (forall ((B_x $$unsorted)) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_B_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (not (= B_x V_t_2)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_D_2) V_t_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat))) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat))) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_t_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) V_B_2)) (not (= V_t_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xa) B_x))))))) (= (tptp.hAPP _let_1 V_D_2) V_t_2)))))) (forall ((V_A_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_t_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) V_A_2)) (not (= V_t_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_xa) B_x))))))) (= (tptp.hAPP _let_1 V_D_2) V_t_2)))))) (forall ((V_A_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_t_2) V_A_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_A_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_D_2)) V_t_2))))))))) (tptp.c_Nat__Transfer_Onat__set _let_197) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (tptp.c_Nat__Transfer_Onat__set (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) V_A_2) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (tptp.c_Nat__Transfer_Onat__set (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) V_A_2) V_B_2))))) _let_196 (forall ((V_C_2 $$unsorted)) (tptp.c_Nat__Transfer_Onat__set (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_C_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat))) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (= (= V_A_2 V_B_2) (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x_2)) (tptp.c_Nat__Transfer_Onat__set (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_x_2 V_y_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x_2))))) (forall ((V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2)) V_A_2))) (forall ((V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2)))))) (forall ((V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat))) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat))) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_a) (tptp.class_Groups_Ocomm__monoid__mult T_a)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_a) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_a tptp.tc_Nat_Onat) V_f_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_t_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_t_2) V_B_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_t_2))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_B_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_D_2)))))))))) (forall ((V_t_2 $$unsorted) (V_A_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_t_2))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_A_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_D_2))))))))) (forall ((V_t_2 $$unsorted) (V_B_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_B_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_D_2)) V_t_2)))))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x_2) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2))))))) (forall ((V_P_2 $$unsorted) (V_d_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_d_2)) (=> (forall ((B_x $$unsorted) (B_k $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) B_k) V_d_2)))))) (= (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_d_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))))) (tptp.c_Nat__Transfer_Ois__nat _let_126) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x) (=> (tptp.c_Nat__Transfer_Ois__nat V_y) (tptp.c_Nat__Transfer_Ois__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x) V_y))))) (forall ((V_z $$unsorted)) (tptp.c_Nat__Transfer_Ois__nat (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_z))) (tptp.c_Nat__Transfer_Ois__nat _let_127) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x) (=> (tptp.c_Nat__Transfer_Ois__nat V_y) (tptp.c_Nat__Transfer_Ois__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_x) V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x) (=> (tptp.c_Nat__Transfer_Ois__nat V_y) (tptp.c_Nat__Transfer_Ois__nat (tptp.c_Nat__Transfer_Otsub V_x V_y))))) _let_195 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x_2) (tptp.c_Nat__Transfer_Onat__set (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_x_2 V_y_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_x_2) V_A_2)) (tptp.c_Nat__Transfer_Ois__nat V_x_2)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_m_2) (=> (tptp.c_Nat__Transfer_Ois__nat V_n_2) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_m_2 V_n_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_m_2) (tptp.hAPP tptp.c_Int_Onat V_n_2))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.c_Nat__Transfer_Ois__nat (tptp.hAPP V_f_2 B_x)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Int_Oint) V_f_2) V_A_2) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Nat_Onat T_a) tptp.c_Int_Onat) V_f_2)) V_A_2))))) (forall ((V_t_2 $$unsorted) (V_A_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_t_2))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_A_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_D_2))))))))) (forall ((V_t_2 $$unsorted) (V_B_2 $$unsorted) (V_D_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_D_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xa) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) V_D_2))) (forall ((B_xb $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_B_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_D_2)) V_t_2)))))))) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_n_2)) (exists ((B_k $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_k) V_n_2)) (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_i) B_k)) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_i))))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_k) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))))) (forall ((V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_k)) (exists ((B_n $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) B_n)) (= V_k (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_n)))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (V_T_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (= (tptp.hAPP _let_1 V_T_2) (tptp.hAPP _let_1 V_S_2)))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (V_T_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (= (tptp.hAPP _let_1 V_S_2) (tptp.hAPP _let_1 V_T_2)))))))) (forall ((V_P_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_F_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_F_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (forall ((B_a $$unsorted) (B_F $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) B_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) B_F)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 B_F))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_F)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) B_a) B_F))))))))) (tptp.hBOOL (tptp.hAPP V_P_2 V_F_2)))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) (tptp.c_HOL_OThe T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_2 T_a) tptp.c_fequal) V_f_2)) V_a_2))) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_d $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_z))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_d))) V_z)))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a T_b) _let_1) V_f_2)) V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a T_b) (tptp.c_Groups_Oabs__class_Oabs T_a)) V_f_2)) V_A_2))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1) _let_1)))) (forall ((T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_fun T_c T_b))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 _let_1) (tptp.c_Fun_Oinj__on T_a _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_2 T_a) (tptp.c_COMBK T_b T_c)) V_f_2) _let_1))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (not (= V_x_2 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (not (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (= (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (= V_x_2 V_y_2))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) _let_194 (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Fun_Oinj__on T_a T_a (tptp.c_COMBI T_a) V_A_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (V_f_H_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_f_H_2) V_f_2) V_A_2) (tptp.c_Fun_Oinj__on T_a T_c V_f_2 V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.hAPP _let_1 _let_2) _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_a)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b)))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2)))) (forall ((V_N_2 $$unsorted)) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc V_N_2)) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m))) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) _let_1) _let_1))) (forall ((V_N_2 $$unsorted) (V_k_2 $$unsorted)) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) V_k_2) V_N_2)) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a T_b) _let_1) V_f_2)) V_A_2))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a_2)) _let_1)) (= V_a_2 _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a) V_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a_2))) (not (= V_a_2 _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a) V_a)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2))) (=> (tptp.c_Fun_Oinj__on T_b T_a V_f_2 V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_b))) (let ((_let_4 (tptp.hAPP _let_2 V_a))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_4) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_3) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_1 _let_4) _let_3)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d)))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2)) (= V_A_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (= (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2)) (= V_A_2 V_B_2))))) (forall ((V_f_H_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (= (tptp.c_Fun_Oinj__on T_b T_c V_f_H_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (tptp.c_Fun_Oinj__on T_a T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_f_H_2) V_f_2) V_A_2)))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_Fun_Oinj__on T_b T_c V_g_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (tptp.c_Fun_Oinj__on T_a T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_g_2) V_f_2) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_g_2) V_f_2) V_A_2) (tptp.c_Fun_Oinj__on T_c T_b V_g_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_c V_f_2) V_A_2)))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_c T_a V_g_2 _let_1) (tptp.c_Fun_Oinj__on T_c T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_b T_c V_f_2) V_g_2) _let_1))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (=> (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_m) V_n)) _let_1) (= (tptp.hAPP _let_2 V_m) _let_1))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Nat_Osemiring__char__0 T_a) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat T_a (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_29 _let_60) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))))) (forall ((V_b_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 _let_1) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) _let_1))) (exists ((B_x $$unsorted)) (and (= V_b_2 (tptp.hAPP V_f_2 B_x)) (forall ((B_y $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 B_y)) (= B_y B_x))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_x)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_x) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_5 (tptp.hAPP _let_4 _let_3))) (=> (tptp.class_Rings_Oordered__ring__abs T_a) (=> (and (or (tptp.hBOOL (tptp.hAPP _let_5 V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_a) _let_3))) (or (tptp.hBOOL (tptp.hAPP _let_5 V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_b) _let_3)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_A_H_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (exists ((B_f $$unsorted)) (and (tptp.c_Fun_Oinj__on T_a T_b B_f V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b B_f) V_A_2)) V_A_H_2)))) (exists ((B_g $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a B_g) V_A_H_2) V_A_2))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_f_2) V_A_2))) (tptp.c_Fun_Oinj__on T_a T_a V_f_2 V_A_2)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_f_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) _let_1) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_a T_a V_f_2 V_A_2) (= _let_1 V_A_2)))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) _let_1)) (=> (tptp.c_Fun_Oinj__on T_a T_a V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_f_2) _let_1) _let_1))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) _let_1)) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_f_2) _let_1) _let_1) (tptp.c_Fun_Oinj__on T_a T_a V_f_2 _let_1))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_3 V_c) V_d)))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d)))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oimage T_a T_b V_f_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_1)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_C_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_C_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_h_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_h_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_c) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_h_2) V_f_2)) V_B_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))) (not (tptp.c_Fun_Oinj__on T_b T_a V_f_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)))) (tptp.c_Fun_Oinj__on T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_x_2) V_y_2) V_A_2)))) (forall ((V_h_2 $$unsorted) (T_b $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_F_2)) (=> (tptp.c_Fun_Oinj__on T_b T_a V_h_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a V_h_2) V_F_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) V_A_2))) (forall ((V_z_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_z_2)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (= V_z_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_z $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w) V_z))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_w))) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_z)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_a T_a V_h_2 V_A_2) (= (tptp.hAPP (tptp.hAPP _let_1 V_g_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_b T_a V_g_2) V_h_2)) V_A_2))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) V_B_2)) (=> (tptp.c_Fun_Oinj__on T_b T_a V_g_2 V_B_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_g_2) V_B_2)) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_B_2))))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) V_B_2)) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (= (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP _let_2 V_A_2)) (and (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_2))) (let ((_let_4 (tptp.c_Set_Oimage T_a T_b V_f_2))) (= (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (and (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2))) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_A_2))) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_B_2))))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (= (exists ((B_f $$unsorted)) (and (tptp.c_Fun_Oinj__on T_a T_b B_f V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b B_f) V_A_2)) V_B_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_B_2))))))) (forall ((V_z_2 $$unsorted) (V_g_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_c))) (=> (tptp.class_Groups_Oab__semigroup__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_h_2 V_A_2) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b _let_1) V_g_2) V_z_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_a _let_1) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_g_2) V_h_2)) V_z_2) V_A_2))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_d $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_z) (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x) V_z))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_d)))))))) (forall ((V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image T_a T_b V_f_2 V_z_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_b T_b V_h_2 V_A_2) (= (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_F_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_b V_g_2) V_h_2)) V_A_2)))))) (forall ((V_j_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_C_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_C_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP V_B_2 B_x)) V_C_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_j_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_B_2)))))))))) _let_193 (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c T_a V_g_2) V_A_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 _let_1) (=> (tptp.c_Fun_Oinj__on T_c T_a V_g_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) _let_1))) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_c T_b V_A_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_b T_c V_f_2) V_g_2)) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_c T_b (tptp.c_Fun_Othe__inv__into T_c T_a V_A_2 V_g_2)) (tptp.c_Fun_Othe__inv__into T_a T_b _let_1 V_f_2)) V_x_2))))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) V_x_2)) V_B_2)))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image T_a T_b V_f_2 V_z_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) V_f_2) V_g_2)) V_h_2)) V_A_2) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_F_2 V_h_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) (tptp.hAPP V_f_2 V_x_2)) V_x_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (= (tptp.hAPP V_f_2 V_x_2) V_y_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) V_y_2) V_x_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (tptp.c_Fun_Oinj__on T_b T_a (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) V_A_2))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b _let_1 V_f_2) (tptp.hAPP V_f_2 V_x_2)) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) V_x_2) (tptp.c_HOL_OThe T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) tptp.c_fequal) V_f_2)) V_x_2))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image T_a T_b V_f_2 V_z_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_a T_b V_f_2) V_g_2) V_z_2) V_A_2))))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (= (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_a T_b V_A_2 V_f_2) V_y_2)) V_y_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_a))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_a) (tptp.class_Groups_Ocomm__monoid__mult T_a)) (=> (= V_A_2 V_B_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_B_2)))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_S_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (not (= V_S_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (=> (forall ((B_A $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) B_A) V_S_2)) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 B_A))) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_S_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (forall ((B_X $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_X) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) B_X) V_B_2))))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2)) V_B_2)))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_n_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (tptp.hAPP V_f_2 B_i)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_k_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) (tptp.hAPP V_f_2 V_n_2))) (exists ((B_i $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_i) V_n_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2)))))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_n_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (tptp.hAPP V_f_2 B_i)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_k_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) (tptp.hAPP V_f_2 V_n_2))) (exists ((B_i $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_i) V_n_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_I_2 $$unsorted)) (=> (not (= V_I_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_i) V_I_2)) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.hAPP V_A_2 B_i)))) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_I_2) V_A_2))))) (forall ((V_i_2 $$unsorted) (V_M_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)))) (let ((_let_2 (tptp.c_member tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_COMBC tptp.tc_Nat_Onat _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) tptp.c_fconj))) (let ((_let_6 (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.c_Set_OCollect tptp.tc_Nat_Onat))) (let ((_let_8 (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_M_2)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_8 (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_3 tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) _let_2) tptp.c_Nat_OSuc)) V_M_2))) (tptp.hAPP _let_1 V_i_2))))) (tptp.hAPP _let_8 (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 _let_2) V_M_2))) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_i_2)))))))))))))))) (forall ((V_i_2 $$unsorted) (V_M_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_member tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_M_2)) (not (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) _let_2) V_M_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP tptp.c_Nat_OSuc V_i_2))))) _let_1)))))) (forall ((V_i_2 $$unsorted) (V_M_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)))) (let ((_let_2 (tptp.c_member tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_COMBC tptp.tc_Nat_Onat _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) tptp.c_fconj))) (let ((_let_6 (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.c_Set_OCollect tptp.tc_Nat_Onat))) (let ((_let_8 (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_M_2))) (= (tptp.hAPP _let_8 (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_3 tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) _let_2) tptp.c_Nat_OSuc)) V_M_2))) (tptp.hAPP _let_1 V_i_2)))) (tptp.hAPP _let_8 (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 _let_2) V_M_2))) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_i_2)))))))))))))))) (forall ((V_k_2 $$unsorted) (V_N_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k_2) B_x)))) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat)) V_k_2) V_N_2))) (forall ((T_a $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (or (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_P_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_Q_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) V_P_2)) V_Q_2)))))))) (forall ((V_k_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) V_k_2))))) (forall ((V_k_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat)) V_k_2))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2)) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2) V_P_2)) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)) V_A_2)) (forall ((V_P_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2))) (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)))) (forall ((V_P_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2))) (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)))) (forall ((V_P_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2))) (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)))) (forall ((T_a $$unsorted)) (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool T_a) tptp.c_fFalse)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fdisj) V_P_2)) V_Q_2)))) (and (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_P_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_Q_2)))))))) (forall ((V_P_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.hAPP _let_1 V_P_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2)))) V_P_2))))) (forall ((T_a $$unsorted)) (= (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool T_a) tptp.c_fTrue)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fdisj) V_P_2)) V_Q_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_P_2)) (tptp.hAPP _let_1 V_Q_2))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) V_P_2)) V_Q_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_P_2)) (tptp.hAPP _let_1 V_Q_2))))) (forall ((V_P_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_P_2)) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_a) V_P_2) V_f_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fdisj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fdisj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fdisj) (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_P_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2)))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)))))) (forall ((T_a $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_P_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) _let_1))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((T_a $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP tptp.c_fequal V_a_2))) V_P_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) _let_1))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP tptp.c_fequal V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP _let_2 V_B_2)))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (let ((_let_2 (tptp.c_Set_OCollect T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_P_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) V_P_2)))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_B_2))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b V_f_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) T_a) (tptp.c_member T_b)) V_f_2)) V_B_2))))) (forall ((V_i_2 $$unsorted) (V_P_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) tptp.c_fconj) V_P_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) V_i_2)))))) (forall ((V_R_2 $$unsorted) (V_f_2 $$unsorted) (V_T_2 $$unsorted) (T_b $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_COMBB _let_1 _let_1 T_b))) (let ((_let_4 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_c))) (let ((_let_5 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c))) (let ((_let_6 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.c_COMBB _let_6 _let_6 T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_T_2)) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_6 T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_c) (tptp.tc_fun _let_6 T_c) T_a) _let_5) V_f_2)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.c_Set_OCollect T_b)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_6 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_T_2)))) V_R_2)))) V_S_2) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a T_c) (tptp.tc_fun _let_1 T_c) T_b) _let_4) (tptp.hAPP (tptp.c_COMBC T_a T_b T_c) V_f_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Set_OCollect T_a)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_S_2)))) (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) V_R_2))))) V_T_2))))))))))))) (forall ((V_S_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2))) (let ((_let_3 (tptp.tc_fun T_a T_a))) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a _let_3) T_b) (tptp.c_If T_a)) V_P_2)) V_f_2)) V_g_2)) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_P_2)))) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_g_2) (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) V_P_2)))))))))) (forall ((V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll T_b)) V_P_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_2) (tptp.c_Orderings_Otop__class_Otop _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) _let_1) (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) V_P_2)))))))) (forall ((V_n_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat T_a V_f_2) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) V_n_2)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (let ((_let_2 (tptp.hAPP _let_1 V_B_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_4 V_f_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP _let_1 V_A_2))) _let_2))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_3) T_a) (tptp.c_If T_b)) _let_2)) V_f_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))) V_A_2))))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_g_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBB _let_1 _let_1 T_c))) (let ((_let_3 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (= (tptp.hAPP _let_3 V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 T_b T_c) _let_3) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Set_OCollect T_a)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_S_2)))) (tptp.hAPP (tptp.c_COMBC T_a T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_a) tptp.c_fequal) V_f_2)))))) (tptp.hAPP (tptp.c_Set_Oimage T_a T_c V_f_2) V_S_2))))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2))) (let ((_let_4 (tptp.tc_Option_Ooption T_b))) (let ((_let_5 (tptp.tc_fun _let_4 _let_4))) (= (tptp.c_Map_Odom T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_4 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_4 _let_5) T_a) (tptp.c_If _let_4)) V_P_2)) V_f_2)) V_g_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Map_Odom T_a T_b V_f_2)) (tptp.hAPP _let_1 V_P_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Map_Odom T_a T_b V_g_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) V_P_2))))))))))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Map_Odom T_a T_b V_g_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC T_a _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (let ((_let_4 (tptp.c_Set_OCollect T_a))) (let ((_let_5 (tptp.c_Groups_Ominus__class_Ominus _let_2))) (= (tptp.c_Map_Odom T_a T_b (tptp.c_Fun_Ooverride__on T_a (tptp.tc_Option_Ooption T_b) V_f_2 V_g_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) (tptp.hAPP (tptp.hAPP _let_5 (tptp.c_Map_Odom T_a T_b V_f_2)) (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_5 V_A_2) _let_1))))) (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) _let_1))))))))))) (forall ((V_k_2 $$unsorted) (V_R_2 $$unsorted) (V_T_2 $$unsorted) (T_b $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_COMBB _let_2 _let_2 T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_T_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_S_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) V_R_2) B_x)))) V_k_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_Nat_Onat T_a) _let_1) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Set_OCollect T_b)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_T_2)))) V_R_2)))) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP _let_1 V_T_2)))))))))) (forall ((V_k_2 $$unsorted) (V_R_2 $$unsorted) (V_t_2 $$unsorted) (T_b $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBB _let_1 _let_1 T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_s_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_t_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_t_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_s_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) V_R_2) B_x)))) (tptp.hAPP V_k_2 B_x)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_Nat_Onat T_a) (tptp.c_Finite__Set_Ocard T_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Set_OCollect T_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_t_2)))) V_R_2)))) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_k_2) V_t_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2))) (let ((_let_3 (tptp.tc_fun T_a T_a))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_a) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2))))))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq _let_1)) V_A_2))))))) (forall ((V_Q_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) V_P_2))) V_Q_2))))) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_a tptp.tc_HOL_Obool) V_Q_2) B_y)))))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a T_b))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_a) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_2 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) V_f_2)))) V_P_2))))))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Ooverride__on T_a T_b V_f_2 V_g_2 V_A_2) V_a_2) (tptp.hAPP V_g_2 V_a_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))) (= (tptp.hAPP (tptp.c_Fun_Ooverride__on T_a T_b V_f_2 V_g_2 V_A_2) V_a_2) (tptp.hAPP V_f_2 V_a_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Fun_Ooverride__on T_a T_b V_f_2 V_g_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_f_2)) (forall ((V_P_2 $$unsorted)) (tptp.c_Nat__Transfer_Onat__set (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) V_P_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Fun_Ooverride__on T_a T_b V_f_2 V_g_2 V_A_2) V_a_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))) (and (=> _let_2 (= _let_1 (tptp.hAPP V_g_2 V_a_2))) (=> (not _let_2) (= _let_1 (tptp.hAPP V_f_2 V_a_2))))))) (forall ((V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) V_P_2)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_1) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_Nat_Onat) V_P_2) _let_1)))))) (forall ((V_P_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) V_P_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_Int_Oint) V_P_2) tptp.c_Int_Onat)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_a) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_7 (tptp.tc_fun T_a T_a))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_a _let_1))) (let ((_let_10 (tptp.tc_fun T_a _let_7))) (let ((_let_11 (tptp.tc_fun T_a _let_3))) (let ((_let_12 (tptp.tc_fun T_a _let_11))) (let ((_let_13 (tptp.tc_fun T_a _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)))) (let ((_let_15 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_16 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_16 V_A_2)) (=> (not (= V_A_2 _let_15)) (=> (tptp.hBOOL (tptp.hAPP _let_16 V_B_2)) (=> (not (= V_B_2 _let_15)) (= (tptp.hAPP (tptp.hAPP _let_6 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_B_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_a) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_a) (tptp.c_COMBS T_a _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_a) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_a) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_a) (tptp.c_COMBB _let_7 _let_1 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a)) (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_A_2)))) (tptp.hAPP _let_2 V_B_2)))))))))))))))))))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2))) (let ((_let_3 (tptp.tc_fun T_a T_a))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_a) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_a) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_7 (tptp.tc_fun T_a T_a))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_a _let_1))) (let ((_let_10 (tptp.tc_fun T_a _let_7))) (let ((_let_11 (tptp.tc_fun T_a _let_3))) (let ((_let_12 (tptp.tc_fun T_a _let_11))) (let ((_let_13 (tptp.tc_fun T_a _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)))) (let ((_let_15 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_16 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (=> (tptp.hBOOL (tptp.hAPP _let_16 V_A_2)) (=> (not (= V_A_2 _let_15)) (=> (tptp.hBOOL (tptp.hAPP _let_16 V_B_2)) (=> (not (= V_B_2 _let_15)) (= (tptp.hAPP (tptp.hAPP _let_6 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a V_B_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_a) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_a) (tptp.c_COMBS T_a _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_a) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_a) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_a) (tptp.c_COMBB _let_7 _let_1 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a)) (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_A_2)))) (tptp.hAPP _let_2 V_B_2)))))))))))))))))))))))))))))) (forall ((V_k_2 $$unsorted) (V_N_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_k_2))) (let ((_let_3 (tptp.tc_fun T_a T_a))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (=> (tptp.class_Groups_Olinordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_a) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_N_2))))))))))))))) (forall ((V_k_2 $$unsorted) (V_N_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_k_2))) (let ((_let_3 (tptp.tc_fun T_a T_a))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (=> (tptp.class_Groups_Olinordered__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_a) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_N_2))))))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (exists ((B_n $$unsorted) (B_f $$unsorted)) (= V_A_2 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat T_a B_f) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) B_n))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2))) (exists ((B_x $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) V_A_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) _let_1) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) tptp.c_fequal) V_f_2)) (tptp.hAPP V_f_2 B_x))))))))))))) (forall ((V_z_2 $$unsorted) (V_z_H_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Int_Onat V_z_2))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_2) (tptp.hAPP tptp.c_Int_Onat V_z_H_2)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg V_z_H_2)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z_2) V_z_H_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_1) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) tptp.c_Int_Onat)))))))))) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_126))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n))))) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_127))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_x_2 $$unsorted)) (= (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x_2)))) (forall ((V_Z_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg V_Z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_Z_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg V_z)) (= (tptp.hAPP tptp.c_Int_Onat V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_z $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg V_z))) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)) V_z))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2)) V_x_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2)) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2)) V_A_2)))))) (forall ((V_N_2 $$unsorted) (V_M_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_M_2) V_N_2)) (=> (not (= V_M_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_M_2))))))))) (forall ((V_N_2 $$unsorted) (V_M_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_M_2) V_N_2)) (=> (not (= V_M_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_M_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_N_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_a _let_3)))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_a _let_3)))))))))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_P_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2))) (let ((_let_4 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b))) (let ((_let_5 (tptp.tc_fun T_b T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_5) T_a) (tptp.c_If T_b)) V_P_2)) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_4 V_f_2) (tptp.hAPP _let_3 _let_1))) (tptp.hAPP (tptp.hAPP _let_4 V_g_2) (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_2) _let_1))))))))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_k_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_C_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) _let_2)) (=> (forall ((B_c $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) B_c) V_C_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) B_c) V_k_2))) (=> (forall ((B_c1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) B_c1) V_C_2)) (forall ((B_c2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_c2) V_C_2)) (=> (not (= B_c1 B_c2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) B_c1) B_c2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard _let_1) V_C_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) _let_2))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (tptp.c_Fun_Oinj__on T_a T_a (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_A_2))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2)))))) (forall ((T_a $$unsorted)) (=> (and (tptp.class_Lattices_Osemilattice__sup T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.c_Orderings_Oord__class_Omax T_a)))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n) V_n)) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (forall ((T_a $$unsorted)) (=> (and (tptp.class_Lattices_Osemilattice__inf T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.c_Orderings_Oord__class_Omin T_a)))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) V_n) _let_1) _let_1))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (let ((_let_3 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (and (tptp.class_Groups_Ocomm__monoid__mult T_a) (tptp.class_Rings_Osemiring__1 T_a)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b tptp.tc_Nat_Onat) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a T_b) _let_1) V_f_2)) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Nat_Onat) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_a) _let_1) V_f_2)) V_A_2)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_a)) V_a)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= V_a_2 (tptp.hAPP _let_1 V_b_2)) (= V_b_2 (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= (tptp.hAPP _let_1 V_a_2) V_b_2) (= (tptp.hAPP _let_1 V_b_2) V_a_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_b_2)) (= V_a_2 V_b_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_A_2)) V_A_2))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_x)) V_x)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_a) V_a))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ouminus T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b T_a)) V_A_2) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y)) V_x))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y)) V_x))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_c)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_c)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2)) (= V_A_2 V_B_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) V_x) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) V_x) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_x) V_x))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_x) V_x))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ouminus T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b T_a)) V_A_2) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) (tptp.hAPP _let_2 V_z))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)))))))) (forall ((V_a $$unsorted) (V_x $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b)) V_x)))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b)) V_x)))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) V_y_2)) V_z_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_z_2))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) V_y_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_y_2))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b_2)) (tptp.hAPP _let_2 V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a_2)) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b_2)) V_a_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) (tptp.hAPP _let_1 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) (tptp.hAPP _let_1 V_a_2)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) V_y_2)) V_z_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_z_2))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) V_y_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_y_2))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2) V_a_2) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) (= _let_1 V_a_2))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b))) (not (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (not (tptp.hBOOL (tptp.hAPP _let_1 V_b))))))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d))))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_y) V_z)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b)))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y) V_x)))) (forall ((V_a $$unsorted) (V_x $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b)) V_x)))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b)) V_x)))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) V_y_2)) V_z_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_y_2) V_z_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) V_y_2) V_x_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y)) V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y)) V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_x)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_x_2)) (tptp.hAPP _let_2 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_a)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b_2)) (tptp.hAPP _let_2 V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a_2)) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b_2)) V_a_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) (tptp.hAPP _let_1 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) (tptp.hAPP _let_1 V_a_2)))))))) (forall ((V_x $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b)) V_x)) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x))))))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d))))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_y) V_z)) V_x))))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b)) V_x))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y) V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y) V_y)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) V_y_2)) V_z_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_z_2))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) V_y_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) V_y_2) V_y_2)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Nat_Onat) V_f_2) V_A_2) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_a) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_f_2)) V_A_2)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2) V_d_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_b_2) V_d_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_b_2) V_d_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_b_2) V_d_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_b_2) V_d_2))))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2))) (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oidom T_a) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_b_2)) (or (= V_a_2 V_b_2) (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b) (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_a))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_b)) V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_b)))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_n $$unsorted) (V_i $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_i)) (tptp.hAPP (tptp.hAPP _let_2 V_n) V_i)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) V_i))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP _let_1 V_a))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_a T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2))))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_2) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a) (tptp.hAPP (tptp.c_COMBK T_a T_b) _let_1)) V_A_2) _let_1)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a T_b) _let_1) V_f_2)) V_A_2)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_a (tptp.c_Orderings_Oord__class_Omax T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_a (tptp.c_Orderings_Oord__class_Omin T_a)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a T_b) _let_2) V_f_2)) V_A_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) V_P_2)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_P_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_a) (tptp.class_Groups_Ocomm__monoid__mult T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Nat_Onat T_a) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Int_Oint) V_f_2) tptp.c_Int_Onat)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_A_2))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_b)))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Rings_Ocomm__semiring__1 T_b) (tptp.class_Rings_Ono__zero__divisors T_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a) V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_g_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a) V_g_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2) V_A_2) (tptp.c_Groups_Oone__class_Oone T_b))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2) V_A_2) (tptp.c_Groups_Oone__class_Oone T_b))))) (forall ((V_f_2 $$unsorted) (V_F_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_F_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Nat_Onat) V_f_2) V_F_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_F_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a_2))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a_2))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a_2)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a_2))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a_2)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a) V_b)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_x) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_y))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_b))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x_2) V_x_2) _let_1) (or (= V_x_2 _let_1) (= V_x_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone T_a))) V_x))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_x)) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_x)) V_x) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_x) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_y))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Orderings_Otop__class_Otop T_a)) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Orderings_Obot__class_Obot T_a)) (tptp.c_Orderings_Otop__class_Otop T_a)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_x)) V_x) (tptp.c_Orderings_Otop__class_Otop T_a)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_x)) (tptp.c_Orderings_Otop__class_Otop T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a_2)) V_b_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) V_b_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_b)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a_2)) V_b_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) V_b_2))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) (tptp.c_Orderings_Otop__class_Otop _let_1)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Orderings_Otop__class_Otop _let_1)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_a) (tptp.class_Groups_Ocomm__monoid__mult T_a)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Int_Oint T_a) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Nat_Onat T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_a tptp.tc_Nat_Onat) V_f_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) V_m)) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat) V_n) V_m))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)) V_A_2) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_x_2 V_y_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a (tptp.hAPP _let_1 V_y_2) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.c_Orderings_Otop__class_Otop _let_1)) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP _let_2 V_A_2)) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) _let_2) V_g_2)) V_h_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_h_2) V_A_2)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_a))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a_2)) _let_1))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a V_x_2 V_y_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_a (tptp.hAPP _let_1 V_y_2) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))))) (forall ((V_k_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_k_2)))) (forall ((V_k_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_k_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fimplies) V_P_2)) V_Q_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_2) (tptp.hAPP _let_1 V_P_2))) (tptp.hAPP _let_1 V_Q_2)))))) (forall ((V_k_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_k_2)))) (forall ((V_k_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_k_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2)))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Nat_Onat) V_f_2) V_S_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP V_f_2 B_x)))))))) (forall ((V_h_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_h_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_c) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_h_2) V_f_2)) V_B_2))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (=> (= V_g_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_h_2) V_f_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_h_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_c) V_g_2) V_A_2))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a))) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (=> (tptp.class_Groups_Oabs__if T_a) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a))) (=> (not _let_2) (= _let_1 V_a))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) _let_2) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_n_2)) V_m_2))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (= (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_m_2) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP _let_1 V_n_2)) V_m_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) T_a) (tptp.c_If T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_b_2)) _let_1)) V_S_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (and (=> _let_4 (= _let_3 (tptp.hAPP V_b_2 V_a_2))) (=> (not _let_4) (= _let_3 _let_1)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) T_a) (tptp.c_If T_b)) (tptp.hAPP tptp.c_fequal V_a_2))) V_b_2)) _let_1)) V_S_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (and (=> _let_4 (= _let_3 (tptp.hAPP V_b_2 V_a_2))) (=> (not _let_4) (= _let_3 _let_1)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oimage T_b T_a V_f_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP _let_2 (tptp.c_Orderings_Otop__class_Otop _let_1)) (tptp.c_Orderings_Otop__class_Otop _let_3)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_3) (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2))))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) _let_1) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) _let_1) _let_1)))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_b_2) V_d_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_b_2) V_d_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_x_2) V_y_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a (tptp.hAPP _let_1 V_y_2) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_a V_x_2 V_y_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a (tptp.hAPP _let_1 V_y_2)) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_a_2 V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_c_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a_2) V_c_2) V_b_2)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_a_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_a (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a_2) V_c_2) V_d_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omax T_a)) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_A_2))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_a) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_a) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_a) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_a) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_a T_a V_h_2 V_A_2) (= (tptp.hAPP (tptp.hAPP _let_1 V_g_2) (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_b T_a V_g_2) V_h_2)) V_A_2))))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a (tptp.c_Groups_Otimes__class_Otimes T_b)) V_g_2) _let_1) V_A_2))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a (tptp.c_Groups_Otimes__class_Otimes T_b)) V_g_2) (tptp.c_Groups_Oone__class_Oone T_b)) V_A_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y) (tptp.c_Orderings_Obot__class_Obot T_a)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y) (tptp.c_Orderings_Otop__class_Otop T_a)) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_x) V_y))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_n_2)) V_m_2)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_m_2) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP _let_1 V_n_2)) V_m_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2) V_A_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_b_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_A_2)) V_a_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_A_2)) V_x_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) B_x) V_x_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_A_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) B_x))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_A_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x_2) B_x))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_A_2)) V_x_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) B_x) V_x_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_B_2)) _let_1) _let_1)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_B_2)) _let_1) _let_1)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_a T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_3)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_3) V_A_2))) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_2) (tptp.hAPP _let_1 V_A_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_B_2)))))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Orderings_Oord__class_Omax T_a)) V_a_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_a_2) V_A_2))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_d_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Ovimage T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_a _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a _let_2) T_b) (tptp.c_If T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_B_2))) V_c_2)) V_d_2)) V_A_2))) (let ((_let_4 (tptp.c_member T_a))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_d_2) V_A_2)))) (let ((_let_6 (not _let_5))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_c_2) V_A_2)))) (and (=> _let_7 (and (=> _let_5 (= _let_3 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> _let_6 (= _let_3 V_B_2)))) (=> (not _let_7) (and (=> _let_5 (= _let_3 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_B_2))) (=> _let_6 (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold1 T_a (tptp.c_Orderings_Oord__class_Omin T_a)))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_B_2)) (tptp.hAPP _let_1 V_A_2)))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_a _let_4)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olinorder__class_OMax T_a _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) _let_2) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_B_2))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_a _let_4)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olinorder__class_OMin T_a _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) _let_2) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_B_2))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Orderings_Oord__class_Omax T_a)) V_x_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a (tptp.c_Orderings_Oord__class_Omin T_a)) V_x_2) V_A_2)))))) (forall ((V_S_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (=> (not (= V_S_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_S_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) V_S_2)))))))) (forall ((V_S_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (=> (not (= V_S_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_S_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.c_Set_Oimage T_a T_a _let_1) V_S_2)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Orderings_Oord_Omin T_a (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (tptp.c_Orderings_Oord__class_Omax T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Orderings_Oord_Omax T_a (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (tptp.c_Orderings_Oord__class_Omin T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.c_Orderings_Oord__class_Omax T_a) (tptp.c_Orderings_Oord_Omax T_a (tptp.c_Orderings_Oord__class_Oless__eq T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.c_Orderings_Oord__class_Omin T_a) (tptp.c_Orderings_Oord_Omin T_a (tptp.c_Orderings_Oord__class_Oless__eq T_a))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_a) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_7 (tptp.tc_fun T_a T_a))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_a _let_1))) (let ((_let_10 (tptp.tc_fun T_a _let_7))) (let ((_let_11 (tptp.tc_fun T_a _let_3))) (let ((_let_12 (tptp.tc_fun T_a _let_11))) (let ((_let_13 (tptp.tc_fun T_a _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)))) (let ((_let_15 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_16 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_17 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_17 V_A_2)) (=> (not (= V_A_2 _let_16)) (=> (tptp.hBOOL (tptp.hAPP _let_17 V_B_2)) (=> (not (= V_B_2 _let_16)) (= (tptp.hAPP (tptp.hAPP _let_6 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_15 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_15 V_B_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_15 (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_a) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_a) (tptp.c_COMBS T_a _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_a) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_a) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_a) (tptp.c_COMBB _let_7 _let_1 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a)) (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_A_2)))) (tptp.hAPP _let_2 V_B_2))))))))))))))))))))))))))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_n))) (tptp.hAPP _let_1 V_m))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (tptp.hAPP _let_1 V_m))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_z)) V_w) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w)))))) (= (tptp.hAPP _let_148 _let_126) _let_126) (= (tptp.c_Lattices_Osemilattice__inf__class_Oinf tptp.tc_Nat_Onat) (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat)) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup tptp.tc_Nat_Onat) (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat)) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z) V_w)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_z)) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_2 V_n_2)) (tptp.hAPP _let_2 V_m_2)) (and (= V_n_2 _let_1) (= V_m_2 _let_1)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_n)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m))))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_w))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_w) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP _let_2 V_n_2)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_2 V_m_2)))) (and (= V_n_2 _let_1) (= V_m_2 _let_1)))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_i))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_i) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_i))) (=> (not _let_2) (= _let_1 V_i)))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Nat_OSuc V_n)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) _let_2) _let_2))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a (tptp.c_Orderings_Oord__class_Omax T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a _let_1) V_A_2)))))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Nat_OSuc V_n))))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_B_2)) _let_2) _let_2))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_a (tptp.c_Orderings_Oord__class_Omax T_a) V_A_2)) V_a_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (not (= V_A_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= V_B_2 _let_3)) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_B_2))))))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_a_2) V_A_2)))))) (forall ((V_z_H $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_z))) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_z_H))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_6 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_5 (tptp.hAPP _let_3 V_A_2)))) (let ((_let_7 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_7 (= _let_6 V_x_2)) (=> (not _let_7) (= _let_6 (tptp.hAPP (tptp.hAPP _let_5 V_x_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_5 _let_4))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (not (= V_A_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= V_B_2 _let_3)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) _let_3) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_B_2)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_B_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_x_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_5 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_4 V_A_2))) (let ((_let_6 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_4 _let_3))))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2))) (let ((_let_3 (tptp.tc_fun T_a T_a))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_6 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_6 (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_a) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))))))))))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.c_Nat__Transfer_Ois__nat (tptp.hAPP V_f_2 B_x)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Int_Oint) V_f_2) V_A_2) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Nat_Onat T_a) tptp.c_Int_Onat) V_f_2)) V_A_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2))) (let ((_let_3 (tptp.tc_fun T_a T_a))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_6 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_6 (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_a) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_a) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_7 (tptp.tc_fun T_a T_a))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_a _let_1))) (let ((_let_10 (tptp.tc_fun T_a _let_7))) (let ((_let_11 (tptp.tc_fun T_a _let_3))) (let ((_let_12 (tptp.tc_fun T_a _let_11))) (let ((_let_13 (tptp.tc_fun T_a _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_a)))) (let ((_let_15 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_16 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_17 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_17 V_A_2)) (=> (not (= V_A_2 _let_16)) (=> (tptp.hBOOL (tptp.hAPP _let_17 V_B_2)) (=> (not (= V_B_2 _let_16)) (= (tptp.hAPP (tptp.hAPP _let_6 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_15 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_15 V_B_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_15 (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_a) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_a) (tptp.c_COMBS T_a _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_a) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_a) (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_a) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_a) (tptp.c_COMBB _let_7 _let_1 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a)) (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_A_2)))) (tptp.hAPP _let_2 V_B_2))))))))))))))))))))))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olattice_OSup__fin T_a (tptp.c_Orderings_Oord__class_Omax T_a) V_A_2)) V_A_2))))))) (forall ((V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_z)) V_z))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) _let_2) _let_2))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a (tptp.c_Orderings_Oord__class_Omin T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a _let_1) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_B_2)) _let_2) _let_2))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_a (tptp.c_Orderings_Oord__class_Omin T_a) V_A_2)) V_a_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (not (= V_A_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= V_B_2 _let_3)) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_B_2))))))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_a_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_6 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_5 (tptp.hAPP _let_3 V_A_2)))) (let ((_let_7 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (=> _let_7 (= _let_6 V_x_2)) (=> (not _let_7) (= _let_6 (tptp.hAPP (tptp.hAPP _let_5 V_x_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_5 _let_4))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (not (= V_A_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= V_B_2 _let_3)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) _let_3) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_B_2)))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_x_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_5 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_4 V_A_2))) (let ((_let_6 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_4 _let_3))))))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Big__Operators_Olattice_OInf__fin T_a (tptp.c_Orderings_Oord__class_Omin T_a) V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a (tptp.c_Orderings_Oord__class_Omax T_a) V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olattice_OInf__fin T_a (tptp.c_Orderings_Oord__class_Omin T_a) V_A_2)) V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2)) V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2)) V_A_2))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_g_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (= (tptp.hAPP V_g_2 B_x) (tptp.c_Groups_Oone__class_Oone T_b)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_f_2 $$unsorted) (V_T_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_S_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_b)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP _let_1 V_S_2)) (tptp.hAPP _let_1 V_T_2)))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_S_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_c_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_3) T_a) (tptp.c_If T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_b_2)) V_c_2)) V_S_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP V_b_2 V_a_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_2 _let_1)))))))))))) (forall ((V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_k))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (=> (tptp.class_Rings_Oring__1 T_a) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_k)))))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat V_k)))))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))) (let ((_let_6 (tptp.hAPP V_f_2 V_a_2))) (=> (tptp.class_Fields_Ofield T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (not (= _let_6 (tptp.c_Groups_Ozero__class_Ozero T_b))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) _let_2) _let_6))) (=> (not _let_5) (= _let_4 _let_2))))))))))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x) (tptp.c_Nat__Transfer_Ois__nat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x) V_n)))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n))) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int tptp.tc_Int_Oint) _let_1) _let_1))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) V_n) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) V_n) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n))))) (forall ((V_n $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_z) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP _let_1 V_z)) V_n))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Int_Oring__char__0 T_a) (= (= (tptp.hAPP _let_1 V_w_2) (tptp.hAPP _let_1 V_z_2)) (= V_w_2 V_z_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) _let_1) V_a) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) _let_1) _let_1)))) (forall ((V_q $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_q) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_q)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_q))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p)) V_q) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_p) V_q))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b)))))) (forall ((V_n $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_z) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) V_n)))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) V_n) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x) V_n))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a))) (=> (tptp.class_Fields_Ofield T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2)) V_r_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_1) V_f_2)) V_r_2)) V_A_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2) V_b_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2) V_a_2) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (= V_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_c)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b) V_c) V_a))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_c) V_b) (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b) V_c)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) V_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_3 (= V_a _let_2))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (and (=> _let_3 (= _let_1 _let_2)) (=> (not _let_3) (= _let_1 (tptp.c_Groups_Oone__class_Oone T_a))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_b_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a_2) V_b_2) (tptp.c_Groups_Oone__class_Oone T_a)) (= V_a_2 V_b_2))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)))))))) (forall ((V_q $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) (tptp.hAPP _let_1 V_q)))))) (forall ((V_q $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) (tptp.hAPP _let_1 V_q)) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_q)))))) (forall ((V_q $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_q)) V_x) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_q)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_p)) (tptp.hAPP _let_1 V_q)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_p) V_q)))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x) V_n)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))) (forall ((V_n_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_a))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) V_n_2)) _let_1))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Oring__char__0 T_a) (= (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z_2)) (= V_z_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Oring__char__0 T_a) (= (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z_2) (tptp.c_Groups_Ozero__class_Ozero T_a)) (= V_z_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_w_2)) (tptp.hAPP _let_1 V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_w_2) V_z_2)))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP _let_1 V_w_2)) (tptp.hAPP _let_1 V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) V_z_2)))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_w) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 V_z)))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 V_z)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_b))) (=> (tptp.class_Fields_Ofield__inverse__zero T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_a) _let_2) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_z $$unsorted) (V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_w) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 V_z)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n)))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_z)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP _let_1 V_z)))))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z_2)) (tptp.c_Int_Oring__1__class_OInts T_a))))) (forall ((V_n_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x_2))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_x_2 _let_1)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) _let_3) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_n_2)) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1))))))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_x)) V_n)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Int_Oint) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_a T_b) _let_1) V_f_2)) V_A_2))))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) (tptp.hAPP (tptp.c_COMBK T_b T_a) V_y_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_y_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (and (tptp.class_Groups_Ocomm__monoid__mult T_a) (tptp.class_Rings_Oring__1 T_a)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b tptp.tc_Int_Oint) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_a T_b) _let_1) V_f_2)) V_A_2))))) (forall ((V_p $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_p))) (let ((_let_3 (tptp.hAPP _let_2 V_m))) (let ((_let_4 (= V_m (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_4 (= _let_3 _let_1)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_p) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) _let_1))))))))))) (forall ((V_n_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_x_2)) V_n_2))) (or (not (= V_x_2 _let_1)) (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2))))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z_2)) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2))))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z_2)) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n)))))))))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_b_2))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_b_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_x_2) V_y_2))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_a) (=> (not (= V_a _let_1)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n) _let_1)))))) (forall ((V_m_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x_2) V_m_2) _let_2) (or (= V_m_2 _let_1) (= V_x_2 _let_2)))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_i)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))))) (forall ((V_n_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_x_2) V_n_2))) (or (tptp.hBOOL (tptp.hAPP _let_2 V_x_2)) (= V_n_2 _let_1)))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_n) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a) (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) V_n) _let_1)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP _let_1 V_m)) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) V_n)))))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_n) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_n)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_a))) (forall ((V_n $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))))))) (forall ((V_n_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (tptp.class_Power_Opower T_a) (tptp.class_Rings_Omult__zero T_a) (tptp.class_Rings_Ono__zero__divisors T_a) (tptp.class_Rings_Ozero__neq__one T_a)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) V_n_2) _let_1) (and (= V_a_2 _let_1) (not (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (tptp.class_Power_Opower T_a) (tptp.class_Rings_Osemiring__0 T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP tptp.c_Nat_OSuc V_n)) _let_1)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a_2)) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_n_2)) (= V_m_2 V_n_2)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_n)) V_a))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Power_Opower T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP _let_1 V_n)))))) (forall ((V_n $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_i)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_i) V_n)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Power_Opower T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_n) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_n)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_n)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_n)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_3 V_a) _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_b) _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (= V_a V_b))))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_a) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_b) _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1)))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) V_n))) (let ((_let_3 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (and (tptp.class_Power_Opower T_a) (tptp.class_Rings_Osemiring__0 T_a)) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP tptp.c_Nat_OSuc V_n)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_b_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) V_b_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x_2) V_y_2)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)))))))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_N)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 V_N))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) V_n))))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_N)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 V_N))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a))) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_n))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1)) _let_1)))))))) (forall ((V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_n)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= V_a V_b))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP tptp.c_Nat_OSuc V_n))) _let_1)))))))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_N)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_N)) (tptp.hAPP _let_1 V_n)))))))))) (forall ((V_a $$unsorted) (V_N $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_N)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_N)) (tptp.hAPP _let_1 V_n)))))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_c_2) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_1) V_c_2)))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_6) V_b_2))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_6))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_3 _let_1)))))))))))))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) _let_1)))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_c_2)))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) V_a_2)) (and (=> _let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_5))) (=> (not _let_6) (and (=> _let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_5) V_b_2))) (=> (not _let_4) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a_2))))))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)))))))) (forall ((V_w $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_2 V_z) V_w)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_w))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a_2) V_b_2))) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a_2) V_b_2)) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2)))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a_2) V_b_2))) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a_2) V_b_2)) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2)))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) _let_1)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) _let_1)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (= V_c_2 _let_1))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) (and (=> (not _let_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2) V_b_2)) (=> _let_2 (= V_a_2 _let_1)))))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (= V_c_2 _let_1))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2) V_a_2) (and (=> (not _let_2) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2))) (=> _let_2 (= V_a_2 _let_1)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_w_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y_2 _let_3)) (=> (not (= V_z_2 _let_3)) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_y_2) (tptp.hAPP (tptp.hAPP _let_2 V_w_2) V_z_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2) (tptp.hAPP (tptp.hAPP _let_1 V_w_2) V_y_2)))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) _let_1))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) V_y)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_y) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) _let_1)))))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_w) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_w)))))))))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_4) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_4) V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_w) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_w))))))))))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_3) V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_w) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_w)))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) _let_1)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_y) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) _let_3)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_c)) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y)) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) V_z))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c_2) _let_1)))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP _let_3 V_c_2)))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) V_a_2)) (and (=> _let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_5))) (=> (not _let_6) (and (=> _let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_5) V_b_2))) (=> (not _let_4) (tptp.hBOOL (tptp.hAPP _let_3 V_a_2))))))))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c_2) _let_1)))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_c_2)))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (and (=> _let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_5) V_b_2))) (=> (not _let_6) (and (=> _let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_5))) (=> (not _let_4) (tptp.hBOOL (tptp.hAPP _let_3 _let_1))))))))))))))) (forall ((V_w $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y _let_4)) (=> (not (= V_z _let_4)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_3 V_w) V_z)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_y))) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_z)) V_y) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_z))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_y (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_y))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_x)) V_y)) V_z))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (=> (not (= V_y (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_y))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_x)) V_y)) V_z))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_z)) V_y) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) V_z))))))) (forall ((V_w $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_3 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y _let_4)) (=> (not (= V_z _let_4)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_3 V_w) V_z)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_y))) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_1))) V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_b)) (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_1)))))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_x)) V_y) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) _let_3)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_c_2)))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y)) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_y))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) V_z))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.class_Fields_Ofield T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (not (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Power_Opower T_a) (= (tptp.c_Power_Opower__class_Opower T_a) (tptp.c_Power_Opower_Opower T_a (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Otimes__class_Otimes T_a))))) (forall ((V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_m)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_k)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) (tptp.hAPP _let_2 V_m)) V_k) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.hAPP tptp.c_Nat_OSuc _let_1))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_a) V_B_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_1 V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_1) V_b) _let_1))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x) (=> (tptp.c_Nat__Transfer_Ois__nat V_y) (tptp.c_Nat__Transfer_Ois__nat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_x) V_y))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) _let_1) V_a) _let_1)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (=> (= (tptp.hAPP _let_1 V_A_2) V_B_2) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun T_a tptp.tc_HOL_Obool) _let_1) (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2)) (tptp.hAPP (tptp.c_Set_OPow T_a) V_B_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Complete__Lattice_OSup__class_OSup (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2)) V_A_2)) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (not (exists ((B_f $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) B_f) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (not (= (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_3 T_b) _let_1) V_B_2))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a_2) V_b_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a_2) V_b_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) _let_1))))))) (forall ((V_a $$unsorted)) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_5 (= V_c _let_4))) (=> (tptp.class_Divides_Osemiring__div T_a) (and (=> _let_5 (= _let_3 _let_4)) (=> (not _let_5) (= _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) V_b) V_a)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a)) V_b) V_a)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.hAPP (tptp.c_Set_OPow T_a) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_a) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_a) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_b $$unsorted) (V_a_H $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_a_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_b $$unsorted) (V_a_H $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_a_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b) V_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) _let_1))))) (forall ((V_b $$unsorted) (V_b_H $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_a) _let_3)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_3) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_H) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b_H)) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) _let_1))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b_2) _let_1)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a_2) V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) _let_1))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) _let_1))))))) (forall ((V_b $$unsorted) (V_b_H $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_3) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_H) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_b_H)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_a_2)) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) V_a_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_b_2))))))))) (forall ((V_i_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_k_2)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_i_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_k_2) V_i_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_b_2)) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a_2) V_b_2))) (tptp.hBOOL (tptp.hAPP _let_2 V_a_2))))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_i_2) V_k_2) _let_2) (or (= V_k_2 _let_2) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_k_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_i_2) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_i_2))))))))) (forall ((V_k $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_x) V_k)) V_x)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_c)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq _let_1)) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_a) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_times_2 $$unsorted) (V_one_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower_Opower T_a V_one_2 V_times_2) V_a_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_one_2)) (forall ((V_n_2 $$unsorted) (V_a_2 $$unsorted) (V_times_2 $$unsorted) (V_one_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower_Opower T_a V_one_2 V_times_2) V_a_2))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.hAPP V_times_2 V_a_2) (tptp.hAPP _let_1 V_n_2))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2))) (let ((_let_4 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_4 tptp.tc_HOL_Obool)) _let_2) (tptp.hAPP (tptp.c_Set_Oimage _let_4 _let_4 _let_3) _let_2)))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_OPow T_a) (tptp.c_Orderings_Otop__class_Otop _let_1)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c))) V_b) (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c) V_b))) V_b) (tptp.hAPP (tptp.hAPP _let_2 V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_b) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.c_Groups_Oone__class_Oone T_a)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) V_b) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.c_Groups_Oone__class_Oone T_a)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_OPow T_a))) (let ((_let_3 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) _let_2) V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_a V_f_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 V_A_2)) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_fun T_b tptp.tc_HOL_Obool) _let_2 _let_1) (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2))) (tptp.hAPP (tptp.c_Set_OPow T_a) V_B_2))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_COMBB _let_3 tptp.tc_HOL_Obool _let_2))) (let ((_let_5 (tptp.c_Groups_Ouminus__class_Ouminus _let_2))) (let ((_let_6 (tptp.tc_fun _let_2 _let_2))) (let ((_let_7 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_2 _let_7))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_5 V_A_2)) (tptp.hAPP (tptp.c_Set_OCollect _let_2) (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_HOL_OEx _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_3 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_3 _let_3) _let_2) (tptp.c_COMBS _let_2 tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_8 _let_2) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_7 _let_2) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_6 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_6 _let_3) _let_2) (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool _let_2)) tptp.c_fequal)) _let_5)))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.c_member _let_2) V_A_2)) _let_1)))))))))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.c_Set_OPow T_a) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) _let_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_OPow T_a))) (let ((_let_3 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_OPowp T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_x_2) (tptp.hAPP (tptp.c_Set_OPow T_a) V_A_2)))))) (forall ((V_r_1 $$unsorted) (V_y $$unsorted) (V_b_1 $$unsorted) (V_a_1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_b_1)))) (=> (= V_a_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b_1) V_y)) V_r_1)) (=> (and (=> _let_4 (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r_1) V_b_1)))) (=> (not _let_4) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_1) V_r_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r_1) _let_1))))) (=> (not (= V_b_1 _let_1)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a_1) V_b_1) V_y))))))))) (forall ((V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_n_2) V_k_2))) (and (=> (= V_k_2 _let_1) (tptp.hBOOL (tptp.hAPP V_P_2 _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_k_2)) (forall ((B_i $$unsorted)) (=> (exists ((B_j $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_j) V_k_2)) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_i))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k_2) _let_1)) (forall ((B_i $$unsorted)) (=> (exists ((B_j $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) B_j) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_i)))))))))) (forall ((V_k $$unsorted) (V_l $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_k))) (let ((_let_3 (tptp.c_SMT_Oz3div V_k V_l))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_l)))) (and (=> _let_4 (= _let_3 (tptp.hAPP _let_2 V_l))) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_l))))))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 V_N_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2))))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 V_N_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2))))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) V_m))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c))))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_m)) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n)))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_5 (= V_k _let_4))) (and (=> _let_5 (= _let_3 _let_4)) (=> (not _let_5) (= _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_a) V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_x) V_y))))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 V_m)))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) V_m)) V_n) V_m))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) V_n) V_m))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) V_m)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m) V_n)))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_5) V_n)) (and (=> _let_4 (= _let_2 _let_5)) (=> (not _let_4) (= _let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_n)))))))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_n)))))))) (forall ((V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (= V_k_2 _let_1))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_n_2) V_k_2))) (and (=> _let_2 (tptp.hBOOL (tptp.hAPP V_P_2 _let_1))) (=> (not _let_2) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) V_k_2)) (=> (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2) B_i)) B_j)) (tptp.hBOOL (tptp.hAPP V_P_2 B_i))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_n))))))) (forall ((V_m_2 $$unsorted) (V_q_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n_2)) (= (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_q_2)) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m_2) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_q_2))))) (= V_q_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m_2) V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m_2) V_n_2))) (or (and (= V_n_2 _let_1) (tptp.hBOOL (tptp.hAPP V_P_2 _let_1))) (exists ((B_q $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 B_q)) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc B_q)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_q))))))))) _let_192 (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.c_Set_Oimage T_a T_a V_h_2) V_N_2)))))))) (forall ((V_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (exists ((B_n $$unsorted)) (= V_x (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Nat_OSuc B_n))))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (= (tptp.hAPP V_g_2 B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_a V_h_2) V_f_2) B_x)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_h_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_c) V_g_2) V_A_2))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP tptp.c_Nitpick_Oint__gcd V_x) V_y) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.c_Nitpick_Onat__gcd (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_x)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_y))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (V_T_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_b)))) (= (tptp.hAPP _let_1 V_S_2) (tptp.hAPP _let_1 V_T_2)))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (V_T_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_b)))) (= (tptp.hAPP _let_1 V_T_2) (tptp.hAPP _let_1 V_S_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (=> (forall ((B_z $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) B_z)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_z) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) B_z) V_x)) V_y)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Int_Oint T_a))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_a) (tptp.class_Groups_Ocomm__monoid__mult T_a)) (=> (= V_A_2 V_B_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_B_2)))))))) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_k_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_n_2) V_k_2)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_n_2 V_k_2))) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_j) V_k_2)) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_i) B_j))))))) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_n_2) V_k_2)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_n_2 V_k_2))) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) B_j) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_i) B_j))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.c_Groups_Osgn__class_Osgn T_a V_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Groups_Oone__class_Oone T_a)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))))) (forall ((V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x) V_m)))) (forall ((V_a_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_b)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a_H) V_b)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c)) V_c)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c)) V_c) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_b) V_c)))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a_H $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_c)) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_b_H V_c)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b_H) V_c))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_b) V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_c))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c))))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a_H $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_c)) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_b_H V_c)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b_H) V_c))))))) (forall ((V_b_H $$unsorted) (V_b $$unsorted) (V_a_H $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_c)) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_b_H V_c)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b_H) V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c)) V_b) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_b V_c)) V_c))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a _let_1 V_b) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Osgn__class_Osgn T_a V_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Osgn__class_Osgn T_a _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a _let_1 V_a) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a V_a (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_1 V_b) _let_1))) (forall ((V_a $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_y $$unsorted) (V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_y) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) V_m)))) (forall ((V_m $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_y V_m)) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_y) V_m)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x) (=> (tptp.c_Nat__Transfer_Ois__nat V_y) (tptp.c_Nat__Transfer_Ois__nat (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_y))))) (forall ((V_y $$unsorted) (V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_y) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) V_m)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Osgn__if T_a) (= (tptp.c_Groups_Osgn__class_Osgn T_a _let_1) _let_1)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (= (tptp.c_Groups_Osgn__class_Osgn T_a V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Osgn__class_Osgn T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Osgn__class_Osgn T_a V_a)) (tptp.c_Groups_Osgn__class_Osgn T_a V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c)) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c) V_b)) V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a V_a (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_m)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_k)) V_m))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_y))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)) V_b))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_d_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m_2 V_d_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (exists ((B_q $$unsorted)) (= V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_d_2) B_q))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_l)) _let_1)) (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l) _let_1))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_k) V_l) _let_1)) (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l) _let_1))))) (forall ((V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (and (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) _let_1) _let_1) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a _let_1) V_a)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)) V_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_c)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c))) V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b))) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_a))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b))) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b)) V_b)) V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b) V_a)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) V_a)))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP _let_2 _let_3)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) _let_3)) (tptp.hBOOL (tptp.hAPP _let_2 _let_1)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)) _let_1))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a) V_b)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) V_a)))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_b)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_3) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Osgn__class_Osgn T_a V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Osgn__class_Osgn T_a V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_a) V_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (= _let_1 _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_b) _let_1))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_b)))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (= _let_1 _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) _let_1) V_b))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) V_c) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c))) V_c)))))) (forall ((V_k $$unsorted) (V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_k)))) (forall ((V_k $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_k)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_m) V_n)) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n)))) (forall ((V_k $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_k) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_k) (tptp.c_Groups_Osgn__class_Osgn T_a V_k))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Osgn__class_Osgn T_a V_x)) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_x)) V_x))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_a) V_b))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_2) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) _let_2)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (= (tptp.c_Groups_Osgn__class_Osgn T_a V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (= (tptp.c_Groups_Osgn__class_Osgn T_a V_a_2) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a_2))))) (forall ((V_y $$unsorted) (V_q_1 $$unsorted) (V_b_1 $$unsorted) (V_a_1 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_b_1)))) (=> (= V_a_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b_1) V_q_1)) V_y)) (=> (and (=> _let_4 (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_y) V_b_1)))) (=> (not _let_4) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_1) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1))))) (=> (not (= V_b_1 _let_1)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a_1 V_b_1) V_y))))))))) (forall ((V_q $$unsorted) (V_r $$unsorted) (V_b $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b) V_r)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_q V_c))) V_r)) _let_1))))))))) (forall ((V_q $$unsorted) (V_r $$unsorted) (V_b $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_r)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_r) _let_3)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_q V_c))) V_r)))))))))) (forall ((V_q $$unsorted) (V_b $$unsorted) (V_r $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_3) V_r)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_q V_c))) V_r)) (tptp.hAPP _let_1 V_c)))))))))) (forall ((V_q $$unsorted) (V_b $$unsorted) (V_r $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_c)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_r)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r) V_b)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_q V_c))) V_r)))))))))) (forall ((V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_n_2 V_k_2))) (and (=> (= V_k_2 _let_1) (tptp.hBOOL (tptp.hAPP V_P_2 V_n_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_k_2)) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_j) V_k_2)) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP V_P_2 B_j))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k_2) _let_1)) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) B_j) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP V_P_2 B_j)))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_c)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_6 (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) _let_5))) (=> (not (= V_b _let_5)) (and (=> _let_6 (= _let_4 _let_3)) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) _let_3) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) V_b))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_6 (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) _let_5))) (=> (not (= V_b _let_5)) (and (=> _let_6 (= _let_4 _let_3)) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) _let_3) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))))))) (forall ((V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Osgn__class_Osgn tptp.tc_Int_Oint V_i))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_3) V_i)))) (let ((_let_5 (= V_i _let_3))) (and (=> _let_5 (= _let_2 _let_3)) (=> (not _let_5) (and (=> _let_4 (= _let_2 _let_1)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) _let_1)))))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Osgn__class_Osgn T_a V_x))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_3) V_x)))) (let ((_let_5 (= V_x _let_3))) (=> (tptp.class_Groups_Osgn__if T_a) (and (=> _let_5 (= _let_2 _let_3)) (=> (not _let_5) (and (=> _let_4 (= _let_2 _let_1)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1))))))))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (= (tptp.c_Groups_Osgn__class_Osgn T_a V_a_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_k $$unsorted) (V_l $$unsorted)) (let ((_let_1 (tptp.c_SMT_Oz3mod V_k V_l))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_l)))) (and (=> _let_2 (= _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l))) (=> (not _let_2) (= _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_l)))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a tptp.tc_Nat_Onat) V_f_2) V_S_2)))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Equiv__Relations_Oquotient T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_a _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_a) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_2)))) V_r_2) V_A_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_r_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2))))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_S_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_S_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_b)))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_a (tptp.c_Groups_Otimes__class_Otimes T_b)) V_f_2) _let_1) V_S_2) _let_1)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (= (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP _let_1 V_m) (tptp.hAPP _let_1 V_n))))) (forall ((V_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_k) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_m))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) V_m))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_n))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2)) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n))) (let ((_let_3 (= _let_1 V_n))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc _let_1)) _let_1))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_n))))) (forall ((V_d_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m_2 V_d_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (exists ((B_q $$unsorted)) (= V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_d_2) B_q))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)))) (and (=> _let_2 (= _let_1 V_m)) (=> (not _let_2) (= _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) V_n))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_k $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k) V_n)) V_m) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n) V_n)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_x V_y))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a V_c)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_b V_c))) V_c)))))) (forall ((V_x $$unsorted) (V_y $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Onat__gcd V_x V_y))) (let ((_let_2 (= V_y (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_2 (= _let_1 V_x)) (=> (not _let_2) (= _let_1 (tptp.c_Nitpick_Onat__gcd V_y (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_x V_y)))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_k $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k) V_n)) V_m)) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_b))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_a) V_b) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a V_b))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_a))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) V_c) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_b V_c))) V_c)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_a))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_3 V_a_2) _let_2))) (let ((_let_5 (tptp.c_Equiv__Relations_Oquotient T_a))) (let ((_let_6 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_7 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_7 _let_6) T_a) _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) _let_3) _let_2))) V_r_2) V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) _let_4)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_6) (tptp.hAPP (tptp.hAPP _let_5 V_A_2) V_r_2)) (tptp.hAPP (tptp.hAPP _let_5 _let_4) V_r_2))))))))))))) (forall ((V_q $$unsorted) (V_b $$unsorted) (V_r $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_q V_c))) V_r)) (tptp.hAPP _let_1 V_c)))))))) (forall ((V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_n_2 V_k_2))) (and (=> _let_1 (tptp.hBOOL (tptp.hAPP V_P_2 V_n_2))) (=> (not _let_1) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) V_k_2)) (=> (= V_n_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2) B_i)) B_j)) (tptp.hBOOL (tptp.hAPP V_P_2 B_j)))))))))) (forall ((V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_k)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k) V_m)) V_k) _let_1)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_x) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_y))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_y)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Int_Onat V_x) (tptp.hAPP tptp.c_Int_Onat V_y))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_Set_Oinsert _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.c_Relation_OImage T_a T_a V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) _let_2) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r_2) _let_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2))) (= (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) _let_1)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2) _let_1))))) (forall ((V_B_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Relation_OImage T_b T_a V_r_2))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_2 V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_3) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_b T_a V_r_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_b T_a V_R_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_b T_a V_r_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_B_2))))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (V_A_H_2 $$unsorted) (V_r_2 $$unsorted) (V_r_H_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_r_H_2) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_H_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Relation_OImage T_a T_b V_r_H_2) V_A_H_2)) (tptp.hAPP (tptp.c_Relation_OImage T_a T_b V_r_2) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_b T_a V_R_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_A_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool)) V_R_2) V_S_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_R_2) V_A_2)) (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_S_2) V_A_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (V_X_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_X_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))) (not (forall ((B_x $$unsorted)) (=> (= V_X_2 (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)))))))) (forall ((V_y $$unsorted) (V_n $$unsorted) (V_x $$unsorted)) (=> (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_x V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_y V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_y) V_x)) (exists ((B_q $$unsorted)) (= V_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) B_q))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) B_x) V_b_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_c_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b_2) B_x)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_c_2) V_A_2)))))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))) (forall ((V_i_2 $$unsorted) (V_P_2 $$unsorted) (V_p_2 $$unsorted)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_p_2)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_i)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc B_i) V_p_2)))))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_i_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) V_p_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_X $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) B_X) V_C_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_A_2) B_X)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_A_2) (tptp.c_Complete__Lattice_OInf__class_OInf (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_C_2))))) (forall ((V_z_2 $$unsorted) (V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold__image T_b T_a (tptp.c_Groups_Otimes__class_Otimes T_b)))) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (= (tptp.hAPP V_g_2 B_x) (tptp.hAPP V_h_2 B_x)))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_z_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 V_h_2) V_z_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_y) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) B_y) V_x_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a V_A_2) V_x_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_y) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) B_y)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a V_A_2) V_x_2)))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b_2) B_x)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_c_2) V_A_2)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) B_x) V_b_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_a _let_1) V_c_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)))))))) (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) _let_1)) (= _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Int_Oring__char__0 T_a)) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_x_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (=> (tptp.class_Int_Onumber T_a) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v)))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))))) (=> (tptp.class_Int_Onumber__ring T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_v)))))))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_v)))) (forall ((V_k $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_k) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_k)))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v)))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))))) (=> (tptp.class_Int_Onumber__ring T_a) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_v))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)))))))) (forall ((V_w_2 $$unsorted) (V_v_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_v_2))) (let ((_let_3 (tptp.hAPP _let_1 V_w_2))) (=> (and (tptp.class_Int_Onumber T_a) (tptp.class_Orderings_Olinorder T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) _let_2) _let_3)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_3) _let_2))))))))) (forall ((V_v $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_v))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Osemiring T_b)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_a) V_b)) _let_1) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_v $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_v)))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Osemiring T_b)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_c))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_v $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_v)))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Oring T_b)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_c))))))) (forall ((V_v $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_v))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Rings_Oring T_b)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_a) V_b)) _let_1) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_x_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_x_2) V_y_2)))))) (forall ((V_v $$unsorted) (V_u $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (let ((_let_3 (tptp.hAPP _let_1 V_u))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) _let_3) _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) _let_3) _let_2)))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Orderings_Oord T_b)) (and (=> _let_5 (= _let_4 _let_3)) (=> (not _let_5) (= _let_4 _let_2)))))))))) (forall ((V_v $$unsorted) (V_u $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_u))) (let ((_let_3 (tptp.hAPP _let_1 V_v))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) _let_2) _let_3))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) _let_2) _let_3)))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Orderings_Oord T_b)) (and (=> _let_5 (= _let_4 _let_3)) (=> (not _let_5) (= _let_4 _let_2)))))))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w)) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) V_w))) V_z)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) V_w)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w)) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w))) V_z)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_v) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP _let_1 V_w)))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))))) (forall ((V_w_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2)) (tptp.c_Int_Oring__1__class_OInts T_a))))) (forall ((V_w_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_3 (= _let_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) _let_2)) (and (=> (not _let_3) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) _let_2) V_b_2)) (=> _let_3 (= V_a_2 _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_3 (= V_c_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) (and (=> (not _let_3) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_2) V_c_2) V_b_2)) (=> _let_3 (= _let_2 _let_1))))))))) (forall ((V_w_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_3 (= V_c_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2) _let_2) (and (=> (not _let_3) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_2) V_c_2))) (=> _let_3 (= _let_2 _let_1))))))))) (forall ((V_a_2 $$unsorted) (V_w_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_3 (= _let_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) _let_2) V_a_2) (and (=> (not _let_3) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) _let_2))) (=> _let_3 (= V_a_2 _let_1))))))))) (forall ((V_c $$unsorted) (V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w)) V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) V_w))) V_c)))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP _let_1 V_w))) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w))) V_z)))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w))))))) (forall ((V_w_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_4) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) _let_4))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_4)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) _let_4))) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_6) V_b_2))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_6))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_3 _let_1)))))))))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_4 (tptp.hAPP _let_3 _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_2) V_c_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_c_2)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_6) V_b_2))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_6))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_4 _let_1)))))))))))))))) (forall ((V_w_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_4 (tptp.hAPP _let_3 _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) _let_2)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_c_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP _let_4 V_c_2)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) _let_1)) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_6))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_6) V_b_2))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_4 _let_1)))))))))))))))) (forall ((V_a_2 $$unsorted) (V_w_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (let ((_let_4 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_4) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) _let_4))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP _let_3 _let_4)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) _let_4)) V_a_2)) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_6))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_6) V_b_2))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)))))))))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_x))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_w $$unsorted) (V_c $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_c) (tptp.hAPP _let_1 V_w))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))) V_c)))))) (forall ((V_a_2 $$unsorted) (V_w_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_3) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) _let_3))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_1) _let_3)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) _let_3)) V_a_2)) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_6))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_6) V_b_2))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a_2)))))))))))))))) (forall ((V_w_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_c_2) _let_2)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_1) V_c_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_2) V_c_2)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2)) _let_1)) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_6))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_6) V_b_2))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)))))))))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_4 (tptp.hAPP _let_3 _let_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 V_c_2) _let_1)))) (let ((_let_7 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_2) V_c_2))) (let ((_let_8 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_1) V_c_2)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) V_c_2))) (and (=> _let_8 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_7) V_b_2))) (=> (not _let_8) (and (=> _let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_7))) (=> (not _let_6) (tptp.hBOOL (tptp.hAPP _let_4 _let_1))))))))))))))))) (forall ((V_w_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_4) _let_1)))) (let ((_let_7 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) _let_4))) (let ((_let_8 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_1) _let_4)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber T_a)) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_b_2) _let_4))) (and (=> _let_8 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_7) V_b_2))) (=> (not _let_8) (and (=> _let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_7))) (=> (not _let_6) (tptp.hBOOL (tptp.hAPP _let_3 _let_1))))))))))))))))) (forall ((V_v_H $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v_H))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (let ((_let_3 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat _let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_H)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_1)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP tptp.c_Int_Onat (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint) _let_1)))))))))) (forall ((V_v_H $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_H)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v_H))))) (forall ((V_v_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_v_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_2))) (let ((_let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w_2))))) (let ((_let_6 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_a T_a) (tptp.hAPP _let_3 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_a) _let_6) (tptp.hAPP _let_6 _let_2))) (tptp.c_COMBI T_a))))) (=> (not _let_5) (= _let_4 (tptp.c_Groups_Oone__class_Oone T_a)))))))))))) (forall ((V_n $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v)) V_n))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_1)))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (=> (not _let_4) (= _let_3 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) _let_1) V_n)))))))))) (forall ((V_w $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_w)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w)))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_v)) (tptp.hAPP _let_2 V_w)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_v) V_w)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_k_2)) (tptp.hAPP _let_2 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2)))))) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_k) V_k)) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2)))) (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP _let_1 V_w_2)))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_v)) (tptp.hAPP _let_2 V_w)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_v) V_w)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_k_2)) (tptp.hAPP _let_2 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2)) (tptp.c_Int_OBit1 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2))))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k1_2)) (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit1 V_k_2) (tptp.c_Int_OBit1 V_l_2)) (= V_k_2 V_l_2))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k1_2)) (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2)) (tptp.c_Int_OBit1 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2))))) (forall ((V_w $$unsorted)) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int tptp.tc_Int_Oint) V_w))) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Int_OBit1 V_w_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_w_2) _let_1)))))) _let_191 (forall ((V_w $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_w))) (forall ((V_v $$unsorted)) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v)))) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w_2))))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_w_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.c_COMBI tptp.tc_Nat_Onat))))))))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Oone__class_Oone T_a)) _let_2)) _let_2))))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_z_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2))))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w_2))))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_a T_a) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_a) _let_5) (tptp.hAPP _let_5 V_z_2))) (tptp.c_COMBI T_a))))) (=> (not _let_4) (= _let_3 (tptp.c_Groups_Oone__class_Oone T_a))))))))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_z_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2))))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w_2))))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint) _let_5) (tptp.hAPP _let_5 V_z_2))) (tptp.c_COMBI tptp.tc_Int_Oint))))) (=> (not _let_4) (= _let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))))) (forall ((V_v $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_v_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v_2))) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2) _let_1) (and (= V_m_2 (tptp.hAPP tptp.c_Int_Onat _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_1)))))) (forall ((V_v_H_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_v_H_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_2)))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_5 (tptp.hAPP _let_1 V_v_2))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_5)))) (let ((_let_7 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (= (= (tptp.hAPP _let_7 V_v_2) (tptp.hAPP _let_7 V_v_H_2)) (and (=> _let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_2) _let_4))) (=> (not _let_6) (and (=> _let_3 (= _let_5 _let_4)) (=> (not _let_3) (= V_v_2 V_v_H_2)))))))))))))) (forall ((V_k $$unsorted) (V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v_H)) V_k))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)))) (let ((_let_5 (tptp.hAPP _let_4 _let_3))) (let ((_let_6 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP _let_6 V_v_H))))) (let ((_let_8 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP _let_6 V_v))))) (and (=> _let_8 (= _let_5 _let_3)) (=> (not _let_8) (and (=> _let_7 (= _let_5 (tptp.hAPP _let_4 V_k))) (=> (not _let_7) (= _let_5 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) V_v_H))) V_k))))))))))))))) (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))) (let ((_let_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_1)))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_v_H))) (let ((_let_3 (tptp.hAPP _let_1 V_v))) (let ((_let_4 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.hAPP _let_4 V_v))) (let ((_let_6 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat _let_5 (tptp.hAPP _let_4 V_v_H)))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_2)))) (let ((_let_8 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_3)))) (and (=> _let_8 (= _let_6 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_8) (and (=> _let_7 (= _let_6 _let_5)) (=> (not _let_7) (= _let_6 (tptp.hAPP tptp.c_Int_Onat (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_3 _let_2)))))))))))))))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (let ((_let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP _let_3 V_v)) (tptp.hAPP _let_3 V_v_H)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_2)))) (and (=> _let_5 (= _let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_5) (= _let_4 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_2) (tptp.hAPP _let_1 V_v_H)))))))))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_1 V_v)) _let_2))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_3)))) (and (=> _let_6 (= _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) _let_2))) (=> (not _let_6) (= _let_5 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) _let_3) _let_2)))))))))))) (forall ((V_n $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) V_n)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_v))) V_n))))))))) (forall ((V_v_2 $$unsorted) (V_v_H_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP _let_2 V_v_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_3) (tptp.hAPP _let_2 V_v_H_2)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v_H_2) tptp.c_Int_OPls)))) (and (=> _let_5 (= _let_4 _let_3)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_v_H_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_1) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) tptp.c_Int_Onat))))))))))) (forall ((V_n_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_2) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Int_Oint) tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_2 tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_n_2)))))))) (= (tptp.c_Int_Osucc tptp.c_Int_OPls) _let_164) (forall ((V_l $$unsorted)) (not (= tptp.c_Int_OPls (tptp.c_Int_OBit1 V_l)))) (forall ((V_k $$unsorted)) (not (= (tptp.c_Int_OBit1 V_k) tptp.c_Int_OPls))) (= (tptp.hAPP _let_148 tptp.c_Int_OPls) tptp.c_Int_OPls) _let_190 (forall ((V_x $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_Opred V_x)) V_x)) (forall ((V_w $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) tptp.c_Int_OPls) V_w) tptp.c_Int_OPls)) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) tptp.c_Int_OPls) V_k) V_k)) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k) tptp.c_Int_OPls) V_k)) (not (tptp.hBOOL (tptp.hAPP _let_169 tptp.c_Int_OPls))) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k) tptp.c_Int_OPls) V_k)) (tptp.hBOOL (tptp.hAPP _let_173 tptp.c_Int_OPls)) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OPls) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OPls)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OPls)) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OPls)) V_a) V_a))) (not (tptp.hBOOL (tptp.hAPP _let_169 _let_126))) (= _let_31 _let_189) (= _let_189 _let_31) (= _let_126 _let_168) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OPls))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2)) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) tptp.c_Int_OPls))))) (forall ((V_m $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OPls) V_v)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP _let_2 V_m) (tptp.hAPP _let_1 (tptp.c_Int_Opred V_v)))))))) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_168))) (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) _let_2) (tptp.hAPP _let_1 V_v)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_v)))))))))) (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_v)) _let_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_v)))))))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_k_2) V_l_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Int_Opred V_k_2)) V_l_2)))) (forall ((V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit1 (tptp.c_Int_Opred (tptp.hAPP _let_1 V_k)))))) _let_188 (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_v)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.c_Int_Opred V_v))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls))) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls))) V_a))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_a) (tptp.class_Int_Onumber__ring T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OPls)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_v_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_v_2) tptp.c_Int_OPls)))) (forall ((V_v_2 $$unsorted)) (= (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_v_2) tptp.c_Int_OPls)))) (tptp.hBOOL (tptp.hAPP _let_181 _let_168)) (forall ((V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OPls) (tptp.c_Int_OBit1 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OPls) V_k_2)))) (forall ((V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Int_OBit1 V_k_2)) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) tptp.c_Int_OPls)))) (forall ((V_v_H_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_v_2) V_v_H_2)))) (let ((_let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_3 V_v_2)) (tptp.hAPP _let_3 V_v_H_2))) (and (=> _let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 tptp.c_Int_OPls) V_v_H_2))) _let_2)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Fields_Ofield T_a) (tptp.class_Int_Onumber__ring T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls))) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Fields_Ofield T_a) (tptp.class_Int_Onumber__ring T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls))) V_a))) (forall ((V_v_H_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_v_2))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_2 V_v_2)) (tptp.hAPP _let_2 V_v_H_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_v_H_2))) (tptp.hBOOL (tptp.hAPP _let_1 tptp.c_Int_OPls))))))) (= _let_127 (tptp.hAPP _let_165 _let_164)) (= _let_186 _let_62) (= _let_62 _let_186) (tptp.c_Nat__Transfer_Ois__nat _let_184) _let_187 (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_x_2)) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_x_2) tptp.c_Int_OPls))))) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OPls) V_y_2))))) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OPls) V_y_2))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_x_2)) (tptp.c_Groups_Ozero__class_Ozero T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_x_2) tptp.c_Int_OPls))))) (forall ((V_v_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OPls) V_v_2)))) (= _let_186 _let_174) (= _let_185 (tptp.hAPP tptp.c_Nat_OSuc _let_175)) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_a)) V_a))))) (tptp.hBOOL (tptp.hAPP _let_181 _let_184)) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_n))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_v_H))) (let ((_let_3 (tptp.hAPP _let_1 V_v))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) _let_3) _let_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 V_v_H) tptp.c_Int_OPls)))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 V_v) tptp.c_Int_OPls)))) (and (=> _let_7 (= _let_4 _let_2)) (=> (not _let_7) (and (=> _let_6 (= _let_4 _let_3)) (=> (not _let_6) (= _let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) V_v_H))))))))))))))) (= _let_184 (tptp.hAPP _let_29 _let_185)) (= _let_185 (tptp.hAPP tptp.c_Int_Onat _let_184)) (forall ((V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_k)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 V_k)))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.c_Int_Opred V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hAPP _let_1 V_w)))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_x_2)) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_x_2) (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) V_y_2))))) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) V_y_2))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_x_2)) (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_x_2) (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_v)) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) V_w)))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_1 tptp.c_Int_OPls)) V_n)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP (tptp.hAPP _let_2 V_m) (tptp.hAPP (tptp.hAPP _let_2 V_n) (tptp.hAPP _let_1 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))))) (forall ((V_k $$unsorted) (V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v_H)) V_k)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_4 (= _let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_v_H))) V_k))))))))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_v_H)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_v_H))))))))) (forall ((V_v_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2)))) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_m))) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_m) V_n))) (forall ((V_v $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_m))) _let_2) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_m) _let_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_m)))) V_n) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_m)) V_n)))) (forall ((V_v $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (let ((_let_3 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_m)))) _let_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_m)) _let_2)))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w))))))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_v)) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))))) (forall ((V_n_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (let ((_let_2 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 tptp.c_fconj) _let_1)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 tptp.c_fimplies) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fequal) tptp.c_Int_Onat)) V_n_2)))))))))) (forall ((V_v_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (let ((_let_2 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (= (tptp.hAPP tptp.c_Nat_OSuc V_n_2) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 tptp.c_fconj) _let_1)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 tptp.c_fimplies) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fequal) tptp.c_Int_Onat)) V_n_2)))))))))) (forall ((V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_a))) (= (tptp.c_Nat_Onat_Onat__case T_a V_a_2 V_f_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a _let_1) tptp.tc_Int_Oint) (tptp.c_If T_a)) tptp.c_Nat__Numeral_Oneg)) V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Int_Oint) V_f_2) tptp.c_Int_Onat)))))) (forall ((V_v_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (let ((_let_3 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fconj) _let_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fimplies) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_n_2)) tptp.c_Int_Onat))))))))))) (forall ((V_n_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_n_2))))))) (forall ((V_n_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_n_2))))))) (forall ((V_v_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (let ((_let_3 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fconj) _let_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fimplies) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_n_2)) tptp.c_Int_Onat))))))))))) (forall ((V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_v)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_v))))))))) (forall ((V_n_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_2) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Int_Oint) tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_2 tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_n_2))))))))) (forall ((V_v_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 _let_3) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_4) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) _let_3)) (tptp.hAPP (tptp.hAPP _let_2 tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n_2)) tptp.c_Int_Onat)))))))))) (forall ((V_n_2 $$unsorted) (V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun T_a T_a))) (= (tptp.c_Nat_Onat_Onat__case T_a V_a_2 V_f_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_a _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a _let_2) tptp.tc_Int_Oint) (tptp.c_If T_a)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Nat_Onat_Onat__case T_a V_a_2 V_f_2 V_n_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Int_Oint) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_n_2)))))))) (forall ((V_v_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Int_Oint))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_3) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.hAPP _let_2 tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n_2)) tptp.c_Int_Onat))))))))) (forall ((V_n_2 $$unsorted) (V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_n_2))) (let ((_let_3 (tptp.c_Nat_Onat_Onat__rec T_a V_a_2 V_f_2))) (let ((_let_4 (tptp.tc_fun T_a T_a))) (let ((_let_5 (tptp.c_COMBS tptp.tc_Int_Oint T_a T_a))) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_a _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a _let_4) tptp.tc_Int_Oint) (tptp.c_If T_a)) tptp.c_Nat__Numeral_Oneg)) (tptp.hAPP _let_3 V_n_2))) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_4 tptp.tc_Int_Oint) V_f_2) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Int_Oint) _let_3) _let_2))))))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Int_Oiszero T_a (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))))) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Int_Oiszero T_a (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_y_2)))))))) (forall ((V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.c_Int_Oiszero T_a V_z_2) (= V_z_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (tptp.c_Int_Oiszero T_a (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (not (tptp.c_Int_Oiszero T_a (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Onat_Onat__rec T_a V_f1_2 V_f2_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_f1_2)) (forall ((V_nat_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Onat_Onat__rec T_a V_f1_2 V_f2_2))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_nat_2)) (tptp.hAPP (tptp.hAPP V_f2_2 V_nat_2) (tptp.hAPP _let_1 V_nat_2))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (tptp.c_Int_Oiszero T_a (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OPls)))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Int_Oring__char__0 T_a)) (not (tptp.c_Int_Oiszero T_a (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 V_w)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (not (tptp.c_Int_Oiszero T_a (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Int_Oiszero T_a (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_y_2)))))))) (forall ((V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Int_Oiszero T_a (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) tptp.c_Int_OPls) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_y_2)))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Int_Oiszero T_a (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) tptp.c_Int_OPls)))))))) (forall ((V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Onat_Onat__rec T_a V_a_2 V_f_2))) (let ((_let_2 (tptp.tc_fun T_a T_a))) (let ((_let_3 (tptp.c_COMBS tptp.tc_Int_Oint T_a T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_a _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a _let_2) tptp.tc_Int_Oint) (tptp.c_If T_a)) tptp.c_Nat__Numeral_Oneg)) V_a_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_2 tptp.tc_Int_Oint) V_f_2) tptp.c_Int_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_a tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP _let_3 V_b)) (tptp.hAPP _let_3 V_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_3 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1) V_a))) _let_1))))))) (forall ((V_v $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP _let_2 V_w))) (let ((_let_4 (tptp.hAPP _let_2 V_v))) (let ((_let_5 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (let ((_let_7 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_2 (tptp.c_Int_OBit1 V_v)) (tptp.hAPP _let_2 (tptp.c_Int_OBit0 V_w))))) (let ((_let_8 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_3)))) (and (=> _let_8 (= _let_7 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_6 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_4 _let_3))) _let_1))) (=> (not _let_8) (= _let_7 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_6 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_5 _let_4) _let_1) _let_3))) _let_1))))))))))))) (forall ((V_n_2 $$unsorted) (V_d_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_a_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Nat_Onat) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Nat_Onat) _let_2) _let_1)) V_d_2))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_n_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) V_d_2))))))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (not (= (tptp.c_Int_OBit0 V_k) (tptp.c_Int_OBit1 V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (not (= (tptp.c_Int_OBit1 V_k) (tptp.c_Int_OBit0 V_l)))) (forall ((V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit0 (tptp.hAPP _let_1 V_k))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) V_l) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit0 V_k_2) (tptp.c_Int_OBit0 V_l_2)) (= V_k_2 V_l_2))) _let_182 (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2)) (tptp.c_Int_OBit0 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2))))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k1_2)) (tptp.c_Int_OBit0 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k1_2)) (tptp.c_Int_OBit0 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2)) (tptp.c_Int_OBit0 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2))))) (= (tptp.c_Int_OBit0 tptp.c_Int_OPls) tptp.c_Int_OPls) (forall ((V_l_2 $$unsorted)) (= (= tptp.c_Int_OPls (tptp.c_Int_OBit0 V_l_2)) (= tptp.c_Int_OPls V_l_2))) (forall ((V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit0 V_k_2) tptp.c_Int_OPls) (= V_k_2 tptp.c_Int_OPls))) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Int_OBit0 V_w_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_w_2) _let_1)))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OPls))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2)) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) tptp.c_Int_OPls))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2)) (tptp.c_Int_OBit1 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2))))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k1_2)) (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2)) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) tptp.c_Int_OPls))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OPls))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k1_2)) (tptp.c_Int_OBit0 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2)) (tptp.c_Int_OBit0 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit1 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit1 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_l $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) tptp.c_Int_OPls))) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP _let_1 V_l))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit1 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_v))) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2)))) (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP _let_1 V_w_2)))))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Opred (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 V_k))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Opred (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit1 (tptp.c_Int_Opred V_k)))) (forall ((V_w_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Int_Oring__char__0 T_a)) (= (tptp.c_Int_Oiszero T_a (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.c_Int_Oiszero T_a (tptp.hAPP _let_1 V_w_2)))))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit1 V_k))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 (tptp.c_Int_Osucc V_k)))) (forall ((V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_w_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.c_COMBI tptp.tc_Nat_Onat)))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Ozero__class_Ozero T_a)) _let_2)) _let_2))))))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Int_OBit1 V_k1_2)) (tptp.c_Int_OBit0 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k1_2) V_k2_2)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Int_OBit1 V_k_2)) (tptp.c_Int_OBit0 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) V_l_2)))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Int_OBit0 V_k1_2)) (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_k1_2) V_k2_2)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Int_OBit0 V_k_2)) (tptp.c_Int_OBit1 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_k_2) V_l_2)))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_HOL_Obool) (tptp.c_Orderings_Otop__class_Otop _let_162)) _let_180) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) V_l) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))) V_l)))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_Opred V_k)) V_l))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) (tptp.c_Int_Osucc V_l)))))) (forall ((V_w_2 $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_z_2))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_a T_a) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a T_a) (tptp.c_Groups_Otimes__class_Otimes T_a)) (tptp.c_COMBI T_a)))))))) (forall ((V_w_2 $$unsorted) (V_z_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_z_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint)) (tptp.c_COMBI tptp.tc_Int_Oint))))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_2) _let_2)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w))))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n))) (and (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_n)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_n))))))))) (tptp.c_Nat__Transfer_Ois__nat _let_177) (forall ((V_w_2 $$unsorted) (V_v_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_v_2)))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_a T_a) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a T_a) (tptp.c_Groups_Otimes__class_Otimes T_a)) (tptp.c_COMBI T_a)))))))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_z) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_z) V_z)))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_z) V_z)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_1) _let_1) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1)))) (= _let_175 _let_180) (= _let_180 _let_175) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a)))) (tptp.hBOOL (tptp.hAPP _let_181 _let_177)) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)))))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_z) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_z) V_z))) (forall ((V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_z) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_z) V_z))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_1) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_n)) _let_1))))))) (= _let_177 (tptp.hAPP _let_29 _let_180)) (= _let_180 _let_178) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) _let_62) _let_62) _let_180) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_m)) _let_1) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m _let_1)))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_m))) _let_1) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_2 V_m) _let_1)))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_v)) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_v) (tptp.hAPP _let_1 V_w)))))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_m)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_m)) (forall ((V_r $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber__ring T_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_r)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_r) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_a) (tptp.class_Int_Onumber__ring T_a)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_r_2) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_3 V_x) _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_y) _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= V_x V_y))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_x) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y) _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (not (= V_a_2 _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))))) (forall ((V_w $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w))) (=> (and (tptp.class_Groups_Omonoid__mult T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) _let_1) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) _let_1) _let_1))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (or (= V_n _let_1) (= V_n (tptp.hAPP tptp.c_Nat_OSuc _let_1)))))) (= _let_178 _let_175) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_a) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a_2) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1) (or (= V_a_2 _let_1) (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1))))))) (forall ((V_n $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) _let_2) _let_2)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_x) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y) _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2))) _let_1)) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2)))) (or (not (= V_x_2 _let_1)) (not (= V_y_2 _let_1))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1))) (tptp.c_Groups_Ozero__class_Ozero T_a)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) _let_1))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_5 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_5 V_x) V_y)) _let_3) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x) _let_3)) (tptp.hAPP (tptp.hAPP _let_4 V_y) _let_3))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) _let_1)) V_x)) V_y)))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)))))) (forall ((V_n $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) (tptp.hAPP (tptp.hAPP _let_3 V_x) (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_2)))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP _let_3 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_1) V_n))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_n)) _let_1)))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)) _let_1)))) (forall ((V_v $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.hAPP _let_1 V_v))) (let ((_let_4 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_v))) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w))))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_2)))) (and (=> _let_6 (= _let_5 (tptp.hAPP (tptp.hAPP _let_4 _let_3) _let_2))) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_3) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) _let_2))))))))))) (forall ((V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) _let_1)) (= _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) _let_1))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_5 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_5 V_x) V_y)) _let_3) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_4 V_x) _let_3)) (tptp.hAPP (tptp.hAPP _let_4 V_y) _let_3))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) _let_1)) V_x)) V_y)))))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))))) (tptp.hBOOL (tptp.hAPP _let_1 V_a)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)))) _let_1))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) _let_1)))) (forall ((V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat V_x)))) (=> (tptp.class_Rings_Osemiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x)) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_x))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_2) _let_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_a)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_a)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP _let_4 V_b))) (tptp.hAPP _let_4 V_a)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_b) _let_1)) V_a)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_a)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) (tptp.hAPP _let_2 V_b))) (tptp.hAPP _let_2 V_a)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)))))) (forall ((V_k $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral) _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) _let_1)) (tptp.hAPP tptp.c_Code__Numeral_Oint__of (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Code__Numeral_Ocode__numeral) V_k) _let_2))))) (let ((_let_4 (tptp.hAPP tptp.c_Code__Numeral_Oint__of V_k))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_6 (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Code__Numeral_Ocode__numeral V_k _let_2) _let_5))) (let ((_let_7 (= V_k _let_5))) (and (=> _let_7 (= _let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not _let_7) (and (=> _let_6 (= _let_4 _let_3)) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_3) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))))))))) (forall ((V_k_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_COMBS T_a T_a T_a))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.c_COMBI T_a)))) (let ((_let_4 (tptp.tc_fun T_a T_a))) (let ((_let_5 (tptp.c_COMBB T_a _let_4 T_a))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_7 (tptp.tc_fun T_a _let_4))) (let ((_let_8 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (let ((_let_9 (tptp.tc_fun _let_4 T_a))) (let ((_let_10 (tptp.tc_fun tptp.tc_Int_Oint T_a))) (let ((_let_11 (tptp.tc_fun tptp.tc_Int_Oint _let_4))) (let ((_let_12 (tptp.hAPP _let_8 V_k_2))) (let ((_let_13 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) _let_6)))) (let ((_let_14 (= V_k_2 _let_6))) (=> (tptp.class_Rings_Oring__1 T_a) (and (=> _let_14 (= _let_12 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_14) (and (=> _let_13 (= _let_12 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP _let_8 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_k_2))))) (=> (not _let_13) (= _let_12 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_11 _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_11 _let_10) tptp.tc_Int_Oint) (tptp.c_COMBB _let_4 T_a tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a _let_9 tptp.tc_Int_Oint) (tptp.c_HOL_OLet T_a T_a)) _let_8))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_4 _let_4) tptp.tc_Int_Oint) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_4 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_4 _let_7) tptp.tc_Int_Oint) _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_7 tptp.tc_Int_Oint) (tptp.c_If T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) tptp.c_fequal) _let_6)))) _let_3))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a T_a) (tptp.hAPP (tptp.hAPP _let_5 _let_1) _let_3)) (tptp.c_Groups_Oone__class_Oone T_a))))) (tptp.c_Divides_Odivmod__int V_k_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))))))))))))))))))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) _let_1)))) (forall ((V_k $$unsorted)) (not (= (tptp.c_Int_OBit0 V_k) tptp.c_Int_OMin))) (forall ((V_l $$unsorted)) (not (= tptp.c_Int_OMin (tptp.c_Int_OBit0 V_l)))) (not (= tptp.c_Int_OPls tptp.c_Int_OMin)) (not (= tptp.c_Int_OMin tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP _let_172 tptp.c_Int_OMin)) (= (tptp.hAPP (tptp.hAPP _let_54 _let_64) _let_63) _let_64) (not (tptp.hBOOL (tptp.hAPP _let_170 tptp.c_Int_OMin))) (forall ((V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit1 V_k_2) tptp.c_Int_OMin) (= V_k_2 tptp.c_Int_OMin))) (forall ((V_l_2 $$unsorted)) (= (= tptp.c_Int_OMin (tptp.c_Int_OBit1 V_l_2)) (= tptp.c_Int_OMin V_l_2))) (= (tptp.c_Int_OBit1 tptp.c_Int_OMin) tptp.c_Int_OMin) (tptp.hBOOL (tptp.hAPP _let_170 _let_126)) (not (tptp.hBOOL (tptp.hAPP _let_173 tptp.c_Int_OMin))) (tptp.hBOOL (tptp.hAPP _let_172 tptp.c_Int_OPls)) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2)) tptp.c_Int_OMin)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) tptp.c_Int_OMin))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (tptp.hBOOL (tptp.hAPP _let_170 tptp.c_Int_OPls)) (not (tptp.hBOOL (tptp.hAPP _let_169 tptp.c_Int_OMin))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2)) tptp.c_Int_OMin)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) tptp.c_Int_OMin))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2)) tptp.c_Int_OMin)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) tptp.c_Int_OMin))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (not (= _let_168 _let_166)) _let_167 (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) tptp.c_Int_OMin) V_k) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_k))) (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_166)) (= (tptp.c_Int_Opred tptp.c_Int_OPls) tptp.c_Int_OMin) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k) tptp.c_Int_OMin) (tptp.c_Int_Opred V_k))) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) tptp.c_Int_OMin) V_k) (tptp.c_Int_Opred V_k))) (= (tptp.c_Int_Opred tptp.c_Int_OMin) (tptp.c_Int_OBit0 tptp.c_Int_OMin)) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (not (tptp.c_Int_Oiszero T_a (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin))))) (= (tptp.c_Int_Osucc tptp.c_Int_OMin) tptp.c_Int_OPls) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k) tptp.c_Int_OMin) (tptp.c_Int_Osucc V_k))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_z)))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_z)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)))) (forall ((T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_a) (tptp.class_Int_Onumber__ring T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_x)))) (forall ((V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OMin) (tptp.c_Int_OBit0 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OMin) V_k_2)))) (forall ((V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Int_OBit0 V_k_2)) tptp.c_Int_OMin)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_k_2) tptp.c_Int_OMin)))) (forall ((V_a $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (forall ((V_l $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 tptp.c_Int_OPls) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit1 (tptp.hAPP (tptp.hAPP _let_1 tptp.c_Int_OMin) V_l))))) (= (tptp.hAPP _let_148 tptp.c_Int_OMin) _let_164) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_m) V_n) _let_1) (or (= V_m _let_1) (= V_m (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_m_2) V_n_2) _let_2) (or (and (= V_m_2 _let_2) (= V_n_2 _let_2)) (and (= V_m_2 _let_1) (= V_n_2 _let_1))))))) (forall ((V_l $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP _let_1 V_l))))) (forall ((V_l $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit1 (tptp.hAPP _let_1 V_l))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)))) (forall ((V_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_a))) (= _let_64 (tptp.hAPP _let_66 tptp.c_Int_OPls)) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_a) (tptp.class_Int_Onumber__ring T_a)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone T_a)) V_x)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) V_n)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_1) V_b) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_a) V_b)) _let_1)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin)))))) (forall ((V_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_b)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_b) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (= _let_63 (tptp.hAPP _let_66 _let_164)) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)) (tptp.c_Groups_Oone__class_Oone T_a)))) _let_163 _let_157 _let_156 (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod T_a T_b))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on _let_1 _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_4 _let_1) (tptp.c_Set_OCollect _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_2) _let_4 _let_1) (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_COMBB T_b _let_2 T_a) (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal)))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) _let_155 (forall ((V_n $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Code__Numeral_Onat__of__aux V_i V_n))) (let ((_let_2 (= V_i (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (and (=> _let_2 (= _let_1 V_n)) (=> (not _let_2) (= _let_1 (tptp.c_Code__Numeral_Onat__of__aux (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP tptp.c_Nat_OSuc V_n)))))))) _let_154 (forall ((V_a $$unsorted) (V_b $$unsorted)) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.c_Divides_Odivmod__int V_a V_b))))) (forall ((V_c $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP _let_2 V_q) V_r))) (=> (not (= V_b _let_3)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_3) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_q) V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_q V_c))) V_r))))))))))) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__size _let_64) _let_31) (forall ((V_w_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OPls) V_w_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_w_2)) (tptp.c_Nat_Osemiring__1__class_ONats T_a)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_a T_b))) (let ((_let_4 (tptp.c_member _let_1))) (let ((_let_5 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_fun T_b _let_5))) (let ((_let_7 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_2 _let_7) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_6 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) _let_4)) _let_3))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a _let_7)) (tptp.hAPP _let_8 V_R_2)) (tptp.hAPP _let_8 V_S_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_R_2) V_S_2))))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_a T_b))) (let ((_let_4 (tptp.c_member _let_1))) (let ((_let_5 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_fun T_b _let_5))) (let ((_let_7 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_2 _let_7) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_6 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) _let_4)) _let_3))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a _let_7)) (tptp.hAPP _let_8 V_R_2)) (tptp.hAPP _let_8 V_S_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_R_2) V_S_2))))))))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_b)))))) (= (= (tptp.hAPP _let_6 V_R_2) (tptp.hAPP _let_6 V_S_2)) (= V_R_2 V_S_2))))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_b)))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a _let_5)) (tptp.hAPP _let_6 V_R_2)) (tptp.hAPP _let_6 V_S_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_R_2) V_S_2)))))))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_a T_b V_r_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_r_2) V_A_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) B_x) V_b_2)) V_r_2)))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odivmod__int__rel V_a V_b))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q_H) V_r_H))) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_r V_r_H))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odivmod__int__rel V_a V_b))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q_H) V_r_H))) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_q V_q_H))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r))) (=> (not (= V_a _let_1)) (= V_r _let_1))))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not (= V_b _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel _let_1 V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_1) _let_1)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_5 (tptp.c_member _let_2))) (let ((_let_6 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun T_c _let_6))) (let ((_let_8 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.tc_fun T_b _let_8))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_9) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_9 T_a) (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_3 _let_8) T_b) (tptp.c_COMBC T_c _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_2) _let_7 T_b) (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 T_c) _let_5)) _let_4)))) V_r_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_3) _let_1) V_r_2)))))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_5 (tptp.c_member _let_2))) (let ((_let_6 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun T_c _let_6))) (let ((_let_8 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.tc_fun T_b _let_8))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_9) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_9 T_a) (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_3 _let_8) T_b) (tptp.c_COMBC T_c _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_2) _let_7 T_b) (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 T_c) _let_5)) _let_4)))) V_r_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_a _let_3) _let_1) V_r_2)))))))))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r))) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_q (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r))) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) V_r)))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r))) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) V_q)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.c_Nat_Osemiring__1__class_ONats T_a))))) _let_153 (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_ONats T_a))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a_2) V_b_2)) _let_1)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_ONats T_a))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a_2) V_b_2)) _let_1)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Nat_Osemiring__1__class_ONats T_a))))) (forall ((V_n_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n_2)) (tptp.c_Nat_Osemiring__1__class_ONats T_a))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_a_2) V_b_2)) V_r_2)))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) T_a) (tptp.c_member _let_1)) (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_a_2))) V_r_2)))))) (forall ((V_br $$unsorted) (V_bq $$unsorted) (V_b $$unsorted) (V_ar $$unsorted) (V_aq $$unsorted) (V_c $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ar) V_br))) (let ((_let_3 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_c) (tptp.hAPP (tptp.hAPP _let_3 V_aq) V_ar))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_b V_c) (tptp.hAPP (tptp.hAPP _let_3 V_bq) V_br))) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_aq) V_bq)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_2) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_2 V_c))))))))))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP (tptp.c_If tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP tptp.c_fequal V_r_2) _let_2)))) (let ((_let_4 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_5 (tptp.hAPP _let_4 V_q_2))) (let ((_let_6 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a_2 V_b_2) (tptp.hAPP (tptp.hAPP _let_6 V_q_2) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel (tptp.hAPP _let_4 V_a_2) V_b_2) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_3 _let_5) (tptp.hAPP (tptp.hAPP _let_1 _let_5) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (tptp.hAPP (tptp.hAPP _let_3 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_r_2))))))))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_b)))) (=> (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) V_q)) V_r)) (=> (and (=> _let_4 (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r) V_b)))) (=> (not _let_4) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b) V_r)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r) _let_1))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r)))))))))) (forall ((V_a $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_c $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_r))) (let ((_let_3 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_b V_c) (tptp.hAPP (tptp.hAPP _let_3 V_q) V_r))) (=> (not (= V_c (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel (tptp.hAPP _let_1 V_b) V_c) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_2) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_2 V_c)))))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_b_2)) (tptp.hAPP (tptp.c_Set_Oimage _let_1 T_c (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) V_f_2)) V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_3 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Nitpick_Onorm__frac V_a_2 V_b_2))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_6 (or (= V_a_2 _let_5) (= V_b_2 _let_5)))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b_2) _let_5)))) (let ((_let_8 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (and (=> _let_7 (= _let_4 (tptp.c_Nitpick_Onorm__frac (tptp.hAPP _let_8 V_a_2) (tptp.hAPP _let_8 V_b_2)))) (=> (not _let_7) (and (=> _let_6 (= _let_4 (tptp.hAPP (tptp.hAPP _let_2 _let_5) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint _let_3) (tptp.hAPP (tptp.hAPP tptp.c_Nitpick_Oint__gcd V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint _let_3) tptp.tc_Int_Oint) _let_2) (tptp.hAPP _let_1 V_a_2))) (tptp.hAPP _let_1 V_b_2))))))))))))))))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_q_2))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_3 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_4 (tptp.tc_fun _let_3 _let_3))) (= (tptp.hAPP (tptp.c_Divides_Oadjust V_b_2) (tptp.hAPP (tptp.hAPP _let_2 V_q_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_r_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_3 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_3 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_3 _let_4) tptp.tc_Int_Oint) (tptp.c_If _let_3)) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_1) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r_2))))))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2))))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_z_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP V_c_2 V_a_2) V_b_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2))))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f1_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_f1_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_f1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_x_2) V_y_2)))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_c_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_a_2) V_b_2) V_c_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_a))) (tptp.c_Fun_Oinj__on (tptp.tc_prod T_a T_b) _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b _let_1) (tptp.hAPP (tptp.c_COMBC T_b T_a _let_1) (tptp.c_Product__Type_OPair T_b T_a))) V_A_2))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)))) (forall ((V_Q_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_x_2) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_P_2) V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 V_x_2) V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_P_2) V_Q_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 V_x_2) V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2)))) (forall ((V_b_H $$unsorted) (V_a_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_b))) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_a_H) V_b_H)) (not (=> (= V_a V_a_H) (not (= V_b V_b_H))))))) (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)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_b))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_H_2) V_b_H_2)) (and (= V_a_2 V_a_H_2) (= V_b_2 V_b_H_2))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_P_2 $$unsorted)) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (forall ((B_a $$unsorted) (B_b $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) B_a) B_b)))))) (forall ((V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_q_2 $$unsorted) (V_p_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) V_c_2))) (=> (= V_p_2 V_q_2) (= (tptp.hAPP _let_1 V_p_2) (tptp.hAPP _let_1 V_q_2))))) (forall ((V_p_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_f_2))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_e _let_2) V_g_2) V_p_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_e T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_e _let_2) (tptp.tc_fun T_e T_a) T_d) (tptp.hAPP (tptp.c_COMBB _let_2 T_a T_e) _let_1)) V_g_2)) V_p_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_b_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_f1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP V_f1_2 V_a_2) V_b_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_b_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) (tptp.tc_fun T_b T_c) T_a) (tptp.hAPP (tptp.c_COMBB _let_1 T_c T_b) V_f_2)) (tptp.c_Product__Type_OPair T_a T_b))) V_f_2))) (forall ((V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.c_HOL_OThe _let_1 V_P_2) (tptp.c_HOL_OThe _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) V_P_2)) (tptp.c_Product__Type_OPair T_a T_b))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_HOL_OThe (tptp.tc_prod T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 T_a) tptp.c_fconj) (tptp.hAPP tptp.c_fequal V_x_2)))) (tptp.hAPP tptp.c_fequal V_y_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2))))) _let_152 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_FunDef_Oin__rel T_a T_b V_R_2) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) V_R_2)))) _let_151 (forall ((V_w $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Divides_OnegDivAlg _let_3 _let_2))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_3) _let_2))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_6) _let_5)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_6) _let_2)) (and (=> _let_7 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_1 tptp.c_Int_OMin)) _let_5))) (=> (not _let_7) (= _let_4 (tptp.hAPP (tptp.c_Divides_Oadjust _let_2) (tptp.c_Divides_OnegDivAlg _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_2))))))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.c_Divides_OnegDivAlg V_a V_b)))))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_l)) (= (tptp.c_Divides_OnegDivAlg V_k V_l) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_k) V_l)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l)))))))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.c_Divides_OnegDivAlg _let_1 V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_b) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_OnegDivAlg V_a V_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_a) V_b))) (let ((_let_6 (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_3) _let_5)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_b) _let_3))))) (and (=> _let_6 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_1 tptp.c_Int_OMin)) _let_5))) (=> (not _let_6) (= _let_2 (tptp.hAPP (tptp.c_Divides_Oadjust V_b) (tptp.c_Divides_OnegDivAlg V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_b))))))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_OnegDivAlg V_a V_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_a) V_b))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_4) _let_3)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_4) V_b)) (and (=> _let_5 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_1 tptp.c_Int_OMin)) _let_3))) (=> (not _let_5) (= _let_2 (tptp.hAPP (tptp.c_Divides_Oadjust V_b) (tptp.c_Divides_OnegDivAlg V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_b))))))))))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (let ((_let_3 (tptp.hAPP _let_1 V_w))) (let ((_let_4 (tptp.c_Divides_OnegDivAlg _let_3 _let_2))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_3) _let_2))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_6) _let_5)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_6) _let_2)) (and (=> _let_7 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_1 tptp.c_Int_OMin)) _let_5))) (=> (not _let_7) (= _let_4 (tptp.hAPP (tptp.c_Divides_Oadjust _let_2) (tptp.c_Divides_OnegDivAlg _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_2))))))))))))))) _let_150 (forall ((V_w $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Divides_OposDivAlg _let_3 _let_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_3) _let_2)))) (let ((_let_7 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_7) _let_2)) (and (=> _let_6 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_7) _let_3))) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.c_Divides_Oadjust _let_2) (tptp.c_Divides_OposDivAlg _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_2))))))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_OposDivAlg V_a V_b))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_b) _let_2))))) (and (=> _let_3 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_2) V_a))) (=> (not _let_3) (= _let_1 (tptp.hAPP (tptp.c_Divides_Oadjust V_b) (tptp.c_Divides_OposDivAlg V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_b)))))))))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_OposDivAlg _let_1 V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_1) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.c_Divides_OposDivAlg V_a V_b))))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_k)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_l)) (= (tptp.c_Divides_OposDivAlg V_k V_l) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_k) V_l)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l))))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (let ((_let_3 (tptp.hAPP _let_1 V_w))) (let ((_let_4 (tptp.c_Divides_OposDivAlg _let_3 _let_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_3) _let_2)))) (let ((_let_7 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 _let_7) _let_2)) (and (=> _let_6 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_7) _let_3))) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.c_Divides_Oadjust _let_2) (tptp.c_Divides_OposDivAlg _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_2))))))))))))))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Divides_OposDivAlg V_a V_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_4) V_b)) (and (=> _let_3 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_4) V_a))) (=> (not _let_3) (= _let_1 (tptp.hAPP (tptp.c_Divides_Oadjust V_b) (tptp.c_Divides_OposDivAlg V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_b)))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (let ((_let_4 (tptp.c_Divides_Odivmod__int V_a V_b))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_5) V_b)))) (let ((_let_7 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_5))) (let ((_let_8 (tptp.hBOOL (tptp.hAPP _let_7 V_a)))) (let ((_let_9 (= V_a _let_5))) (let ((_let_10 (tptp.hBOOL (tptp.hAPP _let_7 V_b)))) (and (=> _let_8 (and (=> _let_10 (= _let_4 (tptp.c_Divides_OposDivAlg V_a V_b))) (=> (not _let_10) (and (=> _let_9 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_5) _let_5))) (=> (not _let_9) (= _let_4 (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OnegDivAlg _let_3 _let_2)))))))) (=> (not _let_8) (and (=> _let_6 (= _let_4 (tptp.c_Divides_OnegDivAlg V_a V_b))) (=> (not _let_6) (= _let_4 (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OposDivAlg _let_3 _let_2)))))))))))))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.c_Divides_Odivmod__nat V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) _let_1) V_n))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat _let_1 V_n))))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Opdivmod V_k_2 V_l_2))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_4 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_5 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_6 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_7 (tptp.tc_fun tptp.tc_Int_Oint _let_6))) (let ((_let_8 (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_7 tptp.tc_Int_Oint) _let_5))) (let ((_let_9 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_10 (tptp.c_If _let_6))) (let ((_let_11 (tptp.tc_fun _let_6 _let_6))) (let ((_let_12 (tptp.tc_fun tptp.tc_Int_Oint _let_11))) (let ((_let_13 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_14 (tptp.c_Divides_Odivmod__int V_k_2 V_l_2))) (let ((_let_15 (= V_l_2 _let_9))) (let ((_let_16 (tptp.hAPP _let_5 _let_9))) (let ((_let_17 (= V_k_2 _let_9))) (and (=> _let_17 (= _let_14 (tptp.hAPP _let_16 _let_9))) (=> (not _let_17) (and (=> _let_15 (= _let_14 (tptp.hAPP _let_16 V_k_2))) (=> (not _let_15) (= _let_14 (tptp.hAPP (tptp.c_Product__Type_Oapsnd tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.c_Groups_Osgn__class_Osgn tptp.tc_Int_Oint V_l_2))) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_10 (tptp.hAPP (tptp.hAPP tptp.c_fdisj (tptp.hAPP (tptp.hAPP tptp.c_fconj (tptp.hAPP (tptp.hAPP _let_13 _let_9) V_l_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_9) V_k_2))) (tptp.hAPP (tptp.hAPP tptp.c_fconj (tptp.hAPP (tptp.hAPP _let_13 V_l_2) _let_9)) (tptp.hAPP (tptp.hAPP _let_13 V_k_2) _let_9)))) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_7 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_7 _let_7) tptp.tc_Int_Oint) (tptp.c_COMBS tptp.tc_Int_Oint _let_6 _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_12 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_6 _let_11) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_6 _let_11) tptp.tc_Int_Oint) _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) tptp.c_fequal) _let_9)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint _let_6) (tptp.hAPP _let_8 _let_3)) _let_9)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_4 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_4 _let_7) tptp.tc_Int_Oint) (tptp.c_COMBB tptp.tc_Int_Oint _let_6 tptp.tc_Int_Oint)) (tptp.hAPP _let_8 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_4 tptp.tc_Int_Oint) _let_2) _let_3)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_l_2))))) _let_1)))))))))))))))))))))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_a V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_a V_g_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_b T_a (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_d V_f_2) V_g_2)) V_x_2))) _let_149 (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_a V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_c) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) (tptp.hAPP V_f_2 V_y_2)))) (forall ((V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_Divides_Odivmod__nat V_m _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_m)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (= (tptp.c_Divides_Odivmod__nat V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)))) (forall ((V_r $$unsorted) (V_q $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q))) (= (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.hAPP _let_1 V_r)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_r))))) _let_147 (forall ((V_qr $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) V_qr)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP tptp.c_Divides_OnegateSnd V_qr)))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Divides_Odivmod__nat V_m_2 V_n_2))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_5 (or (= V_n_2 _let_4) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2))))) (and (=> _let_5 (= _let_3 (tptp.hAPP (tptp.hAPP _let_1 _let_4) V_m_2))) (=> (not _let_5) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat _let_2) tptp.tc_Nat_Onat) _let_1) tptp.c_Nat_OSuc)) (tptp.c_Divides_Odivmod__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m_2) V_n_2) V_n_2))))))))))) _let_146 (forall ((V_k $$unsorted) (V_l $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_k))) (let ((_let_3 (tptp.c_Divides_Opdivmod V_k V_l))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_5 (= V_l _let_4))) (and (=> _let_5 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) _let_4) _let_2))) (=> (not _let_5) (= _let_3 (tptp.c_Divides_OposDivAlg _let_2 (tptp.hAPP _let_1 V_l))))))))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Opdivmod V_k_2 V_l_2))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_4 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_5 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_6 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_7 (tptp.tc_fun tptp.tc_Int_Oint _let_6))) (let ((_let_8 (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_7 tptp.tc_Int_Oint) _let_5))) (let ((_let_9 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_10 (tptp.c_If _let_6))) (let ((_let_11 (tptp.tc_fun _let_6 _let_6))) (let ((_let_12 (tptp.tc_fun tptp.tc_Int_Oint _let_11))) (let ((_let_13 (tptp.c_Groups_Osgn__class_Osgn tptp.tc_Int_Oint V_l_2))) (let ((_let_14 (tptp.c_Divides_Odivmod__int V_k_2 V_l_2))) (let ((_let_15 (= V_l_2 _let_9))) (let ((_let_16 (tptp.hAPP _let_5 _let_9))) (let ((_let_17 (= V_k_2 _let_9))) (and (=> _let_17 (= _let_14 (tptp.hAPP _let_16 _let_9))) (=> (not _let_17) (and (=> _let_15 (= _let_14 (tptp.hAPP _let_16 V_k_2))) (=> (not _let_15) (= _let_14 (tptp.hAPP (tptp.c_Product__Type_Oapsnd tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) _let_13)) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_10 (tptp.hAPP (tptp.hAPP tptp.c_fequal (tptp.c_Groups_Osgn__class_Osgn tptp.tc_Int_Oint V_k_2)) _let_13)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_7 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_7 _let_7) tptp.tc_Int_Oint) (tptp.c_COMBS tptp.tc_Int_Oint _let_6 _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_12 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_6 _let_11) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_6 _let_11) tptp.tc_Int_Oint) _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) tptp.c_fequal) _let_9)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint _let_6) (tptp.hAPP _let_8 _let_3)) _let_9)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_4 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_4 _let_7) tptp.tc_Int_Oint) (tptp.c_COMBB tptp.tc_Int_Oint _let_6 tptp.tc_Int_Oint)) (tptp.hAPP _let_8 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_4 tptp.tc_Int_Oint) _let_2) _let_3)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_l_2))))) _let_1)))))))))))))))))))))))))) (forall ((V_c $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_b))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_a V_b) (tptp.hAPP (tptp.hAPP _let_2 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_b)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_q) V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_q V_c))) V_r))))))))))) (forall ((V_a $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_c $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_r))) (let ((_let_3 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_b V_c) (tptp.hAPP (tptp.hAPP _let_3 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel (tptp.hAPP _let_1 V_b) V_c) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_q)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) _let_2) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat _let_2 V_c)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_OnegDivAlg V_a_2 V_b_2))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_a_2) V_b_2))) (let ((_let_6 (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_3) _let_5)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_b_2) _let_3))))) (let ((_let_7 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OnegDivAlg__rel) (tptp.hAPP (tptp.hAPP _let_7 V_a_2) V_b_2))) (and (=> _let_6 (= _let_2 (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP _let_1 tptp.c_Int_OMin)) _let_5))) (=> (not _let_6) (= _let_2 (tptp.hAPP (tptp.c_Divides_Oadjust V_b_2) (tptp.c_Divides_OnegDivAlg V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_b_2))))))))))))))) (forall ((V_qr_H $$unsorted) (V_qr $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odivmod__nat__rel V_m V_n))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_qr)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_qr_H)) (= V_qr V_qr_H))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_m V_n) (tptp.c_Divides_Odivmod__nat V_m V_n)))) (forall ((V_qr $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_m V_n) V_qr)) (= (tptp.c_Divides_Odivmod__nat V_m V_n) V_qr))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Divides_Odivmod__nat V_m_2 V_n_2) (tptp.c_HOL_OThe (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Divides_Odivmod__nat__rel V_m_2 V_n_2)))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_q) V_r))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) V_r))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_q) V_r))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n) V_q))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n))))) (forall ((V_br $$unsorted) (V_bq $$unsorted) (V_b $$unsorted) (V_ar $$unsorted) (V_aq $$unsorted) (V_c $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ar) V_br))) (let ((_let_3 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_a V_c) (tptp.hAPP (tptp.hAPP _let_3 V_aq) V_ar))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_b V_c) (tptp.hAPP (tptp.hAPP _let_3 V_bq) V_br))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_aq) V_bq)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) _let_2) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat _let_2 V_c))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Divides_OposDivAlg V_a_2 V_b_2))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_b_2) _let_2))))) (let ((_let_4 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OposDivAlg__rel) (tptp.hAPP (tptp.hAPP _let_4 V_a_2) V_b_2))) (and (=> _let_3 (= _let_1 (tptp.hAPP (tptp.hAPP _let_4 _let_2) V_a_2))) (=> (not _let_3) (= _let_1 (tptp.hAPP (tptp.c_Divides_Oadjust V_b_2) (tptp.c_Divides_OposDivAlg V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_b_2)))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_3 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Nitpick_Onorm__frac V_a_2 V_b_2))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_6 (or (= V_a_2 _let_5) (= V_b_2 _let_5)))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_b_2) _let_5)))) (let ((_let_8 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp _let_3 tptp.c_Nitpick_Onorm__frac__rel) (tptp.hAPP (tptp.hAPP _let_2 V_a_2) V_b_2))) (and (=> _let_7 (= _let_4 (tptp.c_Nitpick_Onorm__frac (tptp.hAPP _let_8 V_a_2) (tptp.hAPP _let_8 V_b_2)))) (=> (not _let_7) (and (=> _let_6 (= _let_4 (tptp.hAPP (tptp.hAPP _let_2 _let_5) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint _let_3) (tptp.hAPP (tptp.hAPP tptp.c_Nitpick_Oint__gcd V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint _let_3) tptp.tc_Int_Oint) _let_2) (tptp.hAPP _let_1 V_a_2))) (tptp.hAPP _let_1 V_b_2)))))))))))))))))) (forall ((V_P_2 $$unsorted) (V_a1_2 $$unsorted) (V_a0_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OnegDivAlg__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_a0_2) V_a1_2))) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_P_2 B_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OnegDivAlg__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) B_a) B_b))) (=> (=> (not (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_a) B_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 B_b) _let_2)))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) B_b)))) (tptp.hBOOL (tptp.hAPP _let_1 B_b)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_a0_2) V_a1_2))))) (forall ((V_P_2 $$unsorted) (V_a1_2 $$unsorted) (V_a0_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OposDivAlg__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_a0_2) V_a1_2))) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_P_2 B_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OposDivAlg__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) B_a) B_b))) (=> (=> (not (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_a) B_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) B_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) B_b)))) (tptp.hBOOL (tptp.hAPP _let_1 B_b)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_a0_2) V_a1_2))))) (forall ((V_R2_2 $$unsorted) (V_R1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a _let_1)) V_R1_2) V_R2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Wellfounded_Oaccp T_a V_R2_2)) (tptp.c_Wellfounded_Oaccp T_a V_R1_2)))))) (forall ((V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_3 _let_1) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OnegDivAlg (tptp.hAPP _let_2 _let_3) (tptp.hAPP _let_2 _let_1)))))))))) (forall ((V_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_x) V_y)) V_a) (= V_y V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_a) V_b)) V_b)) (forall ((V_p_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_a) V_p_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_a T_a) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a T_a) T_b) (tptp.c_COMBI T_a))) V_p_2))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_a) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_a T_b V_f_2) V_x_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_c) V_x_2)))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Osnd T_a T_c) V_x_2))) (= (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_a V_f_2) V_x_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_a V_g_2) V_x_2)) (= (tptp.hAPP V_f_2 _let_1) (tptp.hAPP V_g_2 _let_1))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odivmod__int V_a V_b)))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oaccp T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_a V_r_2) V_a_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 B_x) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_a V_r_2) B_x)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_a V_r_2) V_a_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 B_x) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_a V_r_2) B_x)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_OnegDivAlg V_a V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_b)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_OposDivAlg V_a V_b))))))) (forall ((V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_1)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_2 _let_1) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_OposDivAlg _let_2 _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b) _let_2)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OnegDivAlg (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_b) _let_2)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OposDivAlg (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Onat__gcd V_x_2 V_y_2))) (let ((_let_2 (= V_y_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nitpick_Onat__gcd__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_x_2) V_y_2))) (and (=> _let_2 (= _let_1 V_x_2)) (=> (not _let_2) (= _let_1 (tptp.c_Nitpick_Onat__gcd V_y_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_x_2 V_y_2))))))))) (forall ((V_nat $$unsorted)) (= (tptp.c_Nat_Onat_Onat__size (tptp.hAPP tptp.c_Nat_OSuc V_nat)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Nat_Onat_Onat__size V_nat)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_Wellfounded_Omeasure T_a) V_f_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Divides_Odivmod__nat V_m V_n)))) (= (tptp.c_Nat_Onat_Onat__size _let_31) _let_31) _let_145 (forall ((V_P_2 $$unsorted) (V_a1_2 $$unsorted) (V_a0_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Nitpick_Onorm__frac__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_a0_2) V_a1_2))) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Nitpick_Onorm__frac__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) B_a) B_b))) (=> (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP _let_1 B_a)) (tptp.hAPP _let_1 B_b)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_a) B_b)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_a0_2) V_a1_2))))) (forall ((V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_3) _let_1) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OnegDivAlg (tptp.hAPP _let_2 _let_3) (tptp.hAPP _let_2 _let_1)))))))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_t_2 $$unsorted) (V_s_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_Osnd T_a T_b))) (let ((_let_2 (tptp.c_Product__Type_Ofst T_a T_b))) (= (= V_s_2 V_t_2) (and (= (tptp.hAPP _let_2 V_s_2) (tptp.hAPP _let_2 V_t_2)) (= (tptp.hAPP _let_1 V_s_2) (tptp.hAPP _let_1 V_t_2))))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_Osnd T_a T_b))) (let ((_let_2 (tptp.c_Product__Type_Ofst T_a T_b))) (=> (= (tptp.hAPP _let_2 V_p) (tptp.hAPP _let_2 V_q)) (=> (= (tptp.hAPP _let_1 V_p) (tptp.hAPP _let_1 V_q)) (= V_p V_q)))))) (forall ((V_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x) V_y)) V_a) (= V_x V_a))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a) V_b)) V_a)) (forall ((V_p_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) V_p_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_a) (tptp.c_COMBK T_a T_b)) V_p_2))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_a V_f_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_c) V_x_2))) (forall ((V_p $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) V_p)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_a T_b) V_p)) V_p)) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_t $$unsorted)) (= V_t (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) V_t)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_a T_b) V_t)))) (forall ((V_p_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_f_2) V_p_2) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) V_p_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_c) V_p_2)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odivmod__int V_a V_b)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_c))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_2 T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_d _let_1 _let_2) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_d _let_2) V_g_2) (tptp.c_Product__Type_Ofst T_a T_b)))) (tptp.c_Product__Type_Osnd T_a T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_d _let_1 T_a) V_f_2) V_g_2)))))) (forall ((V_z_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_P_2) V_z_2) (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) V_z_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_c) V_z_2)))) (forall ((V_p_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_c_2) V_p_2) (tptp.hAPP (tptp.hAPP V_c_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) V_p_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_c) V_p_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) (tptp.hAPP (tptp.c_Fun_Ocomp T_d (tptp.tc_fun T_c T_a) T_b V_f_2) V_g_2)) V_x_2) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) V_x_2))) (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_c) V_x_2)))) (forall ((V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.c_HOL_OThe _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_P_2)) (tptp.c_HOL_OThe _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool) _let_1) V_P_2) (tptp.c_Product__Type_Ofst T_a T_b))) (tptp.c_Product__Type_Osnd T_a T_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_OnegDivAlg V_a V_b)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_OposDivAlg V_a V_b))))))) (forall ((V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_1)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_2) _let_1) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_OposDivAlg _let_2 _let_1))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b) _let_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OnegDivAlg (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_b) _let_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OposDivAlg (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)))))))))) _let_144 (forall ((T_a $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_x_2) V_y_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_y_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_a) _let_1)) (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_a) _let_1)))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_q_2 $$unsorted) (V_Q_2 $$unsorted) (V_p_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_p_2) V_q_2))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_p_2)) (=> (tptp.hBOOL (tptp.hAPP V_Q_2 V_q_2)) (and (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) _let_1))) (tptp.hBOOL (tptp.hAPP V_Q_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_a T_b) _let_1)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Divides_Odivmod__nat V_m V_n)))) (forall ((V_qr_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_qr_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_1) V_n_2)))) (let ((_let_6 (= V_n_2 _let_1))) (let ((_let_7 (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_qr_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_m_2 V_n_2) V_qr_2)) (and (= V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_7) V_n_2)) _let_2)) (=> _let_6 (= _let_7 _let_1)) (=> (not _let_6) (and (=> _let_5 (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 _let_2) V_n_2)))) (=> (not _let_5) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_n_2) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)))))))))))))))) (forall ((V_z_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun _let_1 _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat _let_2))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_1))) (= (tptp.hAPP tptp.c_Int_Onat V_z_2) (tptp.c_Set_Othe__elem tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) (tptp.hAPP tptp.c_Int_ORep__Integ V_z_2)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_1 _let_4) tptp.tc_Nat_Onat) (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_3 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_2 tptp.tc_Nat_Onat) (tptp.c_Set_Oinsert tptp.tc_Nat_Onat))) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat)))) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_P_2 $$unsorted) (V_a1_2 $$unsorted) (V_a0_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nitpick_Onat__gcd__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_a0_2) V_a1_2))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nitpick_Onat__gcd__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) B_x) B_y))) (=> (=> (not (= B_y (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_y) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat B_x B_y)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_x) B_y))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_a0_2) V_a1_2))))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.tc_fun T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_a _let_1))) (let ((_let_4 (tptp.tc_fun T_a T_b))) (let ((_let_5 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_fun _let_5 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun T_a _let_6))) (let ((_let_8 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_b T_a) V_r_2) V_f_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_5 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_5 _let_8) T_a) (tptp.c_COMBC T_a _let_5 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_7 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_6 T_a) (tptp.c_member _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_a) (tptp.c_COMBB T_b _let_1 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_2 T_a) (tptp.c_Product__Type_OPair T_b T_b)) V_f_2))) V_f_2)))) V_r_2))))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Int_ORep__Integ V_x_2) (tptp.hAPP tptp.c_Int_ORep__Integ V_y_2)) (= V_x_2 V_y_2))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_b T_a) V_r_2) V_f_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) V_r_2)))) _let_143 (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 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_a))) (let ((_let_4 (tptp.c_Relation_OImage T_a T_a V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_a_2) _let_2))) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_2))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2))))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_a))) (let ((_let_4 (tptp.c_Relation_OImage T_a T_a V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_2))) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_a_2) _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2)))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (= (tptp.c_Complete__Lattice_OSup__class_OSup (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2)) V_A_2))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) V_a_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))))) (forall ((V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))) (let ((_let_3 (tptp.c_member _let_1))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_X_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_Y_2) _let_2)) (or (= V_X_2 V_Y_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_X_2) V_Y_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_a))) (let ((_let_3 (tptp.c_Relation_OImage T_a T_a V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2)) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))) (let ((_let_3 (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_X_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_Y_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_X_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_Y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (= V_X_2 V_Y_2))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))) (let ((_let_3 (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_X_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_Y_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_X_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_Y_2)) (= (= V_X_2 V_Y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)))))))))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_a))) (let ((_let_3 (tptp.c_Relation_OImage T_a T_a V_r_2))) (=> (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_a))) (let ((_let_3 (tptp.c_Relation_OImage T_a T_a V_r_2))) (let ((_let_4 (tptp.c_member T_a))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_y_2) V_A_2)) (= (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert T_a))) (let ((_let_4 (tptp.c_Relation_OImage T_a T_a V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (and (= (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2)) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_a))) (let ((_let_4 (tptp.c_Relation_OImage T_a T_a V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_a_2) _let_2))) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_2))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_a))) (let ((_let_3 (tptp.c_Equiv__Relations_Oquotient T_a))) (let ((_let_4 (tptp.c_member T_a))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_y_2) V_A_2)) (= (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1)) V_r_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2))))))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.c_Equiv__Relations_Ocongruent T_a _let_1 V_r_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) V_f_2) (tptp.hAPP V_f_2 V_a_2))))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_r2_2 $$unsorted) (V_A2_2 $$unsorted) (T_b $$unsorted) (V_r1_2 $$unsorted) (V_A1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A1_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A2_2 V_r2_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_a T_b _let_1 V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a1_2) V_A1_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a2_2) V_A2_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a1_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_1 T_a) (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a2_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) V_f_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_a1_2) V_a2_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_A_2) V_B_2)) (tptp.c_Wellfounded_Ofinite__psubset T_a))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_B_2)))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_r2_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_r1_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_a T_b T_c V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.c_Equiv__Relations_Ocongruent T_b T_c V_r2_2 (tptp.hAPP V_f_2 V_a_2)))))) (forall ((V_z2_2 $$unsorted) (V_y2_2 $$unsorted) (V_z1_2 $$unsorted) (V_y1_2 $$unsorted) (V_f_2 $$unsorted) (V_r2_2 $$unsorted) (V_r1_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_a T_b T_c V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_y1_2) V_z1_2)) V_r1_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_y2_2) V_z2_2)) V_r2_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 V_y1_2) V_y2_2) (tptp.hAPP (tptp.hAPP V_f_2 V_z1_2) V_z2_2)))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Ocongruent T_a T_b V_r_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_y_2) V_z_2)) V_r_2)) (= (tptp.hAPP V_f_2 V_y_2) (tptp.hAPP V_f_2 V_z_2))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_r2_2 $$unsorted) (V_A2_2 $$unsorted) (T_b $$unsorted) (V_r1_2 $$unsorted) (V_A1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A1_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A2_2 V_r2_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_a T_b _let_1 V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A2_2)) (tptp.c_Equiv__Relations_Ocongruent T_a _let_1 V_r1_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_1 T_a) (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) V_f_2)))))))) _let_142 (forall ((V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))) (let ((_let_2 (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_3))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.c_Equiv__Relations_Ocongruent T_a _let_3 V_r_2 V_f_2) (=> (= (tptp.hAPP (tptp.hAPP _let_4 V_X_2) V_f_2) (tptp.hAPP (tptp.hAPP _let_4 V_Y_2) V_f_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_X_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_Y_2) _let_1)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_y) V_A_2)) (=> (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_f_2 B_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_y)) V_r_2))))))) (= V_X_2 V_Y_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_X2_2 $$unsorted) (V_X1_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_r2_2 $$unsorted) (V_A2_2 $$unsorted) (T_b $$unsorted) (V_r1_2 $$unsorted) (V_A1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A1_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A2_2 V_r2_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_a T_b _let_1 V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_X1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A1_2) V_r1_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_X2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A2_2) V_r2_2))) (=> (forall ((B_x1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x1) V_A1_2)) (forall ((B_x2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x2) V_A2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP V_f_2 B_x1) B_x2)) V_B_2)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_X1_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_1 T_a) (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_X2_2)) V_f_2))) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_X_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.c_Equiv__Relations_Ocongruent T_a _let_1 V_r_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_X_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP V_f_2 B_x)) V_B_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_1) V_X_2) V_f_2)) V_B_2)))))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_R_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Wellfounded_Omlex__prod T_a V_f_2 V_R_2))))))) (forall ((V_code__numeral $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__size (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__size V_code__numeral)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_Wellfounded_Omlex__prod T_a V_f_2 V_R_2))))) (forall ((V_code__numeral_H_2 $$unsorted) (V_code__numeral_2 $$unsorted)) (= (= (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_2) (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_H_2)) (= V_code__numeral_2 V_code__numeral_H_2))) (forall ((V_code__numeral_H $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_H)))) (forall ((V_code__numeral_H_1 $$unsorted)) (not (= (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_H_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Code__Numeral_OSuc__code__numeral V_n)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)) V_n)) (forall ((V_code__numeral $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize tptp.tc_Code__Numeral_Ocode__numeral))) (= (tptp.hAPP _let_1 (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_code__numeral)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_nat $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_nat)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_nat)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Oprod_Oprod__rec T_b T_c T_a V_f1_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP V_f1_2 V_a_2) V_b_2))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat) V_n) V_n)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat) _let_31) _let_31) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Code__Numeral_Ocode__numeral) _let_64) _let_31) (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 (tptp.c_Product__Type_OPair T_a T_b))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_H_2) V_b_H_2))) (tptp.c_Wellfounded_Olex__prod T_a T_b V_r_2 V_s_2))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_a_H_2)) V_r_2)) (and (= V_a_2 V_a_H_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_b_2) V_b_H_2)) V_s_2)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Equiv__Relations_Oequiv T_a (tptp.c_Orderings_Otop__class_Otop _let_5) V_A_2) (tptp.c_Equiv__Relations_Oequivp T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_a _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_a) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_a)))) V_A_2))))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_c T_a T_b V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_b T_c V_g_2) V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_d) V_x_2))) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) V_x_2))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_c T_a T_b V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_c T_b V_g_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_a T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_a T_d V_f_2) V_g_2)) V_x_2))) (forall ((T_a $$unsorted)) (tptp.c_Equiv__Relations_Oequivp T_a tptp.c_fequal)) _let_141 (forall ((V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_a V_R_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_x_2) V_x_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_a V_R_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_y_2) V_x_2))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP V_R_2 V_x_2))) (=> (tptp.c_Equiv__Relations_Oequivp T_a V_R_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_y_2) V_z_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_z_2))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_a) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_c T_b T_a V_f_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_a) V_x_2))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_b) V_x_2))) (= (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_c T_a T_b V_f_2) V_x_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_c T_a T_b V_g_2) V_x_2)) (= (tptp.hAPP V_f_2 _let_1) (tptp.hAPP V_g_2 _let_1))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_c T_a T_b V_f_2) V_x_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_b) V_x_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_c T_a T_b V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP V_f_2 V_x_2)) V_y_2))) (forall ((V_p_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_a V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_a T_c V_g_2) V_p_2)) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_a T_b V_g_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_d V_f_2) V_p_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_c T_b T_a V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_a T_c V_g_2) V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_d T_c) V_x_2))) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_d T_c) V_x_2))))) (forall ((T_a $$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 (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_y_H_2) V_y_2)) (tptp.hAPP V_R_2 V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP _let_1 V_y_H_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.c_Recdef_Osame__fst T_a T_b V_P_2 V_R_2)))))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_s_2) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_s_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_t_2))) tptp.c_FunDef_Opair__less))))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_r_2) V_A_2))) (not (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) B_x) V_b_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)))))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_s_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_t_2))) tptp.c_FunDef_Opair__less)))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_s_2) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_s_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_t_2))) tptp.c_FunDef_Opair__leq)))))))) (forall ((V_X_2 $$unsorted) (V_x_2 $$unsorted) (V_Y_2 $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_y_2) V_Y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) tptp.c_FunDef_Opair__less)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_X_2) V_Y_2)) tptp.c_FunDef_Omax__strict)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_x_2) V_X_2)) V_Y_2)) tptp.c_FunDef_Omax__strict)))))))))) (forall ((V_YS_2 $$unsorted) (V_y_2 $$unsorted) (V_XS_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) V_XS_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_x_2) V_XS_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) tptp.c_FunDef_Opair__less)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_YS_2)) tptp.c_FunDef_Omin__strict)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_y_2) V_YS_2))) tptp.c_FunDef_Omin__strict)))))))))) (forall ((V_X_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not (= V_X_2 _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_X_2) _let_2)) tptp.c_FunDef_Omin__strict)))))) (forall ((V_Y_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_Y_2)) (=> (not (= V_Y_2 _let_3)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_3) V_Y_2)) tptp.c_FunDef_Omax__strict)))))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_s_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_t_2))) tptp.c_FunDef_Opair__leq)))))) (forall ((V_YS_2 $$unsorted) (V_y_2 $$unsorted) (V_XS_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) V_XS_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_x_2) V_XS_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) tptp.c_FunDef_Opair__leq)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_YS_2)) tptp.c_FunDef_Omin__weak)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_y_2) V_YS_2))) tptp.c_FunDef_Omin__weak)))))))))) (forall ((V_XS_2 $$unsorted) (V_x_2 $$unsorted) (V_YS_2 $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_y_2) V_YS_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) tptp.c_FunDef_Opair__leq)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_XS_2) V_YS_2)) tptp.c_FunDef_Omax__weak)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_x_2) V_XS_2)) V_YS_2)) tptp.c_FunDef_Omax__weak)))))))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Set_Oinsert T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_a_2) (tptp.hAPP (tptp.hAPP _let_3 V_b_2) (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP _let_1 V_r_2))))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_3 V_R_2)) (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 (tptp.hAPP (tptp.c_Set_Oimage _let_3 _let_2 _let_1) V_R_2))))))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OField T_a) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 V_s_2)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 V_s_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_a)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Relation_OField T_a) V_r_2))))) (forall ((V_X_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_X_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) V_X_2)) tptp.c_FunDef_Omax__weak)))))) (forall ((V_X_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_X_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) tptp.c_FunDef_Omin__weak)))) _let_140 _let_139 _let_136 (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a _let_1))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Relation_OId__on T_a V_A_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun tptp.tc_HOL_Obool _let_3) T_a) (tptp.c_COMBC T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 T_a) tptp.c_fconj)) tptp.c_fequal))) V_A_2))))))) (forall ((T_a $$unsorted)) (= (tptp.c_Relation_OId__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_a T_a (tptp.c_Relation_OId__on T_a V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_Relation_OId__on T_a V_A_2))) (and (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (=> (= V_a_2 V_b_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) (tptp.c_Relation_OId__on T_a V_A_2)))))) _let_135 (forall ((V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_R_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Omax__ext T_a V_R_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_A_2) V_B_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_C_2) V_D_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_D_2))) _let_1)))))))))) _let_134 (forall ((V_xa_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (= (tptp.c_Wellfounded_Omax__extp T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_3 _let_1) T_a) (tptp.c_COMBC T_a _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_2) _let_5 T_a) (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 T_a) (tptp.c_member _let_2))) (tptp.c_Product__Type_OPair T_a T_a)))) V_R_2) V_x_2 V_xa_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_xa_2)) (tptp.c_Wellfounded_Omax__ext T_a V_R_2)))))))))) (tptp.c_FunDef_Oreduction__pair _let_119 (tptp.hAPP (tptp.hAPP _let_133 tptp.c_FunDef_Omin__strict) tptp.c_FunDef_Omin__weak)) (tptp.c_FunDef_Oreduction__pair _let_119 (tptp.hAPP (tptp.hAPP _let_133 tptp.c_FunDef_Omax__strict) tptp.c_FunDef_Omax__weak)) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_FunDef_Oreduction__pair T_a V_P_2) (tptp.c_FunDef_Oreduction__pair T_b (tptp.hAPP (tptp.hAPP (tptp.c_FunDef_Orp__inv__image T_a T_b) V_P_2) V_f_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert _let_1))) (let ((_let_4 (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel))) (= (= (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2)) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) tptp.c_Int_Ointrel)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_x_2) V_y_2)) tptp.c_Wellfounded_Oless__than)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x_2) V_y_2)))) (forall ((V_v_2 $$unsorted) (V_u_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_3 _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_3 _let_3) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP _let_2 V_u_2) V_v_2))) tptp.c_Int_Ointrel)) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_v_2) (tptp.hAPP (tptp.hAPP _let_1 V_u_2) V_y_2))))))) (tptp.c_Equiv__Relations_Oequiv _let_93 _let_120 tptp.c_Int_Ointrel) _let_130 _let_129 _let_128 (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP (tptp.c_Relation_OImage _let_2 _let_2 tptp.c_Int_Ointrel) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_i_2) V_j_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)))))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_i_2)) (tptp.hAPP _let_1 V_j_2))))))) (= _let_127 (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_125 (tptp.hAPP (tptp.hAPP _let_124 (tptp.hAPP (tptp.hAPP _let_123 _let_62) _let_31)) _let_122)))) (forall ((V_v_2 $$unsorted) (V_u_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hAPP _let_3 V_y_2))) (let ((_let_5 (tptp.hAPP _let_3 V_x_2))) (let ((_let_6 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_7 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_8 (tptp.c_Set_Oinsert _let_1))) (let ((_let_9 (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_9 (tptp.hAPP (tptp.hAPP _let_8 (tptp.hAPP (tptp.hAPP _let_7 V_x_2) V_y_2)) _let_2)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_9 (tptp.hAPP (tptp.hAPP _let_8 (tptp.hAPP (tptp.hAPP _let_7 V_u_2) V_v_2)) _let_2)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_9 (tptp.hAPP (tptp.hAPP _let_8 (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 V_u_2)) (tptp.hAPP _let_4 V_v_2))) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 V_v_2)) (tptp.hAPP _let_4 V_u_2)))) _let_2)))))))))))))) (forall ((V_x_2 $$unsorted)) (= (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP tptp.c_Int_ORep__Integ V_x_2)) V_x_2)) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.c_Set_Oinsert _let_1))) (let ((_let_5 (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) _let_2)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_x_2)) _let_2)))))))))) (= _let_126 (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_125 (tptp.hAPP (tptp.hAPP _let_124 (tptp.hAPP (tptp.hAPP _let_123 _let_31) _let_31)) _let_122)))) (forall ((V_m_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_m_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) (forall ((V_v_2 $$unsorted) (V_u_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.c_Set_Oinsert _let_1))) (let ((_let_6 (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)) _let_2)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_u_2) V_v_2)) _let_2)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_u_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_v_2))) _let_2))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_x_2) V_y_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_x_2) V_y_2)))) (forall ((V_z_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 _let_2))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (let ((_let_5 (tptp.tc_fun tptp.tc_Nat_Onat _let_2))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_z_2) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_2) (tptp.hAPP tptp.c_Int_ORep__Integ V_z_2)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_5 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) tptp.tc_Nat_Onat) (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Nat_Onat _let_1) _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 tptp.tc_Nat_Onat) (tptp.c_Set_Oinsert _let_1))) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_1) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))))))))) (forall ((V_v_2 $$unsorted) (V_u_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)))) (let ((_let_4 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.c_Set_Oinsert _let_2))) (let ((_let_6 (tptp.c_Relation_OImage _let_2 _let_2 tptp.c_Int_Ointrel))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)) _let_3)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_u_2) V_v_2)) _let_3))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_v_2)) (tptp.hAPP (tptp.hAPP _let_1 V_u_2) V_y_2))))))))))) (forall ((V_v_2 $$unsorted) (V_u_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)))) (let ((_let_4 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.c_Set_Oinsert _let_2))) (let ((_let_6 (tptp.c_Relation_OImage _let_2 _let_2 tptp.c_Int_Ointrel))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)) _let_3)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_u_2) V_v_2)) _let_3))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_v_2)) (tptp.hAPP (tptp.hAPP _let_1 V_u_2) V_y_2))))))))))) (forall ((V_z_2 $$unsorted)) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (not (= V_z_2 (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) B_x) B_y)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))))))) _let_121 (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) V_c_2) (tptp.c_Relation_OId__on T_a V_A_2))) (not (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (not (= V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)))))))) (forall ((V_x_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool)) (tptp.hAPP tptp.c_Int_ORep__Integ V_x_2)) tptp.c_Int_OInteg))) (tptp.c_Typedef_Otype__definition tptp.tc_Int_Oint _let_119 tptp.c_Int_ORep__Integ tptp.c_Int_OAbs__Integ tptp.c_Int_OInteg) (forall ((V_code__numeral_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__case T_a V_f1_2 V_f2_2 (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_2)) (tptp.hAPP V_f2_2 V_code__numeral_2))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_a _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_a) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_a)))) V_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Wellfounded_Oacc T_a V_r_2)))))))))) (forall ((V_R2_2 $$unsorted) (V_R1_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_R1_2) V_R2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Wellfounded_Oacc T_a V_R2_2)) (tptp.c_Wellfounded_Oacc T_a V_R1_2))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_a V_r_2))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.c_Wellfounded_Oacc T_a V_r_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) V_a_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.c_Wellfounded_Oacc T_a V_r_2))))))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__case T_a V_f1_2 V_f2_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)) V_f1_2)) (forall ((V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_a T_b V_Rep_2 V_Abs_2 V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_Rep_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_A_2))) (forall ((V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_a T_b V_Rep_2 V_Abs_2 V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_Abs_2) V_A_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (=> (tptp.c_Typedef_Otype__definition T_a T_b V_Rep_2 V_Abs_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (= (= (tptp.hAPP V_Abs_2 V_x_2) (tptp.hAPP V_Abs_2 V_y_2)) (= V_x_2 V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_a T_b V_Rep_2 V_Abs_2 V_A_2) (= (= (tptp.hAPP V_Rep_2 V_x_2) (tptp.hAPP V_Rep_2 V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_a T_b V_Rep_2 V_Abs_2 V_A_2) (= (tptp.hAPP V_Abs_2 (tptp.hAPP V_Rep_2 V_x_2)) V_x_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_a T_b V_Rep_2 V_Abs_2 V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_Rep_2 V_x_2)) V_A_2)))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_a T_b V_Rep_2 V_Abs_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) V_A_2)) (= (tptp.hAPP V_Rep_2 (tptp.hAPP V_Abs_2 V_y_2)) V_y_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_c _let_3))) (let ((_let_5 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_prod T_a T_c))) (let ((_let_7 (tptp.tc_fun _let_6 tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_7 tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.tc_fun T_c _let_8))) (let ((_let_10 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_11 (tptp.tc_fun T_c _let_10))) (let ((_let_12 (tptp.tc_fun _let_5 _let_5))) (let ((_let_13 (tptp.tc_fun T_b _let_5))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) V_r_2) V_s_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) (tptp.hAPP (tptp.c_COMBB _let_5 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_c))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_13 _let_13) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_13 _let_13) T_a) (tptp.c_COMBB _let_5 _let_5 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 T_a) (tptp.c_COMBS T_c tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_11 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_10 T_c) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_7 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_7 _let_5) T_a) (tptp.c_COMBC T_c _let_7 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_6) _let_9 T_a) (tptp.hAPP (tptp.c_COMBB _let_6 _let_8 T_c) (tptp.c_member _let_6))) (tptp.c_Product__Type_OPair T_a T_c)))) V_r_2))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_b) (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) (tptp.c_member _let_1))) (tptp.hAPP (tptp.c_COMBC T_c T_b _let_1) (tptp.c_Product__Type_OPair T_c T_b))))) V_s_2)))))))))))))))))))) _let_118 (forall ((V_R_2 $$unsorted) (V_Y_2 $$unsorted) (V_X_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_X_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_Y_2)) (=> (not (= V_Y_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_X_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_Y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_xa)) V_R_2)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_X_2) V_Y_2)) (tptp.c_Wellfounded_Omax__ext T_a V_R_2)))))))))) (forall ((V_s_2 $$unsorted) (V_c_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_b_2) V_c_2)) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_c) V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_b T_c) V_r_2) V_s_2)))))) (forall ((V_r_2 $$unsorted) (V_I_2 $$unsorted) (T_d $$unsorted) (V_s_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) V_s_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_3) V_I_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_2) V_I_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_r_2))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (V_I_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_a T_c T_b))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun (tptp.tc_prod T_a T_c) tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_4) V_I_2) V_r_2)) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_2) V_I_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_2) T_d) _let_1) V_r_2)) V_s_2)))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_c) tptp.tc_HOL_Obool))) V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)))) (forall ((V_R_2 $$unsorted) (V_T_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_a T_c T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_c) tptp.tc_HOL_Obool)) V_S_2) V_T_2)) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_S_2) V_R_2)) (tptp.hAPP (tptp.hAPP _let_1 V_T_2) V_R_2))))) (forall ((V_T_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) V_R_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_S_2)) (tptp.hAPP _let_1 V_T_2))))) (forall ((V_s_2 $$unsorted) (V_s_H_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (V_r_H_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_a T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_r_H_2) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) V_s_H_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_c) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_r_H_2) V_s_H_2)) (tptp.hAPP (tptp.hAPP _let_1 V_r_2) V_s_2))))))) (forall ((V_T_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_d T_c) V_R_2) V_S_2)) V_T_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_d T_b) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_d T_c T_b) V_S_2) V_T_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_A_2))) (=> (= (tptp.hAPP _let_3 V_B_2) _let_2) (=> (= (tptp.hAPP _let_3 V_C_2) _let_2) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2) V_C_2)) _let_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Relation_Orel__comp T_a T_a T_a))) (=> (= (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2) _let_2) (=> (= (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2) _let_2) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) V_C_2) _let_2))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Omax__ext T_a V_R_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod _let_2 _let_2))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_R_2) V_S_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp _let_2 _let_2 _let_2) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_4) (tptp.c_Wellfounded_Omax__ext T_a V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_5) _let_5)) (tptp.c_Orderings_Obot__class_Obot _let_4))))) _let_1))))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Omin__ext T_a V_R_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod _let_2 _let_2))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_R_2) V_S_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp _let_2 _let_2 _let_2) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_4) (tptp.c_Wellfounded_Omin__ext T_a V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_5) _let_5)) (tptp.c_Orderings_Obot__class_Obot _let_4))))) _let_1))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_c _let_3))) (let ((_let_5 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_prod T_a T_b))) (let ((_let_7 (tptp.tc_fun _let_6 tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_7 tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.tc_fun T_b _let_8))) (let ((_let_10 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Opred__comp T_a T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_7 _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_7 _let_10) T_a) (tptp.c_COMBC T_b _let_7 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_6) _let_9 T_a) (tptp.hAPP (tptp.c_COMBB _let_6 _let_8 T_b) (tptp.c_member _let_6))) (tptp.c_Product__Type_OPair T_a T_b)))) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_b) (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_b T_c)))) V_s_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_c) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_b T_c) V_r_2) V_s_2))))))))))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_R_2) V_S_2)) V_R_2)) (tptp.c_FunDef_Oreduction__pair T_a (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_R_2) V_S_2)))))) _let_117 (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than) (forall ((T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_c_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_s_2 V_b_2) V_c_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Opred__comp T_a T_b T_c V_r_2 V_s_2) V_a_2) V_c_2))))) (forall ((T_a $$unsorted)) (tptp.c_Wellfounded_Owf T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_a T_b) V_r_2) V_f_2)))) (forall ((V_rb_2 $$unsorted) (T_b $$unsorted) (V_ra_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_ra_2) (=> (tptp.c_Wellfounded_Owf T_b V_rb_2) (tptp.c_Wellfounded_Owf (tptp.tc_prod T_a T_b) (tptp.c_Wellfounded_Olex__prod T_a T_b V_ra_2 V_rb_2))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.c_Wellfounded_Omeasure T_a) V_f_2))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Owf T_a V_R_2) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_R_2) V_R_2)))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_a_2)) V_r_2))))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_x_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_a_2)) V_r_2)))))))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_x_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_a_2)) V_r_2)))))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_a_2)) V_r_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Omax__ext T_a V_r_2)))) (tptp.c_Wellfounded_Owf _let_93 tptp.c_FunDef_Opair__less) (forall ((V_a3_2 $$unsorted) (V_a1_2 $$unsorted) (V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Opred__comp T_a T_b T_c V_r_2 V_s_2) V_a1_2) V_a3_2)) (exists ((B_b $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a1_2) B_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_s_2 B_b) V_a3_2)))))) (forall ((V_p_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_p_2) V_r_2)) (tptp.c_Wellfounded_Owf T_a V_p_2)))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_r_H_2)))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_H_2) V_r_2)))) (forall ((T_a $$unsorted)) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Ofinite__psubset T_a))) (forall ((V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (tptp.c_Wellfounded_Owf T_a (tptp.c_Wellfounded_Omlex__prod T_a V_f_2 V_R_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Omin__ext T_a V_r_2)))) (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat (tptp.hAPP _let_111 (tptp.hAPP _let_110 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)))) (forall ((V_d_2 $$unsorted)) (tptp.c_Wellfounded_Owf tptp.tc_Int_Oint (tptp.c_Int_Oint__ge__less__than V_d_2))) (forall ((V_d_2 $$unsorted)) (tptp.c_Wellfounded_Owf tptp.tc_Int_Oint (tptp.c_Int_Oint__ge__less__than2 V_d_2))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Wellfounded_Oacc T_a V_r_2))))) _let_116 (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_R_2) V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (tptp.c_Wellfounded_Owf T_a V_R_2))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_a T_a T_a))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (= (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2)) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_R_2) V_R_2)) (tptp.hAPP (tptp.hAPP _let_1 V_S_2) V_R_2))) V_S_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Owf T_a V_r_2) (not (exists ((B_f $$unsorted)) (forall ((B_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) (tptp.hAPP B_f (tptp.hAPP tptp.c_Nat_OSuc B_i))) (tptp.hAPP B_f B_i))) V_r_2))))))) (forall ((V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_R_2) V_A_2))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (=> (tptp.c_Wellfounded_Owf T_a V_S_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) V_R_2) V_S_2)) V_R_2)) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_R_2) V_S_2))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Owellorder T_a) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless T_a)))))) _let_115 (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.c_FunDef_Oreduction__pair T_a V_P_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_R_2) (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) V_P_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_S_2) (tptp.hAPP (tptp.c_Product__Type_Osnd _let_1 _let_1) V_P_2))) (=> (tptp.c_Wellfounded_Owf T_a V_S_2) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_R_2) V_S_2))))))))) _let_114 _let_113 (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nitpick_Ofrac T_a) V_a) V_b) (tptp.c_Nitpick_OAbs__Frac T_a (tptp.c_Nitpick_Onorm__frac V_a V_b)))) _let_112 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_List_Olist T_a) (tptp.c_List_Olenlex T_a V_r_2)))) (forall ((T_a $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP V_R_2 B_x)))) (tptp.c_Wellfounded_Owf (tptp.tc_prod T_a T_b) (tptp.c_Recdef_Osame__fst T_a T_b V_P_2 V_R_2)))) (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_List_Olist T_a) (tptp.c_List_Olex T_a V_r_2)))) (forall ((V_n_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_List_Olist T_a) (tptp.hAPP (tptp.c_List_Olexn T_a V_r_2) V_n_2)))) (forall ((T_a $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a T_a))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 B_x))) (tptp.hAPP V_f_2 B_x))))) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) V_P_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) V_g_2))))))))) _let_109 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_List_Olexn T_a V_r_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_a tptp.tc_HOL_Obool) V_P_2))) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.c_COMBC T_b T_a tptp.tc_HOL_Obool) V_P_2))))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.c_Set_Oimage T_c _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b _let_3) T_c) (tptp.c_Product__Type_OPair T_a T_b)) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.hAPP (tptp.c_Set_Oimage T_c T_a V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) (tptp.c_Set_Oimage T_c T_b V_g_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 T_a) (tptp.c_Set_Ovimage T_c T_a V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_Set_Oinsert T_a)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) V_A_2)))))))) _let_108 (forall ((T_a $$unsorted)) (tptp.c_Wellfounded_OwfP T_a (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool T_a) tptp.c_fFalse)))) (forall ((V_B_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP V_B_2 V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2)))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Relation_ORange T_a T_b) V_r_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_b) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_b T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_C_2)) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) _let_1)) (= V_A_2 V_B_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_OwfP T_a V_r_2) (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_a V_r_2) B_x1))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_OwfP T_a V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_a V_r_2) V_x_2)))) (forall ((V_C_2 $$unsorted) (V_J_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_I_2) V_J_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_I_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_1 V_J_2) V_C_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) _let_1)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_I_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_Groups_Ominus__class_Ominus _let_2)) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_a T_a) (tptp.c_Relation_OId__on T_a V_A_2)) V_A_2)) (forall ((V_B_2 $$unsorted) (V_X_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_a T_b))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_3 V_X_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_3 _let_2) V_X_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) _let_2) _let_1) V_B_2))))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_a) (tptp.tc_fun _let_1 T_a) T_b) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_a)) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum (tptp.tc_prod T_b T_c) T_a) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_B_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (or (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBK _let_1 T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBK _let_1 T_a) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (let ((_let_3 (tptp.c_Product__Type_OSigma T_a T_b))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_3 V_A_2) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2)) _let_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_a) (tptp.tc_fun _let_1 T_a) T_b) (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_a)) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod (tptp.tc_prod T_b T_c) T_a) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_a) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_B_2))))))) (forall ((V_p_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_OwfP T_a V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_p_2) V_r_2)) (tptp.c_Wellfounded_OwfP T_a V_p_2)))) (forall ((V_C_2 $$unsorted) (V_J_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_I_2) V_J_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_I_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_1 V_J_2) V_C_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) _let_1)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_I_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2)) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_s_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_B_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_B_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_C_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_c) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_b T_c) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_C_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_b T_a) _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_a _let_1) (tptp.hAPP (tptp.c_COMBC T_a T_b _let_1) (tptp.c_Product__Type_OPair T_a T_b)))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_b) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_B_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_A_2))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_B_2))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_I_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2)) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) _let_1)))))) (forall ((V_C_2 $$unsorted) (V_J_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_I_2) V_J_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_I_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_1 V_J_2) V_C_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) (tptp.c_Relation_OId__on T_a V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_b T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) (tptp.c_Orderings_Otop__class_Otop _let_1))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP V_B_2 V_a_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP V_B_2 V_a_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2))) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP V_B_2 V_a_2)))))))) (forall ((V_S_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_b T_a))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) _let_1))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_r_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool))))) (forall ((T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_b T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_b)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Relation_ORange T_a T_b) V_r_2))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_b T_a) T_a (tptp.c_Product__Type_Osnd T_b T_a)) V_R_2) (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) V_R_2))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_A_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_b) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_b T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_C_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_B_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_a T_b V_r_2) V_C_2)) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) V_B_2)))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_B_2)))) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_a) tptp.c_fconj) V_P_2))) V_Q_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2)) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Q_2))))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_a) tptp.c_fconj) V_P_2))) V_Q_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.hAPP (tptp.c_Set_OCollect T_a) V_P_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.c_Set_OCollect T_b)) V_Q_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_b T_a) T_b (tptp.c_Product__Type_Ofst T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_B_2))))) (let ((_let_3 (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (not _let_3) (= _let_2 V_A_2))))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_a T_b) T_b (tptp.c_Product__Type_Osnd T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_B_2))))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_3) (= _let_2 V_B_2))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) V_r_2))) (exists ((B_y $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) B_y) V_a_2)) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_b T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_a_2) V_b_2)) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) (tptp.hAPP _let_1 V_r_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_b T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_a T_b))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) V_B_2)))) (let ((_let_5 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP _let_3 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_5 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_4)) (tptp.hAPP _let_3 (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_2 T_a) V_A_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) V_X_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_X_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a _let_1 V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_b (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 T_b T_a (tptp.c_Product__Type_Ofst T_b T_c)) V_f_2)) V_A_2)) (tptp.hAPP (tptp.c_Set_Ovimage T_a T_c (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 T_c T_a (tptp.c_Product__Type_Osnd T_b T_c)) V_f_2)) V_B_2))))) (forall ((V_F_2 $$unsorted) (V_E_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBK _let_1 T_a))) (let ((_let_3 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Product__Type_OSigma T_a T_b))) (let ((_let_5 (tptp.tc_fun T_a _let_1))) (let ((_let_6 (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun _let_5 _let_6))) (let ((_let_8 (tptp.tc_fun T_d _let_6))) (let ((_let_9 (tptp.tc_fun T_d _let_5))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR (tptp.tc_prod T_c T_d) _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_d) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_d tptp.tc_HOL_Obool) T_c) V_B_2))) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_9 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_9 _let_8) T_c) (tptp.c_COMBB _let_5 _let_6 T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_7 T_c) _let_4) V_E_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_d) _let_2) V_F_2)))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_3) V_A_2) V_E_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_1) V_B_2) V_F_2)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 _let_2))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b _let_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_5 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_5 _let_2) T_a) (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_2)) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_b _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_Set_Oinsert _let_1))) (tptp.c_Product__Type_OPair T_a T_b)))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_A_2)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_a) V_A_2) V_r_2))))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (tptp.c_Wellfounded_OwfP T_a (tptp.c_FunDef_Oin__rel T_a T_a V_R_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Wellfounded_OwfP T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_a _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_a) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_a)))) V_r_2)) (tptp.c_Wellfounded_Owf T_a V_r_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_1 _let_1))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_c))) (let ((_let_6 (tptp.tc_fun T_c T_c))) (let ((_let_7 (tptp.tc_fun T_b T_c))) (let ((_let_8 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_7 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_7 _let_7) T_a) (tptp.c_COMBB T_c T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_6 T_a) _let_5) V_f_2))) V_g_2))) (let ((_let_9 (tptp.tc_fun _let_7 _let_1))) (let ((_let_10 (tptp.tc_fun T_a _let_1))) (let ((_let_11 (tptp.tc_fun T_a _let_7))) (let ((_let_12 (tptp.tc_fun T_b _let_3))) (let ((_let_13 (tptp.tc_fun T_a _let_12))) (let ((_let_14 (tptp.tc_fun T_a _let_4))) (=> (tptp.class_Rings_Osemiring__0 T_c) (=> (tptp.c_Fun_Oinj__on (tptp.tc_prod T_a T_b) T_c (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) V_B_2))) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_c) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_c) (tptp.c_Fun_Oid T_c)) (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_c) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_2 T_c) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_10 _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_14 (tptp.tc_fun _let_10 _let_10) T_c) (tptp.c_COMBS T_a _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 _let_14 T_c) (tptp.hAPP (tptp.c_COMBB _let_12 _let_4 T_a) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_13 T_c) (tptp.hAPP (tptp.c_COMBB _let_1 _let_12 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_b) tptp.c_fconj))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_11 _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_11 _let_10) T_c) (tptp.c_COMBB _let_7 _let_1 T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) _let_9 T_c) (tptp.c_COMBB T_c tptp.tc_HOL_Obool T_b)) tptp.c_fequal))) _let_8))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_4 T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_B_2))))))))))))))))))))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ORangeP T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_b)))) V_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_Relation_ORange T_a T_b) V_r_2)))))))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (exists ((B_f $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_I_2) V_A_2))) (let ((_let_2 (tptp.tc_prod T_b T_a))) (and (tptp.c_Fun_Oinj__on T_a _let_2 B_f _let_1) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_a _let_2 B_f) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_a) V_I_2) V_A_2)))))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ORangeP T_a T_b V_r_2) V_b_2)))) (forall ((V_Y_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a (tptp.c_Fun_Oid T_a)) V_Y_2) V_Y_2)) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Product__Type_Oapfst T_a T_a T_b (tptp.c_Fun_Oid T_a)) (tptp.c_Fun_Oid (tptp.tc_prod T_a T_b)))) (forall ((T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Oapsnd T_b T_b T_a (tptp.c_Fun_Oid T_b)) (tptp.c_Fun_Oid (tptp.tc_prod T_a T_b)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_a T_a (tptp.c_Fun_Oid T_a)) V_A_2) V_A_2)) (forall ((V_x_1 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Oid T_a) V_x_1) V_x_1)) (forall ((V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Oid T_a) V_x) V_x)) (forall ((V_a2_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ORangeP T_a T_b V_r_2) V_a2_2)) (exists ((B_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 B_a) V_a2_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Fun_Oinj__on T_a T_a (tptp.c_Fun_Oid T_a) V_A_2)) (forall ((V_g_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_a (tptp.c_Fun_Oid T_b)) V_g_2) V_g_2)) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_b T_a V_f_2) (tptp.c_Fun_Oid T_a)) V_f_2)) (forall ((V_v_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_a_2) V_b_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_a (tptp.c_Fun_Oid T_b)) V_c_2)) (= (tptp.hAPP V_a_2 (tptp.hAPP V_b_2 V_v_2)) (tptp.hAPP V_c_2 V_v_2)))) (= (tptp.c_Int_Oring__1__class_Oof__int tptp.tc_Int_Oint) (tptp.c_Fun_Oid tptp.tc_Int_Oint)) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.c_Set_Oimage T_a T_a (tptp.c_Fun_Oid T_a)) _let_1) _let_1))) (forall ((V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_a T_b V_f_2 V_F_2) (= (tptp.hAPP V_F_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Fun_Oid T_b)))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b _let_1) (tptp.c_Product__Type_OPair T_a T_b)) (tptp.c_Fun_Oid _let_1)))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_b) (tptp.c_Fun_Oid T_b)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_b) (tptp.c_Fun_Oid T_b)) (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_B_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_1 T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_c) (tptp.tc_fun _let_1 T_c) T_a) (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c)) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod (tptp.tc_prod T_a T_b) T_c) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2)))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_1 T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_c) (tptp.tc_fun _let_1 T_c) T_a) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c)) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum (tptp.tc_prod T_a T_b) T_c) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2)))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_c))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_c))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_3 T_c))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_a _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_3 T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_c _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_c) (tptp.tc_fun T_c _let_4) T_a) (tptp.c_Finite__Set_Ofold__image T_c T_b _let_2)) V_g_2)) _let_1)) V_B_2)) _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c (tptp.tc_prod T_a T_b) _let_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b T_c) V_g_2)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_Nat_Onat T_a) (tptp.c_Finite__Set_Ocard T_b)) V_B_2)) V_A_2))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (=> (forall ((B_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP V_B_2 B_a))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) V_r_2))) (not (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) B_x) V_b_2)) V_r_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) V_A_2) V_B_2))) (not (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_y) (tptp.hAPP V_B_2 B_x))) (not (= V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) B_x) B_y)))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_OwfP T_a V_r_2) (tptp.c_Wellfounded_Oacyclic T_a (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) V_r_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.c_Wellfounded_Owf T_a V_s_2) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_a) V_r_2)) (tptp.hAPP (tptp.c_Relation_ORange T_a T_a) V_s_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_r_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_a) (tptp.c_Relation_OId__on T_a V_A_2)) V_A_2)) (forall ((V_r_2 $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Oacyclic T_a V_s_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.c_Wellfounded_Oacyclic T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_r_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 V_s_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_a T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_b)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_r_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_a T_b) T_a (tptp.c_Product__Type_Ofst T_a T_b)) V_R_2) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_R_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_r_2))) (exists ((B_y $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) B_y)) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_a T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) (tptp.hAPP _let_1 V_r_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_a)) V_r_2)) (=> (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (tptp.c_Wellfounded_Owf T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_a)) V_r_2)) (= (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Oacyclic T_a V_r_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_a T_b))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_a T_b))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OField T_a) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_a) V_r_2)) (tptp.hAPP (tptp.c_Relation_ORange T_a T_a) V_r_2)))) (forall ((V_S_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_a T_b))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_S_2) _let_1))))) (forall ((V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_P_2))) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx T_b)) V_P_2)))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ODomainP T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_b)))) V_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_r_2)))))))))) _let_107 (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_r_2))) (not (forall ((B_y $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) B_y)) V_r_2))))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ODomainP T_a T_b V_r_2) V_a_2)))) (forall ((V_a1_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ODomainP T_a T_b V_r_2) V_a1_2)) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a1_2) B_x1))))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_c))) (let ((_let_2 (tptp.tc_prod T_b T_d))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_d T_c V_g_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Oimage _let_2 _let_1 (tptp.c_Product__Type_Omap__pair T_b T_a T_d T_c V_f_2 V_g_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) (forall ((V_B_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (tptp.c_Fun_Oinj__on T_c T_d V_g_2 V_B_2) (tptp.c_Fun_Oinj__on (tptp.tc_prod T_a T_c) (tptp.tc_prod T_b T_d) (tptp.c_Product__Type_Omap__pair T_a T_b T_c T_d V_f_2 V_g_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_a) V_B_2)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_d))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP V_g_2 V_b_2))) (tptp.hAPP (tptp.c_Set_Oimage _let_2 _let_1 (tptp.c_Product__Type_Omap__pair T_a T_c T_b T_d V_f_2 V_g_2)) V_R_2))))))) (forall ((V_prod_2 $$unsorted) (V_i_2 $$unsorted) (V_h_2 $$unsorted) (T_f $$unsorted) (T_e $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_c T_a T_d T_b V_f_2 V_g_2) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_e T_c T_f T_d V_h_2 V_i_2) V_prod_2)) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_e T_a T_f T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_a T_e V_f_2) V_h_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_b T_f V_g_2) V_i_2)) V_prod_2))) (forall ((V_g2_2 $$unsorted) (V_g1_2 $$unsorted) (T_f $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (= (tptp.c_Product__Type_Omap__pair T_a T_c T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_c T_a V_f1_2) V_f2_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_d T_b V_g1_2) V_g2_2)) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_e T_f) (tptp.tc_prod T_c T_d) (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_Omap__pair T_e T_c T_f T_d V_f1_2 V_g1_2)) (tptp.c_Product__Type_Omap__pair T_a T_e T_b T_f V_f2_2 V_g2_2)))) (forall ((V_i_2 $$unsorted) (V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_f $$unsorted) (T_e $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_e T_f) (tptp.tc_prod T_c T_d) (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_Omap__pair T_e T_c T_f T_d V_f_2 V_g_2)) (tptp.c_Product__Type_Omap__pair T_a T_e T_b T_f V_h_2 V_i_2)) (tptp.c_Product__Type_Omap__pair T_a T_c T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_c T_a V_f_2) V_h_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_d T_b V_g_2) V_i_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_d T_c) T_c _let_1 (tptp.c_Product__Type_Osnd T_d T_c)) (tptp.c_Product__Type_Omap__pair T_a T_d T_b T_c V_f_2 V_g_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c _let_1 V_g_2) (tptp.c_Product__Type_Osnd T_a T_b))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_c T_d) T_c _let_1 (tptp.c_Product__Type_Ofst T_c T_d)) (tptp.c_Product__Type_Omap__pair T_a T_c T_b T_d V_f_2 V_g_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_c _let_1 V_f_2) (tptp.c_Product__Type_Ofst T_a T_b))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_c T_a T_d T_b V_f_2 V_g_2) V_x_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_d) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_a) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_c T_b T_d T_a V_f_2 V_g_2) V_x_2)) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) V_x_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_c T_a T_d T_b V_f_2 V_g_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP V_g_2 V_b_2)))) (forall ((V_z_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_a T_a T_b T_b (tptp.c_COMBI T_a) (tptp.c_COMBI T_b)) V_z_2) V_z_2)) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Product__Type_Omap__pair T_a T_a T_b T_b (tptp.c_COMBI T_a) (tptp.c_COMBI T_b)) (tptp.c_Fun_Oid (tptp.tc_prod T_a T_b)))) _let_106 _let_105 (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Product__Type_Omap__pair T_a T_a T_b T_b (tptp.c_Fun_Oid T_a) (tptp.c_Fun_Oid T_b)) (tptp.c_Fun_Oid (tptp.tc_prod T_a T_b)))) _let_104 (forall ((V_B_H_2 $$unsorted) (V_B_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (V_A_H_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) V_A_2) V_A_H_2) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_d T_c V_g_2) V_B_2) V_B_H_2) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_b T_d) (tptp.tc_prod T_a T_c) (tptp.c_Product__Type_Omap__pair T_b T_a T_d T_c V_f_2 V_g_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_d) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_d tptp.tc_HOL_Obool) T_b) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_c) V_A_H_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_a) V_B_H_2)))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_a T_a) (tptp.tc_prod T_b T_b) (tptp.c_Product__Type_Omap__pair T_a T_b T_a T_b V_f_2 V_f_2)) V_r_2))))) (forall ((V_R_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_c_2) (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_c T_d) _let_1 (tptp.c_Product__Type_Omap__pair T_c T_a T_d T_b V_f_2 V_g_2)) V_R_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (= V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_y))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) B_x) B_y)) V_R_2))))))))) _let_103 (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Orefl__on T_a V_A_2 (tptp.c_Relation_OId__on T_a V_A_2))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Orefl__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_s_2 $$unsorted) (V_B_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.c_Relation_Orefl__on T_a V_B_2 V_s_2) (tptp.c_Relation_Orefl__on T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))) (forall ((V_s_2 $$unsorted) (V_B_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.c_Relation_Orefl__on T_a V_B_2 V_s_2) (tptp.c_Relation_Orefl__on T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_y_2) V_A_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_a_2)) V_r_2))))) _let_102 (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) V_A_2)))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) V_r_2)))) (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2)))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Predicate_Oreflp T_a V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_x_2) V_x_2)))) (forall ((V_P_2 $$unsorted) (V_y_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool)) V_y_2) tptp.c_Int_OInteg)) (=> (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_ORep__Integ B_x)))) (tptp.hBOOL (tptp.hAPP V_P_2 V_y_2))))) (forall ((V_y_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool)) V_y_2) tptp.c_Int_OInteg)) (not (forall ((B_x $$unsorted)) (not (= V_y_2 (tptp.hAPP tptp.c_Int_ORep__Integ B_x))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (and (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) B_x) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_a) tptp.c_fconj) _let_2))) _let_2)) B_x))))))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) V_r_2))))))) (forall ((V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool)) B_y) tptp.c_Int_OInteg)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP tptp.c_Int_OAbs__Integ B_y))))) (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))))) _let_101 (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_OPowp T_a V_A_2) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_B_2)) (tptp.hBOOL (tptp.hAPP V_A_2 B_x)))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((V_P_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_3 T_a) (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))) V_P_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) _let_1) (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) V_P_2)))))))) _let_100 (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_a) (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) T_a) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b)) (tptp.c_member T_a))) V_B_2))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a V_A_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))) (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq T_a)))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a V_A_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_a (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)))) (tptp.c_Orderings_Oord__class_Oless__eq T_a))))))))) (forall ((V_x_2 $$unsorted)) (not (forall ((B_y $$unsorted)) (=> (= V_x_2 (tptp.hAPP tptp.c_Int_OAbs__Integ B_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool)) B_y) tptp.c_Int_OInteg))))))) (forall ((T_a $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_i $$unsorted)) (tptp.c_Wellfounded_OwfP T_b (tptp.hAPP V_r_2 B_i))) (=> (forall ((B_i $$unsorted) (B_j $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP V_r_2 B_j))) (let ((_let_3 (tptp.hAPP V_r_2 B_i))) (=> (not (= _let_3 _let_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_Predicate_ODomainP T_b T_b _let_3)) (tptp.c_Predicate_ORangeP T_b T_b _let_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))) (tptp.c_Wellfounded_OwfP T_b (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_a (tptp.tc_fun T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_r_2))))) (forall ((V_n_2 $$unsorted) (V_ts_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple T_a)) B_x) V_ts_2)) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a (tptp.hAPP tptp.c_Nat_OSuc V_n_2) B_x))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_Hoare__Mirabelle_Otriple T_a)) B_x) V_ts_2)) (tptp.c_Hoare__Mirabelle_Otriple__valid T_a V_n_2 B_x))))) _let_99 (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_a) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.c_member T_a)))))))) (forall ((V_c_2 $$unsorted) (V_z_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_p_2 $$unsorted)) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (=> (= V_p_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) B_a) B_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_z_2) (tptp.hAPP (tptp.hAPP V_c_2 B_a) B_b))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_c_2) V_p_2))))) (forall ((V_p_2 $$unsorted) (V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_z_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_c_2) V_p_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (= V_p_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) B_x) B_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_z_2) (tptp.hAPP (tptp.hAPP V_c_2 B_x) B_y))))))))) (forall ((V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.c_Hilbert__Choice_OEps _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_P_2)) (tptp.c_Hilbert__Choice_OEps _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool) _let_1) V_P_2) (tptp.c_Product__Type_Ofst T_a T_b))) (tptp.c_Product__Type_Osnd T_a T_b)))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_A $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) B_A) (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_R_2) B_A))) (= B_A (tptp.c_Orderings_Obot__class_Obot _let_1))))) (tptp.c_Wellfounded_Owf T_a V_R_2))) (forall ((T_a $$unsorted) (V_c_2 $$unsorted) (V_P_2 $$unsorted)) (=> (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (=> (= V_c_2 (tptp.c_Hilbert__Choice_OEps T_a V_P_2)) (tptp.hBOOL (tptp.hAPP V_P_2 V_c_2))))) (forall ((T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Hilbert__Choice_OEps T_a V_P_2))))) (forall ((T_a $$unsorted) (B_P $$unsorted)) (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP B_P B_x))) (tptp.hBOOL (tptp.hAPP B_P (tptp.c_Hilbert__Choice_OEps T_a B_P))))) (forall ((T_a $$unsorted) (V_P_2 $$unsorted)) (=> (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Hilbert__Choice_OEps T_a V_P_2))))) (forall ((T_a $$unsorted) (V_P_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Hilbert__Choice_OEps T_a V_P_2))) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Hilbert__Choice_OEps T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_x_2)) V_x_2)) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Hilbert__Choice_OEps T_a (tptp.hAPP tptp.c_fequal V_x_2)) V_x_2)) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Hilbert__Choice_OEps (tptp.tc_prod T_a T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 T_a) tptp.c_fconj) (tptp.hAPP tptp.c_fequal V_x_2)))) (tptp.hAPP tptp.c_fequal V_y_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2))))) (forall ((V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.c_Hilbert__Choice_OEps _let_1 V_P_2) (tptp.c_Hilbert__Choice_OEps _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) V_P_2)) (tptp.c_Product__Type_OPair T_a T_b))))))) (forall ((V_P_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_a T_b))) (let ((_let_4 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a))) (let ((_let_5 (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.c_Hilbert__Choice_OLeastM T_a T_b V_m_2 V_P_2) (tptp.c_Hilbert__Choice_OEps T_a (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 tptp.c_fconj) V_P_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_a) (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 tptp.c_fimplies) V_P_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_2) T_a) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_1 T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_b)) V_m_2))) V_m_2))))))))))))) (forall ((V_P_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a))) (let ((_let_3 (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hAPP (tptp.c_Hilbert__Choice_OGreatestM T_a T_b V_m_2) V_P_2) (tptp.c_Hilbert__Choice_OEps T_a (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 tptp.c_fconj) V_P_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OAll T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 tptp.c_fimplies) V_P_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_1 T_a) (tptp.hAPP (tptp.c_COMBC T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_b)) V_m_2))) V_m_2))))))))))) (forall ((V_m_2 $$unsorted) (T_a $$unsorted) (V_k_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_k_2)) (and (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Hilbert__Choice_OLeastM T_a tptp.tc_Nat_Onat V_m_2 V_P_2))) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP V_m_2 (tptp.c_Hilbert__Choice_OLeastM T_a tptp.tc_Nat_Onat V_m_2 V_P_2))) (tptp.hAPP V_m_2 B_y)))))))) (forall ((T_a $$unsorted) (V_m_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP V_m_2 (tptp.c_Hilbert__Choice_OLeastM T_a tptp.tc_Nat_Onat V_m_2 V_P_2))) (tptp.hAPP V_m_2 V_x_2))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (= (tptp.c_Hilbert__Choice_OGreatest T_a) (tptp.c_Hilbert__Choice_OGreatestM T_a T_a (tptp.c_COMBI T_a))))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_C_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP V_B_2 B_x)) (tptp.hAPP V_D_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 V_C_2) V_D_2))))))) (forall ((V_m_2 $$unsorted) (T_a $$unsorted) (V_k_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_k_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Hilbert__Choice_OLeastM T_a tptp.tc_Nat_Onat V_m_2 V_P_2))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_y) V_x_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_y) (tptp.c_Wellfounded_Oacc T_a V_r_2))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Wellfounded_Oacc T_a V_r_2))))) (forall ((V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Wellfounded_Oacc T_a V_R_2)))) (not (forall ((B_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_z) V_x_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_z) (tptp.c_Wellfounded_Oacc T_a V_R_2)))))))) (forall ((V_R_2 $$unsorted) (V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_a1_2) V_a2_2)) (tptp.c_Wellfounded_Omax__ext T_a V_R_2))) (and (tptp.hBOOL (tptp.hAPP _let_2 V_a1_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_a2_2)) (not (= V_a2_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_a1_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) V_a2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_xa)) V_R_2))))))))))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (not (forall ((B_k $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) (tptp.hAPP V_f_2 (tptp.hAPP tptp.c_Nat_OSuc B_k))) (tptp.hAPP V_f_2 B_k))) V_r_2)))))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted)) (= (tptp.c_Com_Ocom_Ocom__size (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Com_Ocom_Ocom__size V_com_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize tptp.tc_Com_Ocom))) (= (tptp.hAPP _let_1 (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_com_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_c_2 $$unsorted) (V_a_2 $$unsorted) (V_Y_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OLocal V_Y_2 V_a_2 V_c_2))) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c_2)))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted) (V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2) (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2)))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2) (tptp.c_Com_Ocom_OWhile V_fun_H_2 V_com_H_2)))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_loc_H_2 $$unsorted) (V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted)) (= (= (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2) (tptp.c_Com_Ocom_OLocal V_loc_H_2 V_fun_H_2 V_com_H_2)) (and (= V_loc_2 V_loc_H_2) (= V_fun_2 V_fun_H_2) (= V_com_2 V_com_H_2)))) (forall ((V_pname_H_2 $$unsorted) (V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2) (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H_2)))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted) (V_pname_H_2 $$unsorted)) (not (= (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pname_H_2) (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2)))) (forall ((V_a_2 $$unsorted) (V_Y_2 $$unsorted) (V_c_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_c_2)) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OLocal V_Y_2 V_a_2 V_c_2))))) (forall ((V_a1_2 $$unsorted) (V_a3_2 $$unsorted) (V_a2_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.c_Com_Ocom_OLocal V_a2_2 V_a3_2 V_a1_2))) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_a1_2)))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_loc_H_2 $$unsorted)) (not (= tptp.c_Com_Ocom_OSKIP (tptp.c_Com_Ocom_OLocal V_loc_H_2 V_fun_H_2 V_com_H_2)))) (forall ((V_com_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_loc_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OLocal V_loc_H_2 V_fun_H_2 V_com_H_2) tptp.c_Com_Ocom_OSKIP))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted) (V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OSemi V_com1_H_2 V_com2_H_2) (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2)))) (forall ((V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2) (tptp.c_Com_Ocom_OSemi V_com1_H_2 V_com2_H_2)))) (forall ((V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_fun_H_2 $$unsorted) (V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2) (tptp.c_Com_Ocom_OCond V_fun_H_2 V_com1_H_2 V_com2_H_2)))) (forall ((V_com_2 $$unsorted) (V_fun_2 $$unsorted) (V_loc_2 $$unsorted) (V_com2_H_2 $$unsorted) (V_com1_H_2 $$unsorted) (V_fun_H_2 $$unsorted)) (not (= (tptp.c_Com_Ocom_OCond V_fun_H_2 V_com1_H_2 V_com2_H_2) (tptp.c_Com_Ocom_OLocal V_loc_2 V_fun_2 V_com_2)))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 V_A_2) (=> (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_a T_b V_f_2) V_A_2)) (=> (forall ((B_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_a) V_A_2)) (= (tptp.hAPP V_g_2 B_a) (tptp.hAPP V_h_2 (tptp.hAPP V_f_2 B_a))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_h_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_c) V_g_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_c T_d T_a) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_d) _let_1)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) _let_1))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_f $$unsorted) (T_e $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_f T_b))) (let ((_let_2 (tptp.tc_fun T_c (tptp.tc_fun T_d T_b)))) (let ((_let_3 (tptp.tc_prod T_c T_d))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_a T_c T_d T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_a T_e T_f _let_3) V_f_2) V_g_2)) V_h_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_a T_e T_f T_b) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_e _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_f _let_3) (tptp.tc_fun _let_2 _let_1) T_e) (tptp.c_Product__Type_Oscomp T_f T_c T_d T_b)) V_g_2)) V_h_2))))))) (forall ((V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_a T_c T_a T_b) (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_a) V_x_2)) V_f_2) (tptp.hAPP V_f_2 V_x_2))) (forall ((V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_a T_b T_c (tptp.tc_prod T_b T_c)) V_x_2) (tptp.c_Product__Type_OPair T_b T_c)) V_x_2)) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_c T_d T_a) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_a) V_g_2) (tptp.hAPP V_f_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_c T_d T_a) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_a) V_g_2) (tptp.hAPP V_f_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Random_Oiterate T_b T_a V_k_2 V_f_2) V_x_2))) (let ((_let_2 (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_x_2))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_a T_b T_a (tptp.tc_prod T_b T_a)) (tptp.hAPP V_f_2 V_x_2)) (tptp.c_Random_Oiterate T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_k_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)) V_f_2)))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Orderings_Oord T_a)) (=> (= V_a_2 V_c_2) (=> (= V_b_2 V_d_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_c_2) B_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) B_x) V_d_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))))) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_a V_c_2) V_d_2))))))))) (forall ((V_i $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_2 (tptp.c_Random_Olog V_b V_i))) (let ((_let_3 (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) V_i) V_b))))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Code__Numeral_Ocode__numeral) _let_1) (tptp.c_Random_Olog V_b (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Code__Numeral_Ocode__numeral) V_i) V_b)))))))))) (forall ((V_r $$unsorted) (V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_2 (tptp.c_Random_Ominus__shift V_r V_k V_l))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) V_k) V_l)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Code__Numeral_Ocode__numeral) V_r) V_k)) V_l))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l)))))))) (forall ((V_k $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_2 (tptp.c_Random_Oinc__shift V_v V_k))) (let ((_let_3 (= V_v V_k))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Code__Numeral_Ocode__numeral) V_k) _let_1)))))))) (forall ((V_s $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP (tptp.c_Random_Orange V_k) V_s))) V_k))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_m_2) V_n_2)) (tptp.c_Transitive__Closure_Otrancl tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (forall ((B_i $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) B_i)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_n_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (tptp.hAPP V_f_2 B_i)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_n_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_m_2)) V_k_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) (tptp.hAPP V_f_2 V_n_2))) (exists ((B_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) B_i)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_i) V_n_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2)))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf T_a (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))) _let_98 _let_97 (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Relation_OField T_a) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) _let_1) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) _let_1)))))) (forall ((V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (= (tptp.c_Transitive__Closure_Otrancl T_a _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_a T_a))) (= (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (tptp.hAPP _let_1 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_a T_a))) (= (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (tptp.hAPP _let_1 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_a)))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_r_2))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_b_2)) V_R_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c_2)) (tptp.c_Transitive__Closure_Otrancl T_a V_R_2))))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_a V_s_2)))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) _let_1) V_r_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) V_A_2)))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) _let_1)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r_2) V_s_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) _let_1) V_s_2)) V_r_2)) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_s_2)))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.tc_prod T_a T_a))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 tptp.tc_HOL_Obool) T_a) (tptp.c_member _let_3)))) (let ((_let_6 (tptp.c_COMBC T_a _let_4 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Transitive__Closure_Otrancl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_3) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_x_2)) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_4) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (tptp.hAPP (tptp.c_Set_OCollect _let_3) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_8 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_8 _let_8) T_a) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_7 T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a _let_3) _let_2) V_y_2))) _let_1)))) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_2 V_x_2))) _let_1))))))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_4 (tptp.c_member _let_1))) (let ((_let_5 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_fun T_a _let_5))) (let ((_let_7 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a _let_7)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_2 _let_7) T_a) (tptp.c_COMBC T_a _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_1) _let_6 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_a) _let_4)) _let_3))) V_r_2)) tptp.c_fequal) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_r_2) (tptp.c_Relation_OId T_a))))))))))))) (forall ((V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_a_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_a_2)) (tptp.c_Relation_OId T_a)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))))) _let_96 (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Otrancl T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (tptp.c_Relation_OId T_a)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Otrancl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_a))) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (= (tptp.c_Transitive__Closure_Otrancl T_a _let_1) _let_1))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Transitive__Closure_Otrancl T_a V_R_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (forall ((V_r_2 $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Relation_OId T_a)) V_s_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) _let_1) V_s_2)) V_r_2)) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_s_2)))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_s_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) _let_1)))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_S_2) _let_1)) (= (tptp.c_Transitive__Closure_Ortrancl T_a V_S_2) _let_1)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_s_2)))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_S_2))) (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2))))))) (forall ((V_X_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r_2) V_X_2)) V_X_2)) (= (tptp.hAPP (tptp.c_Relation_OImage T_a T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) V_X_2) V_X_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Orefl__on T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_a))) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (= (tptp.c_Transitive__Closure_Ortrancl T_a _let_1) _let_1))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_R_2) (tptp.c_Relation_OId T_a))) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_S_2))) (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP _let_1 V_R_2) V_S_2))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) _let_1) (tptp.c_Relation_OId T_a)) _let_1))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) V_x_2))) (=> (or (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_S_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_R_2) V_S_2)))))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) _let_1) _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Relation_Orel__comp T_a T_a T_a))) (= (tptp.hAPP (tptp.hAPP _let_2 V_r_2) _let_1) (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) (tptp.c_Relation_OId T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) _let_1) V_r_2))))) _let_95 (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_a T_a) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_a) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_b T_b) V_R_2) (tptp.c_Relation_OId T_b)) V_R_2)) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_b) (tptp.c_Relation_OId T_a)) V_R_2) V_R_2)) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_a T_a (tptp.c_Relation_OId T_a)) V_A_2) V_A_2)) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) (tptp.c_Relation_OId T_a))) (= V_a_2 V_b_2))) (forall ((V_z_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_x_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_z_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_z_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_b_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c_2)) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))))))))) (forall ((V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)))) (let ((_let_2 (= V_a_2 V_b_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (or _let_2 (and (not _let_2) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_R_2))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_b_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c_2)) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))))))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)))) (let ((_let_2 (= V_x_2 V_y_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (or _let_2 (and (not _let_2) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_R_2))))))))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_a) (tptp.c_Relation_OId T_a)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_a T_a) (tptp.c_Relation_OId T_a)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Orefl__on T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Relation_OId T_a))) _let_94 (forall ((V_y_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_a) V_R_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (= V_x_2 V_y_2)))) (forall ((V_b_2 $$unsorted) (V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_a V_r_2))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_b_2) V_a_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_a V_r_2))) (let ((_let_2 (tptp.c_member T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_b_2) V_a_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (= (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) V_r_2)) (and (tptp.c_Wellfounded_Owf T_a V_r_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (= (tptp.c_Wellfounded_Oacyclic T_a (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) V_r_2)) (and (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Orefl__on T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_a)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_a _let_1))) (= (tptp.c_Relation_OId T_a) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool _let_1) (tptp.c_HOL_OEx T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) _let_1) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a _let_1) (tptp.c_Product__Type_OPair T_a T_a)) (tptp.c_COMBI T_a)))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Oirrefl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_a)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_m_2) V_n_2)) (tptp.c_Transitive__Closure_Ortrancl tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) V_A_2)))) (or (= V_a_2 V_b_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (tptp.c_Orderings_Obot__class_Obot _let_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_x_2)) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (V_r_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) (tptp.hAPP V_f_2 (tptp.hAPP tptp.c_Nat_OSuc B_i))) (tptp.hAPP V_f_2 B_i))) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i_2) V_k_2))) (tptp.hAPP V_f_2 V_i_2))) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_a T_a))) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OId T_a))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (V_S_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_a))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_3) V_S_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Orefl__on T_a V_A_2 (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2))) (forall ((V_r_2 $$unsorted) (V_S_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_a))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_3) V_S_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_2) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_b_2) V_a_2)) V_r_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_b_2) V_a_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_b) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_b_2) V_a_2)) V_r_2)))) (forall ((V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oinv__image T_b T_a))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) (tptp.hAPP (tptp.hAPP _let_1 V_R_2) V_f_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_R_2)) V_f_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) (tptp.hAPP _let_1 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_a)) V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_b) V_r_2)) V_r_2)) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 V_s_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OId__on T_a V_A_2))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) _let_1) _let_1))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 V_s_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_c T_a) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_a) V_s_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_c) V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Oacyclic T_a (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) (tptp.c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_a T_a))) (= (tptp.c_Transitive__Closure_Otrancl T_a (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_a T_a))) (=> (tptp.c_Wellfounded_Owf T_a (tptp.hAPP _let_1 V_r_2)) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) V_r_2) V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) V_r_2) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2)) (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_a) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_b) V_r_2)) (tptp.hAPP (tptp.c_Relation_ODomain T_a T_b) V_r_2))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_a T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_a (tptp.hAPP _let_1 V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_a T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_a (tptp.hAPP _let_1 V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_b T_a V_r_2) V_A_2)) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_2) (tptp.hAPP (tptp.c_Relation_OImage T_a T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_B_2))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) V_r_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) V_r_2) V_r_2) (=> (= (tptp.hAPP (tptp.c_Relation_ODomain T_a T_a) V_r_2) V_A_2) (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_a T_a)) V_r_2)) (=> (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.hAPP (tptp.c_COMBC T_b T_a _let_1) (tptp.c_Product__Type_OPair T_b T_a))))) V_r_2)))))))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_b T_a V_r_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Relation_Osingle__valued T_a T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_c) _let_1) V_B_2))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Ototal__on T_a V_A_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_a))) (tptp.c_Relation_Ototal__on T_a V_A_2 V_r_2))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Osingle__valued T_a T_a (tptp.c_Relation_OId T_a))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Osingle__valued T_a T_a (tptp.c_Relation_OId__on T_a V_A_2))) (forall ((V_s_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Osingle__valued T_a T_b V_r_2) (=> (tptp.c_Relation_Osingle__valued T_b T_c V_s_2) (tptp.c_Relation_Osingle__valued T_a T_c (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_b T_c) V_r_2) V_s_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (=> (tptp.c_Relation_Osingle__valued T_a T_b V_s_2) (tptp.c_Relation_Osingle__valued T_a T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Ototal__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_r_2)) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Ototal__on T_a V_A_2 (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) (tptp.c_Relation_Ototal__on T_a V_A_2 V_r_2))) (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 (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_b)))) (=> (tptp.c_Relation_Osingle__valued T_a T_b V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z_2)) V_r_2)) (= V_y_2 V_z_2))))))) _let_92 _let_91 (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_a T_a)))) (let ((_let_4 (tptp.hAPP _let_2 V_x_2))) (=> (tptp.c_Relation_Osingle__valued T_a T_a V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_4 V_y_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_4 V_z_2)) _let_1)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_z_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_z_2) V_y_2)) _let_1))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_b T_a V_R_2))) (=> (tptp.c_Relation_Osingle__valued T_a T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_R_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_R_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_member _let_1) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool)) V_R_2))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))))) (tptp.c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((V_n_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Osingle__valued T_a T_a V_R_2) (tptp.c_Relation_Osingle__valued T_a T_a (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_n_2) V_R_2)))) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 _let_1) V_n_2) (tptp.c_Fun_Ocomp T_b T_b T_a V_f_2)) (tptp.c_Fun_Ocomp T_b T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow T_b T_b) V_n_2) V_f_2))))) (forall ((V_x_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow T_a T_a) V_n_2) V_f_2))) (= (tptp.hAPP V_f_2 (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_x_2))))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow T_a T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_n_2) (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_f_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_n_2)) V_f_2)))) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_n_2) V_R_2)) (tptp.c_Wellfounded_Owf T_a V_R_2))) (forall ((V_n_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_n_2) V_R_2))) (let ((_let_2 (tptp.c_Relation_Orel__comp T_a T_a T_a))) (= (tptp.hAPP (tptp.hAPP _let_2 V_R_2) _let_1) (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_R_2))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_R_2) V_R_2)) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_n_2) V_R_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))))))) (forall ((V_R_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) V_p_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (exists ((B_n $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_p_2) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) B_n) V_R_2))))))) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_R_2)) V_R_2)))) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2)) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_a T_a) (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_R_2)) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_R_2))))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_R_2) (tptp.c_Relation_OId T_a))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_R_2))) (= V_x_2 V_y_2)))) (forall ((V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_R_2))))) (forall ((V_n_2 $$unsorted) (V_z_2 $$unsorted) (V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_4 (tptp.hAPP _let_3 V_x_2))) (let ((_let_5 (tptp.c_member _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_y_2)) V_R_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_z_2)) (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_R_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_z_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_R_2))))))))))) (forall ((V_z_2 $$unsorted) (V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_4 (tptp.hAPP _let_3 V_x_2))) (let ((_let_5 (tptp.c_member _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_y_2)) (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_R_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_z_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_z_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_R_2))))))))))) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow T_a T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_a T_a V_f_2) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_f_2))))) (forall ((V_f_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow T_a T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2)) V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_a T_a (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_f_2)) (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_f_2))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow T_a T_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_f_2) (tptp.c_Fun_Oid T_a))) (forall ((V_R_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) V_p_2) (tptp.c_Transitive__Closure_Otrancl T_a V_R_2))) (exists ((B_n $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) B_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_p_2) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) B_n) V_R_2)))))))) _let_90 _let_89 (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_n_2) V_R_2))) (=> (=> (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (not (= V_x_2 V_z_2))) (not (forall ((B_y $$unsorted) (B_m $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (let ((_let_3 (tptp.c_member _let_2))) (=> (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc B_m)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_2 tptp.tc_HOL_Obool) B_m) V_R_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 B_y) V_z_2)) V_R_2)))))))))))))) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_n_2) V_R_2))) (=> (=> (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (not (= V_x_2 V_z_2))) (not (forall ((B_y $$unsorted) (B_m $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.c_member _let_1))) (=> (= V_n_2 (tptp.hAPP tptp.c_Nat_OSuc B_m)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) B_y)) V_R_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 B_y) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) B_m) V_R_2))))))))))))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_P_2) V_Q_2)))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) B_x)) (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2))) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_y)) V_Q_2)) (= B_x B_y))))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2)))))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_P_2) V_Q_2)))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) V_b_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2))) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_y) B_x)) V_Q_2)) (= B_y B_x))))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2)))))))) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_R_2))) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (let ((_let_3 (tptp.c_member _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_2 tptp.tc_HOL_Obool) V_n_2) V_R_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 B_y) V_z_2)) V_R_2)))))))))))) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_R_2))) (exists ((B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.c_member _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) B_y)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 B_y) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_n_2) V_R_2))))))))))) (forall ((V_R_2 $$unsorted) (V_n_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)) V_R_2))) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_3 (tptp.c_member _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) B_y)) V_R_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 B_y) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_n_2) V_R_2))))))))))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Otrancl T_a V_R_2))) (exists ((B_z $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_z)) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_z) V_y_2)) V_R_2)))))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Otrancl T_a V_R_2))) (exists ((B_z $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_z)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_z) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))))))))) (forall ((V_p_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) V_p_2) (tptp.c_Relation_OId T_a))) (not (forall ((B_x $$unsorted)) (not (= V_p_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Owf T_a V_r_2) (forall ((B_Q $$unsorted)) (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) B_Q))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) B_Q)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_y) B_x)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_y) B_Q))))))))))) (forall ((V_P_H_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))) (let ((_let_2 (tptp.c_Set_OCollect tptp.tc_Int_Oint))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (= (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_H_2 B_x))))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_P_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_P_H_2))))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_A_2) V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Q_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) (tptp.hAPP _let_1 V_P_2))) (tptp.hAPP (tptp.hAPP _let_3 V_B_2) (tptp.hAPP _let_1 V_Q_2))))))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (V_xz_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) V_xz_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_a T_c T_b) V_r_2) V_s_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted) (B_z $$unsorted)) (=> (= V_xz_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) B_x) B_z)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_c) B_x) B_y)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) B_y) B_z)) V_s_2))))))))) (forall ((V_r_2 $$unsorted) (V_yx_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_b)) V_yx_2) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_a) V_r_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (= V_yx_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) B_y) B_x)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) B_x) B_y)) V_r_2)))))))) _let_88 _let_87 (forall ((V_q $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_b))) (=> (and (tptp.class_Rings_Oinverse T_b) (tptp.class_Rings_Oring__1 T_b)) (= (tptp.c_Nitpick_Oof__frac T_a T_b V_q) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Nitpick_Onum T_a) V_q))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Nitpick_Odenom T_a) V_q))))))) _let_86 _let_85 _let_84 _let_83 _let_82 _let_81 (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_b) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2) V_A_2)))))) _let_80 (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Ointernal__split T_b T_c T_a) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP V_c_2 V_a_2) V_b_2))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_b) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_a T_b) V_f_2) V_A_2)))))) (= (tptp.c_HOL_Obool_Obool__size tptp.c_fTrue) _let_31) (= (tptp.c_HOL_Obool_Obool__size tptp.c_fFalse) _let_31) (forall ((V_u_2 $$unsorted) (V_f_2 $$unsorted)) (=> (forall ((B_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_n) (tptp.hAPP V_f_2 B_n)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat)) V_f_2)) V_u_2)))))) _let_79 (forall ((V_n_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.c_List_Olexn T_a V_r_2) V_n_2))) (and (= (tptp.hAPP _let_2 V_xs_2) V_n_2) (= (tptp.hAPP _let_2 V_ys_2) V_n_2)))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (not (= (tptp.hAPP _let_1 V_xs) (tptp.hAPP _let_1 V_ys))) (not (= V_xs V_ys))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_prod _let_1 _let_1))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun _let_1 _let_4))) (let ((_let_6 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_8 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.c_COMBB tptp.tc_Nat_Onat _let_8 _let_1))) (let ((_let_10 (tptp.tc_fun _let_1 tptp.tc_Nat_Onat))) (let ((_let_11 (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_6) _let_1) (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1)))) (let ((_let_12 (tptp.c_COMBC _let_1 _let_10 _let_6))) (let ((_let_13 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_14 (tptp.c_COMBB tptp.tc_HOL_Obool _let_13 _let_1))) (let ((_let_15 (tptp.tc_fun _let_1 _let_13))) (let ((_let_16 (tptp.c_COMBB _let_6 _let_15 _let_1))) (let ((_let_17 (tptp.hAPP (tptp.c_COMBB _let_15 (tptp.tc_fun _let_6 _let_6) _let_1) (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool)))) (let ((_let_18 (tptp.c_COMBS _let_1 _let_6 _let_6))) (= (tptp.c_List_Olenlex T_a V_r_2) (tptp.hAPP (tptp.c_Set_OCollect _let_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP _let_18 (tptp.hAPP _let_17 (tptp.hAPP (tptp.hAPP _let_16 (tptp.hAPP _let_14 tptp.c_fdisj)) (tptp.hAPP (tptp.hAPP _let_12 (tptp.hAPP _let_11 (tptp.hAPP (tptp.hAPP _let_9 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) _let_7))) _let_7)))) (tptp.hAPP (tptp.hAPP _let_18 (tptp.hAPP _let_17 (tptp.hAPP (tptp.hAPP _let_16 (tptp.hAPP _let_14 tptp.c_fconj)) (tptp.hAPP (tptp.hAPP _let_12 (tptp.hAPP _let_11 (tptp.hAPP (tptp.hAPP _let_9 tptp.c_fequal) _let_7))) _let_7)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_3 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_3 _let_6) _let_1) (tptp.c_COMBC _let_1 _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 _let_2) _let_5 _let_1) (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 _let_1) (tptp.c_member _let_2))) (tptp.c_Product__Type_OPair _let_1 _let_1)))) (tptp.c_List_Olex T_a V_r_2)))))))))))))))))))))))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.c_List_Osublist T_a V_xs_2 V_I_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP _let_1 V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member tptp.tc_Nat_Onat)) V_I_2))))))) (forall ((V_n_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_2) tptp.c_fequal) (tptp.c_Nat_Osize__class_Osize _let_2))) _let_1))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_prod _let_2 _let_2))) (let ((_let_7 (tptp.c_List_Olexn T_a V_r_2))) (let ((_let_8 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a _let_2 _let_2) (tptp.c_List_Olist_OCons T_a)))) (let ((_let_9 (tptp.tc_prod T_a _let_2))) (= (tptp.hAPP _let_7 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun _let_6 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod _let_9 _let_9) _let_6 (tptp.c_Product__Type_Omap__pair _let_9 _let_2 _let_9 _let_2 _let_8 _let_8)) (tptp.c_Wellfounded_Olex__prod T_a _let_2 V_r_2 (tptp.hAPP _let_7 V_n_2)))) (tptp.hAPP (tptp.c_Set_OCollect _let_6) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_2 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_5 _let_5) _let_2) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 _let_2) tptp.c_fconj) _let_3))) _let_3))))))))))))))) (forall ((V_x $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 V_ys))) (not (= V_xs (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_ys)))))) (forall ((V_list_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_list_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_list_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_H_2) V_list_H_2)) (and (= V_a_2 V_a_H_2) (= V_list_2 V_list_H_2))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs) V_xs))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (not (= V_xs (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)))) (forall ((V_list $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a) V_list)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_list)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olex T_a V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.c_Nat_Osize__class_Osize _let_2))) (let ((_let_6 (tptp.c_List_Olist_OCons T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_6 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_6 V_y_2) V_ys_2))) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (= (tptp.hAPP _let_5 V_xs_2) (tptp.hAPP _let_5 V_ys_2))) (and (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) V_ys_2)) _let_1)))))))))))) _let_78 (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Code__Numeral_Ocode__numeral T_a) V_x))) (let ((_let_2 (tptp.hAPP (tptp.c_Random_Opick T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a)) V_x) V_xs)) V_i))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) V_i) _let_1)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Code__Numeral_Ocode__numeral T_a) V_x))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.c_Random_Opick T_a V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_i) _let_1))))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Random_Oselect__weight T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral T_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)) V_x_2)) V_xs_2)) (tptp.c_Random_Oselect__weight T_a V_xs_2))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_b_2 $$unsorted) (V_x_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_a V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.c_List_Olist_OCons T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_5 V_a_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_5 V_b_2) V_y_2))) _let_1)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2)) (and (= V_a_2 V_b_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) _let_1))))))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)))) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olex T_a V_r_2))) (and (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olexord T_a V_r_2))) (= (tptp.hAPP _let_2 V_x_2) (tptp.hAPP _let_2 V_y_2)))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))) (let ((_let_3 (tptp.c_List_Olist_OCons T_a))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_1 _let_1))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_1 _let_1)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_ys_2))) _let_2)))))))))) (forall ((V_list_2 $$unsorted) (V_a_2 $$unsorted) (V_fa_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.c_List_Olist_Olist__size T_a V_fa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a_2) V_list_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_fa_2 V_a_2)) (tptp.c_List_Olist_Olist__size T_a V_fa_2 V_list_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Wellfounded_Oacc _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))) (let ((_let_3 (tptp.c_member _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Wellfounded_Oacc T_a V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) _let_2)))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel1 T_a V_r_2)) (tptp.c_List_Olistrel1 T_a V_s_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Olistrel1 T_a (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)) (tptp.hAPP (tptp.c_Relation_Oconverse _let_1 _let_1) (tptp.c_List_Olistrel1 T_a V_r_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel1 T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_1 _let_1))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_1 _let_1)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_xs_2)) (tptp.hAPP _let_3 V_ys_2))) _let_2))))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_a V_r_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_3 _let_3))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_3 _let_3)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 V_ys_2))) _let_1))))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))) (= (tptp.hAPP _let_2 V_x_2) (tptp.hAPP _let_2 V_y_2)))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel1 T_a V_r_2))) (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olistrel1 T_a (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_a) V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Relation_Oconverse _let_1 _let_1) (tptp.c_List_Olistrel1 T_a V_r_2)))))))) (forall ((V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_xs_2))) (tptp.c_List_Olistrel1 T_a V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_a V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.c_List_Olist_OCons T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_5 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_5 V_y_2) V_ys_2))) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (= V_xs_2 V_ys_2)) (and (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) V_ys_2)) _let_1))))))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert _let_1))) (let ((_let_4 (tptp.c_Relation_OImage _let_1 _let_1 (tptp.c_List_Olistrel T_a V_r_2)))) (= (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) _let_2)) (tptp.c_List_Oset__Cons T_a (tptp.hAPP (tptp.c_Relation_OImage T_a T_a V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) _let_2))))))))) (forall ((V_y_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_x)) V_r_2)))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_y_2) V_y_2)) (tptp.c_List_Olexord T_a V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_xs_2)) (tptp.c_List_Olistrel T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel T_a V_r_2)) (tptp.c_List_Olistrel T_a V_s_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel T_a V_r_2)) (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel T_a V_r_2))) (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)))))) (forall ((V_zs_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.hAPP _let_3 V_xs_2))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_ys_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_3 V_ys_2) V_zs_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_zs_2)) _let_1)))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olistrel T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (tptp.c_Transitive__Closure_Ortrancl (tptp.tc_List_Olist T_a) (tptp.c_List_Olistrel1 T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olistrel1 T_a V_r_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olistrel T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))))))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) V_r_H_2)) (=> (tptp.c_Relation_Orefl__on T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_r_H_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel1 T_a V_r_2)) (tptp.c_List_Olistrel T_a V_r_H_2))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olistrel T_a V_r_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel T_a V_r_2))) (let ((_let_2 (tptp.c_List_Olist_OCons T_a))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_3 _let_3))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_3 _let_3)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_ys_2))) _let_1)))))))))) (forall ((V_xa_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (let ((_let_6 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_List_Olistrelp T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_3 _let_6) T_a) (tptp.c_COMBC T_a _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_2) _let_5 T_a) (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 T_a) (tptp.c_member _let_2))) (tptp.c_Product__Type_OPair T_a T_a)))) V_r_2) V_x_2 V_xa_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_xa_2)) (tptp.c_List_Olistrel T_a V_r_2))))))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y_2) V_ys_2))) (tptp.c_List_Olistrel T_a V_r_2))) (not (forall ((B_x $$unsorted) (B_xs $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) B_x) B_xs)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) V_y_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) B_xs) V_ys_2)) (tptp.c_List_Olistrel T_a V_r_2)))))))))))) (forall ((V_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (= (tptp.c_List_Olistrelp T_a V_r_2 (tptp.hAPP (tptp.hAPP _let_1 V_a1_2) V_a3_2) (tptp.hAPP (tptp.hAPP _let_1 V_a2_2) V_a4_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a1_2) V_a2_2)) (tptp.c_List_Olistrelp T_a V_r_2 V_a3_2 V_a4_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_x_2) V_y_2)) (=> (tptp.c_List_Olistrelp T_a V_r_2 V_xs_2 V_ys_2) (tptp.c_List_Olistrelp T_a V_r_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_ys_2)))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (V_ys_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y_2) V_ys_2)) V_xs_2)) (tptp.c_List_Olistrel T_a V_r_2))) (not (forall ((B_y $$unsorted) (B_ys $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) B_y) B_ys)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_y_2) B_y)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_ys_2) B_ys)) (tptp.c_List_Olistrel T_a V_r_2)))))))))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olists T_a V_A_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_a) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_a) V_A_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_2 _let_2) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel T_a V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma _let_2 _let_2) _let_1) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_2) _let_1)))))))) (forall ((T_a $$unsorted)) (= (tptp.c_List_Olists T_a (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun (tptp.tc_List_Olist T_a) tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_List_Olist T_a) tptp.tc_HOL_Obool)) (tptp.c_List_Olists T_a V_A_2)) (tptp.c_List_Olists T_a V_B_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (tptp.c_Equiv__Relations_Oequiv (tptp.tc_List_Olist T_a) (tptp.c_List_Olists T_a V_A_2) (tptp.c_List_Olistrel T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (tptp.c_Relation_Orefl__on (tptp.tc_List_Olist T_a) (tptp.c_List_Olists T_a V_A_2) (tptp.c_List_Olistrel T_a V_r_2)))) (forall ((V_A_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olists T_a V_A_2))) (let ((_let_2 (tptp.c_member (tptp.tc_List_Olist T_a)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) _let_1)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) _let_1))))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Wellfounded_Oacc _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olists T_a (tptp.c_Wellfounded_Oacc T_a V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olists T_a (tptp.c_Wellfounded_Oacc T_a V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Wellfounded_Oacc _let_1 (tptp.c_List_Olistrel1 T_a V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel T_a V_r_2))) (and (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)) (forall ((B_n $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) B_n)) (tptp.hAPP (tptp.hAPP _let_1 V_ys_2) B_n))) V_r_2)))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_a V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_a T_a)))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_a) B_b)) V_r_2)) (= B_a B_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_b) B_a)) V_r_2)))))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) _let_1)) (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_x_2)) _let_1))))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((T_a $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (= V_xs_2 V_ys_2) (and (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) B_i) (tptp.hAPP (tptp.hAPP _let_1 V_ys_2) B_i))))))))) (forall ((V_n $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_n)))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_x)) (forall ((V_n_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) V_n_2) (tptp.c_Nat_Onat_Onat__case T_a V_x_2 (tptp.hAPP _let_1 V_xs_2) V_n_2)))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) V_n))) (let ((_let_3 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_3 (= _let_2 V_x)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v))) (let ((_let_2 (tptp.c_List_Onth T_a))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) _let_1))) (let ((_let_4 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_4 (= _let_3 V_x)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.hAPP _let_2 V_x_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) (tptp.c_List_Olexord T_a V_r_2))) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_3) (tptp.hAPP _let_2 V_y_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) _let_3) V_y_2) V_x_2)) (exists ((B_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Otake T_a) B_i))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) (tptp.hAPP _let_3 V_x_2)) (tptp.hAPP _let_3 V_y_2)))) (= (tptp.hAPP _let_2 V_x_2) (tptp.hAPP _let_2 V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_i)) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) B_i))) V_r_2))))))))))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 tptp.tc_Nat_Onat) tptp.c_fconj))) (let ((_let_4 (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun tptp.tc_Nat_Onat T_a))) (let ((_let_6 (tptp.tc_fun tptp.tc_Nat_Onat _let_2))) (= (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_List_Osublist T_a V_xs_2 V_I_2)) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_1 _let_1) T_a) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_6 T_a) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_5 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.tc_fun _let_5 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool tptp.tc_Nat_Onat)) tptp.c_fequal)) (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2))))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 tptp.tc_HOL_Obool) (tptp.c_member tptp.tc_Nat_Onat)) V_I_2))))))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)))))) (forall ((V_n $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs) V_xs))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.c_List_Oset T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_n_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.c_List_Osublist T_a V_xs_2 V_I_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_n)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs)) V_i) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_i))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n) V_xs)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_y_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)))) (or (= V_y_2 V_x_2) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Osublist T_a V_xs_2 V_I_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n_2) V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))) (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Osublist T_a V_xs_2 V_I_2))))))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) V_n)))) (forall ((V_xs $$unsorted) (V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) V_n) V_m)) V_xs)))) (forall ((V_n_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Osublist T_a V_l_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n_2) V_l_2))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_i))))))) (forall ((V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_y_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_x_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Olist_Olist__size T_a V_f_2 V_xs_2))))))) (forall ((V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_y_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_x_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Olist_Olist__size T_a V_f_2 V_xs_2))))))) (forall ((V_A_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.c_List_Olists T_a V_A_2))) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_a) B_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_n_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_n_2)) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (exists ((B_i $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_i) V_x_2))))) _let_77 (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat T_a))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) (tptp.c_HOL_OEx tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.tc_fun _let_2 _let_1) T_a) (tptp.c_COMBB T_a tptp.tc_HOL_Obool tptp.tc_Nat_Onat)) tptp.c_fequal)) (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))))))))))) (forall ((V_n_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_1) (tptp.c_Orderings_Oord__class_Oless__eq _let_2)) (tptp.c_List_Oset T_a))) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1) tptp.c_fequal) (tptp.c_Nat_Osize__class_Osize _let_1))) V_n_2))))))))) (forall ((V_n_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_1) (tptp.c_Orderings_Oord__class_Oless__eq _let_2)) (tptp.c_List_Oset T_a))) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat)) (tptp.c_Nat_Osize__class_Osize _let_1))) V_n_2))))))))) _let_76 (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel T_a V_r_2))) (and (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (let ((_let_2 (tptp.c_member _let_1))) (let ((_let_3 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_a _let_4))) (let ((_let_6 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_a) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_3 _let_6) T_a) (tptp.c_COMBC T_a _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_1) _let_5 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 T_a) _let_2)) (tptp.c_Product__Type_OPair T_a T_a)))) V_r_2)) B_x))))))))))))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (V_n $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake (tptp.tc_prod T_a T_b)) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_n) V_ys))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist (tptp.tc_prod T_a T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys)))) (forall ((V_ys $$unsorted) (V_y $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_y) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys))))) (forall ((T_a $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (= V_xs_2 V_ys_2) (and (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_a) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) B_x))))))))) (forall ((V_xs_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_a) V_xs_2) V_xs_2)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= V_a_2 V_b_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)))) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_ys_2))))))))) (forall ((V_ys $$unsorted) (T_b $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs) V_ys)) V_i) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_i)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_ys) V_i))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat _let_1))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun tptp.tc_Nat_Onat _let_4))) (= (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool _let_1) (tptp.c_HOL_OEx tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_2 _let_2) _let_1) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_1) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 tptp.tc_Nat_Onat) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) _let_1) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool tptp.tc_Nat_Onat)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b _let_1) tptp.tc_Nat_Onat) (tptp.c_Product__Type_OPair T_a T_b)) (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2))) (tptp.hAPP (tptp.c_List_Onth T_b) V_ys_2)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2))))))))))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (= (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Int_Oint V_i_2 V_j_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) (tptp.c_List_Oupto (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_j_2) _let_1)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)))))) (forall ((V_ns_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Nat_Onat) V_ns_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) V_ns_2))) (= B_x (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) (tptp.c_List_Oupto V_i_2 V_j_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_i_2 V_j_2))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_i_2 V_j_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) (tptp.c_List_Oupto V_i_2 V_j_2)))) (forall ((V_ns $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_k) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist tptp.tc_Nat_Onat)) V_ns))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth tptp.tc_Nat_Onat) V_ns) V_k)) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Nat_Onat) V_ns))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_x) (tptp.hAPP _let_1 V_xs)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint V_i_2) V_j_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) (tptp.c_List_Oupto V_i_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_j_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Int_Oint V_i_2 V_j_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) (tptp.c_List_Oupto (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_j_2)))) _let_75 (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_List_Olist_Olist__size T_a V_f_2 V_xs_2)) (tptp.c_List_Olist_Olist__size T_a V_g_2 V_xs_2))))) (forall ((V_n $$unsorted) (V_ns $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_k) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist tptp.tc_Nat_Onat)) V_ns))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update tptp.tc_Nat_Onat V_ns) V_k) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_ns)) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth tptp.tc_Nat_Onat) V_ns) V_k)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.c_List_Obutlast T_a (tptp.hAPP (tptp.hAPP _let_1 V_n) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_xs))))) (forall ((V_x $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_i)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x) V_xs))) (forall ((V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_i)) V_xs)) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (=> (not (= V_i V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x)) V_j) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_j))))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_List_Obutlast T_a V_xs))) (let ((_let_2 (tptp.c_List_Obutlast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_k) V_x)))) (let ((_let_3 (= V_k (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a _let_1) V_k) V_x)))))))) (forall ((V_x $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x)) (tptp.hAPP _let_1 V_xs)))) (forall ((V_y $$unsorted) (V_ys $$unsorted) (V_x $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_ys) V_i) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update (tptp.tc_prod T_a T_b) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys)) V_i) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x) V_y))))) (forall ((V_xy $$unsorted) (V_i $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update (tptp.tc_prod T_a T_b) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys)) V_i) V_xy) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) V_xy))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_ys) V_i) (tptp.hAPP (tptp.c_Product__Type_Osnd T_a T_b) V_xy)))))) (forall ((V_x_H $$unsorted) (V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_i_H $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Olist__update T_a V_xs))) (=> (not (= V_i V_i_H)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP (tptp.hAPP _let_1 V_i) V_x)) V_i_H) V_x_H) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP (tptp.hAPP _let_1 V_i_H) V_x_H)) V_i) V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_1 V_x)) V_i) V_y) (tptp.hAPP _let_1 V_y)))) (forall ((V_y $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_1 V_xs)) (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_y) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_y))))) (forall ((V_v_2 $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_OCons T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_x_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_3 V_xs_2)) V_i_2) V_v_2) (tptp.c_Nat_Onat_Onat__case _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_v_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 tptp.tc_Nat_Onat) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat T_a _let_1) (tptp.c_List_Olist__update T_a V_xs_2)) V_v_2)) V_i_2)))))) (forall ((V_y $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) V_xs)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_xs)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_i_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_i_2) V_x_2))) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_i_2) V_x_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x)) V_i) V_x))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_i_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_i_2) V_x_2) V_xs_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_i_2) V_x_2)))) (forall ((V_x $$unsorted) (V_j $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x)) V_j))) (let ((_let_3 (= V_i V_j))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (and (=> _let_3 (= _let_2 V_x)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_j))))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Otake T_a) V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs)) (tptp.hAPP _let_1 V_xs))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_n_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_n_2) V_x_2)))))) _let_74 (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel1 T_a V_r_2))) (exists ((B_y $$unsorted) (B_n $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_n)) B_y)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) B_n) B_y))))))) (forall ((V_i_2 $$unsorted) (V_a_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_i_2)) (tptp.c_Orderings_Obot__class_Obot _let_2)))))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_i_2) V_a_2)))))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct tptp.tc_Int_Oint) (tptp.c_List_Oupto V_i V_j)))) (forall ((V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_i) V_xs)))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Osublist T_a V_xs_2 V_I_2)))))) (forall ((V_xs $$unsorted) (T_b $$unsorted) (V_ys $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_ys)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_a) V_xs) V_ys))))) (forall ((V_ys $$unsorted) (T_b $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs) V_ys))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (and (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_i_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_j_2) _let_2)) (= (= (tptp.hAPP _let_1 V_i_2) (tptp.hAPP _let_1 V_j_2)) (= V_i_2 V_j_2))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (forall ((B_j $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (=> (not (= B_i B_j)) (not (= (tptp.hAPP _let_1 B_i) (tptp.hAPP _let_1 B_j))))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_a) (tptp.c_COMBI T_a)) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Ocard_H T_a V_A_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (let ((_let_4 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_3) (tptp.c_Hilbert__Choice_OEps _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_3 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_3) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 _let_4 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_4 tptp.tc_HOL_Obool) _let_3) tptp.c_fequal) (tptp.c_List_Oset T_a))) V_A_2))) (tptp.c_List_Odistinct T_a)))))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Odom T_a T_b (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) V_xs_2))) (tptp.c_Map_Odom T_a T_b V_m_2)))) (forall ((V_ys_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (= (tptp.hAPP (tptp.c_Map_Omap__upds T_a T_b V_f_2 V_xs_2 V_ys_2) V_x_2) (tptp.hAPP V_f_2 V_x_2)))) (forall ((V_y_2 $$unsorted) (V_m_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)) V_i_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2))) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_ys_2) V_i_2) V_y_2)) (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 V_ys_2))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Otake T_a) V_k))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ys))) (=> (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_k)) (= (tptp.hAPP (tptp.hAPP _let_1 V_xs) B_i) (tptp.hAPP (tptp.hAPP _let_1 V_ys) B_i))))) (= (tptp.hAPP _let_1 V_xs) (tptp.hAPP _let_1 V_ys))))))))) (forall ((V_m_2 $$unsorted) (V_D_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) _let_1) V_D_2)) (= (tptp.c_Map_Orestrict__map T_a T_b (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 V_ys_2) V_D_2) (tptp.c_Map_Omap__upds T_a T_b (tptp.c_Map_Orestrict__map T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_D_2) _let_1)) V_xs_2 V_ys_2))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Orestrict__map T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)) (tptp.c_Map_Orestrict__map T_a T_b V_f_2 V_A_2)) V_a_2) (tptp.hAPP V_f_2 V_a_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Orestrict__map T_a T_b (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_A_2) V_B_2) (tptp.c_Map_Orestrict__map T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)))) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_A_2) V_x_2) (tptp.hAPP V_m_2 V_x_2)))) (forall ((V_A_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Odom T_a T_b (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Map_Odom T_a T_b V_m_2)) V_A_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_D_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_D_2)) V_x_2) V_y_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_D_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))) V_x_2) V_y_2))))) (forall ((V_y_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_D_2)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_D_2)) V_x_2) V_y_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_D_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))) V_x_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (let ((_let_2 (tptp.c_Map_Orestrict__map T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_x_2) V_y_2) V_D_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_D_2)))) (let ((_let_4 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_4) V_D_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_4))))) V_x_2) V_y_2))) (=> (not _let_3) (= _let_2 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_D_2))))))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (= (tptp.hAPP _let_2 (tptp.c_List_Oremove1 T_a V_x_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v))) (let ((_let_2 (tptp.c_List_Otake T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP _let_3 V_xs)))) (let ((_let_5 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_5 (= _let_4 (tptp.c_List_Olist_ONil T_a))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_xs))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_a)) (tptp.c_List_Olist_ONil T_a)) (tptp.c_List_Olists T_a V_A_2)))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) V_xs_2) _let_1)) (tptp.c_List_Olistrel T_a V_r_2))) (= V_xs_2 _let_1))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_1) V_xs_2)) (tptp.c_List_Olistrel T_a V_r_2))) (= V_xs_2 _let_1))))) (forall ((V_y $$unsorted) (V_i $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a _let_1) V_i) V_y) _let_1))) (forall ((V_v $$unsorted) (V_i $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a _let_1) V_i) V_v) _let_1))) (forall ((V_x_2 $$unsorted) (V_k_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_k_2) V_x_2) _let_1) (= V_xs_2 _let_1)))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_List_Olist_ONil T_a)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (= V_xs_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_xs_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_xs_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.c_List_Olist_ONil T_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.c_List_Olist_ONil T_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_ys $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) (tptp.c_List_Olist_ONil T_a)) V_ys) (tptp.c_List_Olist_ONil (tptp.tc_prod T_a T_b)))) (forall ((V_xs $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs) (tptp.c_List_Olist_ONil T_b)) (tptp.c_List_Olist_ONil (tptp.tc_prod T_a T_b)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (tptp.c_List_Olistrelp T_a V_r_2 _let_1 _let_1))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (tptp.c_List_Olistrelp T_a V_r_2 _let_1 _let_1))) (forall ((V_fa_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist_Olist__size T_a V_fa_2 (tptp.c_List_Olist_ONil T_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_zs $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oremove1 T_a V_x (tptp.c_List_Oremove1 T_a V_y V_zs)) (tptp.c_List_Oremove1 T_a V_y (tptp.c_List_Oremove1 T_a V_x V_zs)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Oremove1 T_a V_x _let_1) _let_1))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Osublist T_a _let_1 V_A_2) _let_1))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n_2) V_xs_2) _let_1) (or (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_xs_2 _let_1))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_xs) (tptp.c_List_Olist_ONil T_a))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) _let_1) _let_1))) (forall ((V_i $$unsorted) (V_j $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_j) V_i)) (= (tptp.c_List_Oupto V_i V_j) (tptp.c_List_Olist_ONil tptp.tc_Int_Oint)))) (forall ((V_list_H $$unsorted) (V_a_H $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_List_Olist_ONil T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a_H) V_list_H)))) (forall ((V_list_H $$unsorted) (V_a_H $$unsorted) (T_a $$unsorted)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a_H) V_list_H) (tptp.c_List_Olist_ONil T_a)))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y))) (let ((_let_2 (tptp.c_List_Oremove1 T_a V_x (tptp.hAPP _let_1 V_xs)))) (let ((_let_3 (= V_x V_y))) (and (=> _let_3 (= _let_2 V_xs)) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_x V_xs))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (let ((_let_2 (tptp.c_List_Obutlast T_a (tptp.hAPP _let_1 V_xs)))) (let ((_let_3 (tptp.c_List_Olist_ONil T_a))) (let ((_let_4 (= V_xs _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs)))))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Obutlast T_a _let_1) _let_1))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_a_2))) (=> (not (= V_a_2 V_b_2)) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_b_2 V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))))) (forall ((V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))) (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_y_2 V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (= (tptp.c_List_Oremove1 T_a V_x_2 V_xs_2) V_xs_2))) (forall ((V_bs_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 (tptp.c_List_Olist_ONil T_a) V_bs_2) V_m_2)) (forall ((V_as_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_as_2 (tptp.c_List_Olist_ONil T_b)) V_m_2)) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_x V_xs)))))) (forall ((T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.c_List_Olist_ONil T_a)))) (forall ((T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Osublist T_a V_xs_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.c_List_Olist_ONil T_a))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_x_2 V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 (tptp.c_List_Olist_ONil T_a))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) (tptp.c_List_Olist_ONil T_a))) (tptp.c_List_Olistrel1 T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.c_List_Olist_ONil T_a)) V_xs_2)) (tptp.c_List_Olistrel1 T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_1) _let_1)) (tptp.c_List_Olistrel T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) (tptp.c_List_Olist_ONil T_a))) (tptp.c_List_Olexord T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) (tptp.c_List_Olist_ONil T_a))) (tptp.c_List_Olex T_a V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.c_List_Olist_ONil T_a)) V_ys_2)) (tptp.c_List_Olex T_a V_r_2)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_n_2) (tptp.hAPP _let_3 V_xs_2)) (tptp.c_Nat_Onat_Onat__case _let_2 (tptp.c_List_Olist_ONil T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) _let_1) V_xs_2)) V_n_2)))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Oupto V_i V_j))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_i) V_j)))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Int_Oint) V_i) (tptp.c_List_Oupto (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_j)))) (=> (not _let_2) (= _let_1 (tptp.c_List_Olist_ONil tptp.tc_Int_Oint))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) _let_1))) (let ((_let_3 (tptp.c_List_Osublist T_a _let_2 V_A_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_A_2)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 _let_1)))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Olists T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_List_Olist_ONil T_a)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP _let_2 V_xs)))) (let ((_let_4 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_4 (= _let_3 (tptp.c_List_Olist_ONil T_a))) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_xs)))))))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (let ((_let_3 (tptp.hAPP _let_1 V_m))) (let ((_let_4 (tptp.c_List_Oupto _let_3 _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_3) _let_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Int_Oint) _let_3) (tptp.c_List_Oupto (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_3) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) _let_2)))) (=> (not _let_5) (= _let_4 (tptp.c_List_Olist_ONil tptp.tc_Int_Oint)))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_List_Olist_ONil T_a)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))) (= (tptp.hAPP (tptp.c_Relation_OImage _let_1 _let_1 (tptp.c_List_Olistrel T_a V_r_2)) _let_2) _let_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Oset__Cons T_a V_A_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_2) _let_1) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)))) (tptp.hAPP (tptp.c_Set_Oimage T_a _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.c_List_Olist_OCons T_a)) _let_1)) V_A_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (let ((_let_2 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_3 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_x_2 V_xs_2)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (=> (not _let_4) (= _let_3 _let_2)))))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Oupto V_i_2 V_j_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_i_2) V_j_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_List_Oupto__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_i_2) V_j_2))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Int_Oint) V_i_2) (tptp.c_List_Oupto (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_j_2)))) (=> (not _let_2) (= _let_1 (tptp.c_List_Olist_ONil tptp.tc_Int_Oint)))))))) (forall ((V_s_2 $$unsorted) (T_a $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Product__Type_Ofst T_a (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP (tptp.c_Random_Oselect T_a V_xs_2) V_s_2))) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Olistset T_a (tptp.c_List_Olist_ONil (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_List_Olist_ONil T_a)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (let ((_let_3 (tptp.tc_List_Olist _let_2))) (let ((_let_4 (tptp.tc_List_Olist T_a))) (let ((_let_5 (tptp.tc_fun _let_3 _let_3))) (let ((_let_6 (tptp.tc_fun _let_4 _let_3))) (= (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_y_2) V_ys_2)) (tptp.hAPP (tptp.c_List_Olist_Olist__case _let_3 T_a (tptp.c_List_Olist_ONil _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_6 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_6 _let_6) T_a) (tptp.c_COMBB _let_3 _let_3 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 T_a) (tptp.c_List_Olist_OCons _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_b _let_2) (tptp.c_Product__Type_OPair T_a T_b)) V_y_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_4 (tptp.tc_List_Olist T_b) _let_3) _let_1) V_ys_2))) V_xs_2))))))))) (forall ((V_list_2 $$unsorted) (V_a_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Olist_Olist__case T_a T_b V_f1_2 V_f2_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_a_2) V_list_2)) (tptp.hAPP (tptp.hAPP V_f2_2 V_a_2) V_list_2))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Olist_Olist__case T_a T_b V_f1_2 V_f2_2) (tptp.c_List_Olist_ONil T_b)) V_f1_2)) (forall ((V_As_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olistset T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_As_2)) (tptp.c_List_Oset__Cons T_a V_A_2 (tptp.c_List_Olistset T_a V_As_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (let ((_let_3 (tptp.tc_List_Olist _let_2))) (let ((_let_4 (tptp.tc_fun _let_3 _let_3))) (let ((_let_5 (tptp.tc_List_Olist T_b))) (let ((_let_6 (tptp.tc_fun _let_5 _let_3))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) V_ys_2) (tptp.hAPP (tptp.c_List_Olist_Olist__case _let_3 T_b (tptp.c_List_Olist_ONil _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_6 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_6 _let_6) T_b) (tptp.c_COMBB _let_3 _let_3 _let_5)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 T_b) (tptp.c_List_Olist_OCons _let_2)) (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2)))) (tptp.hAPP _let_1 V_xs_2))) V_ys_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (tptp.c_List_Oremove1 T_a V_x_2 (tptp.c_List_Olinorder__class_Osorted__list__of__set T_a V_A_2))))))) (forall ((V_P_2 $$unsorted) (V_a1_2 $$unsorted) (V_a0_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_List_Oupto__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_a0_2) V_a1_2))) (=> (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_List_Oupto__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) B_i) B_j))) (=> (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) B_i) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) B_j))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 B_i) B_j))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_a0_2) V_a1_2))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_List_Olist_ONil T_a)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Oanamorph T_b T_a V_f_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 T_b))) (let ((_let_4 (tptp.tc_prod _let_2 T_b))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (let ((_let_6 (tptp.tc_fun _let_2 _let_5))) (let ((_let_7 (tptp.tc_fun _let_4 _let_4))) (let ((_let_8 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_List_Olist_ONil T_a)) V_x_2))) (let ((_let_9 (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_x_2))) (let ((_let_10 (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (and (=> _let_10 (= _let_9 _let_8)) (=> (not _let_10) (= _let_9 (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case _let_4 (tptp.tc_prod T_a T_b) _let_8 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_5 _let_5) T_a) (tptp.c_COMBB _let_4 _let_4 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_7 T_a) (tptp.c_Product__Type_Oprod_Oprod__case _let_2 T_b _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_2 _let_2) _let_6 T_a) (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_2) _let_3)) (tptp.c_List_Olist_OCons T_a))))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_k_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)))))) (tptp.hAPP V_f_2 V_x_2)))))))))))))))) (forall ((V_A_2 $$unsorted) (V_l_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_member tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_List_Olist_ONil T_a))) (let ((_let_4 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (= (tptp.c_List_Osublist T_a (tptp.hAPP _let_4 V_l_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_If (tptp.tc_List_Olist T_a)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_A_2)) (tptp.hAPP _let_4 _let_3)) _let_3)) (tptp.c_List_Osublist T_a V_l_2 (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) _let_1) tptp.c_Nat_OSuc)) V_A_2)))))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_ys_2) V_zs_2) _let_1) (or (and (= V_ys_2 (tptp.c_List_Olist_ONil T_a)) (= V_zs_2 _let_1)) (exists ((B_ys_H $$unsorted)) (and (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) B_ys_H)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) B_ys_H) V_zs_2) V_xs_2))))))) (forall ((V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_ys_2) V_zs_2)) (or (and (= V_ys_2 (tptp.c_List_Olist_ONil T_a)) (= _let_1 V_zs_2)) (exists ((B_ys_H $$unsorted)) (and (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) B_ys_H) V_ys_2) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) B_ys_H) V_zs_2)))))))) (forall ((V_y_2 $$unsorted) (V_ys_2 $$unsorted) (V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.c_List_Olist_OCons T_a))) (let ((_let_3 (tptp.c_List_Oappend T_a))) (= (= (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_3 V_ys_2) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1))) (and (= V_xs_2 V_ys_2) (= V_x_2 V_y_2))))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs)) V_ys) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys)))))) (forall ((V_zs $$unsorted) (V_xs $$unsorted) (V_ys $$unsorted) (V_xs1 $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (=> (= (tptp.hAPP _let_2 V_xs1) V_ys) (=> (= V_xs (tptp.hAPP (tptp.hAPP _let_1 V_xs1) V_zs)) (= (tptp.hAPP _let_2 V_xs) (tptp.hAPP (tptp.hAPP _let_1 V_ys) V_zs))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 V_ys_2))))) (forall ((V_vs $$unsorted) (V_ys $$unsorted) (V_us $$unsorted) (T_b $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (= (tptp.hAPP _let_3 V_xs) (tptp.hAPP _let_2 V_us)) (=> (= (tptp.hAPP _let_3 V_ys) (tptp.hAPP _let_2 V_vs)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_us) V_vs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_us)) (tptp.hAPP (tptp.hAPP _let_1 V_ys) V_vs))))))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 V_ys))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 V_ys)))))) (forall ((V_us $$unsorted) (V_ys $$unsorted) (V_zs $$unsorted) (V_xs1 $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_xs))) (=> (= (tptp.hAPP _let_2 V_xs1) V_zs) (=> (= V_ys (tptp.hAPP (tptp.hAPP _let_1 V_xs1) V_us)) (= (tptp.hAPP _let_2 V_ys) (tptp.hAPP (tptp.hAPP _let_1 V_zs) V_us))))))) (forall ((V_zs_2 $$unsorted) (V_xs_2 $$unsorted) (V_ys_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_ys_2) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_zs_2) V_xs_2)) (= V_ys_2 V_zs_2)))) (forall ((V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2))) (= (= (tptp.hAPP _let_1 V_ys_2) (tptp.hAPP _let_1 V_zs_2)) (= V_ys_2 V_zs_2)))) (forall ((V_ts_2 $$unsorted) (V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2) (tptp.hAPP (tptp.hAPP _let_1 V_zs_2) V_ts_2)) (exists ((B_us $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP _let_1 B_us))) (or (and (= V_xs_2 (tptp.hAPP (tptp.hAPP _let_1 V_zs_2) B_us)) (= (tptp.hAPP _let_2 V_ys_2) V_ts_2)) (and (= (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) B_us) V_zs_2) (= V_ys_2 (tptp.hAPP _let_2 V_ts_2)))))))))) (forall ((V_zs $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_xs))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ys)) V_zs) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ys) V_zs)))))) (forall ((V_A_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olists T_a V_A_2))) (let ((_let_2 (tptp.c_member (tptp.tc_List_Olist T_a)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) _let_1)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_ys_2) _let_1))))))) (forall ((V_zs_2 $$unsorted) (V_m_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_ys_2) V_zs_2)) (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 V_ys_2)))) (forall ((V_zs_2 $$unsorted) (V_m_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_zs_2) V_ys_2) (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 V_ys_2)))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_ys $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs))) (let ((_let_2 (tptp.c_List_Obutlast T_a (tptp.hAPP _let_1 V_ys)))) (let ((_let_3 (= V_ys (tptp.c_List_Olist_ONil T_a)))) (and (=> _let_3 (= _let_2 (tptp.c_List_Obutlast T_a V_xs))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_ys))))))))) (forall ((V_ys $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_Olist_ONil T_a)) V_ys) V_ys)) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (and (= V_xs_2 _let_1) (= V_ys_2 _let_1))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) (tptp.c_List_Olist_ONil T_a)) V_xs)) (forall ((V_ys_2 $$unsorted) (T_a $$unsorted) (V_xs_2 $$unsorted)) (= (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (= V_ys_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_ys_2 $$unsorted)) (= (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (= V_xs_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2) _let_1) (and (= V_xs_2 _let_1) (= V_ys_2 _let_1))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2) V_xs_2) (= V_ys_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2) V_ys_2) (= V_xs_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((T_a $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted)) (=> (= V_xs V_ys) (= V_xs (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_Olist_ONil T_a)) V_ys)))) (forall ((V_ys $$unsorted) (V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_ys))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_x)) (forall ((V_n $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_ys) V_n)))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (let ((_let_3 (tptp.c_List_Oappend T_a))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_xs) V_ys)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) V_ys))))))) (forall ((V_x $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys)) V_i) V_x) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x)) V_ys))))) (forall ((V_y $$unsorted) (V_ys $$unsorted) (V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_ys))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_y) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_ys)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_3 (tptp.c_List_Oremove1 T_a V_x_2 (tptp.hAPP _let_2 V_ys_2)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_x_2 V_xs_2)) V_ys_2))) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 (tptp.c_List_Oremove1 T_a V_x_2 V_ys_2)))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (or (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_ys_2))))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2))))))))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Obutlast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) (tptp.c_List_Olist_ONil T_a)))) V_xs)) (forall ((V_vs_2 $$unsorted) (V_us_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_a V_r_2))) (let ((_let_2 (tptp.c_List_Oappend T_a))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_3 _let_3))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_3 _let_3)))) (=> (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_1)) (= V_us_2 V_vs_2)) (and (= V_xs_2 V_ys_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_us_2) V_vs_2)) _let_1)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) V_us_2)) (tptp.hAPP (tptp.hAPP _let_2 V_ys_2) V_vs_2))) _let_1))))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_v_2 $$unsorted) (V_u_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_a V_r_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Oappend T_a) V_x_2))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_3 _let_3))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_3 _let_3)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_u_2) V_v_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_u_2)) (tptp.hAPP _let_2 V_v_2))) _let_1))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_a))) (let ((_let_3 (tptp.c_List_Odistinct T_a))) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2))) (and (tptp.hBOOL (tptp.hAPP _let_3 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_3 V_ys_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 V_ys_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (let ((_let_2 (tptp.c_List_Onth T_a))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) V_n))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) _let_1)))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_xs) V_n))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_ys) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) _let_1)))))))))) (forall ((V_x $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (let ((_let_2 (tptp.c_List_Oappend T_a))) (let ((_let_3 (tptp.hAPP _let_2 V_xs))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_3 V_ys)) V_n) V_x))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) _let_1)))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_n) V_x)) V_ys))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_ys) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) _let_1)) V_x))))))))))) (forall ((V_A_2 $$unsorted) (V_l_H_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oappend T_a))) (= (tptp.c_List_Osublist T_a (tptp.hAPP (tptp.hAPP _let_2 V_l_2) V_l_H_2) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_List_Osublist T_a V_l_2 V_A_2)) (tptp.c_List_Osublist T_a V_l_H_2 (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) (tptp.c_member tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_l_2)))) V_A_2)))))))) (forall ((V_ys_2 $$unsorted) (V_vs_2 $$unsorted) (V_us_2 $$unsorted) (V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_OCons T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Oappend T_a) V_us_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2)) (=> (= V_xs_2 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_vs_2))) (=> (= V_ys_2 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_vs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel1 T_a V_r_2)))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_u_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Oappend T_a) V_u_2))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_3 _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_3 _let_3) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_x_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_y_2)))) (tptp.c_List_Olexord T_a V_r_2)))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP _let_1 V_i) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_i)) (tptp.c_List_Olist_ONil T_a))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_ys_2 $$unsorted) (V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_a V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.c_List_Olist_ONil T_a))) (let ((_let_6 (tptp.c_List_Olist_OCons T_a))) (let ((_let_7 (tptp.c_List_Oappend T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_7 V_xs_2) (tptp.hAPP (tptp.hAPP _let_6 V_x_2) _let_5))) (tptp.hAPP (tptp.hAPP _let_7 V_ys_2) (tptp.hAPP (tptp.hAPP _let_6 V_y_2) _let_5)))) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) V_ys_2)) _let_1)) (= V_x_2 V_y_2)) (and (= V_xs_2 V_ys_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) V_r_2))))))))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel1 T_a V_r_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_x) B_y)) V_r_2)) (forall ((B_us $$unsorted) (B_vs $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Oappend T_a) B_us))) (=> (= V_xs_2 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_vs))) (not (= V_ys_2 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_y) B_vs)))))))))))))) (forall ((V_r_2 $$unsorted) (V_v_2 $$unsorted) (V_u_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_a V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.hAPP (tptp.c_List_Oappend T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_5 V_u_2)) (tptp.hAPP _let_5 V_v_2))) _let_1)) (=> (forall ((B_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) B_a) B_a)) V_r_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_u_2) V_v_2)) _let_1)))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs_2) (tptp.hAPP (tptp.c_List_Olist_Olist__case _let_2 T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 _let_2) (tptp.c_List_Oappend T_a))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.c_List_Olist_OCons T_a)) _let_1))) V_xs_2))))) (forall ((V_x_2 $$unsorted) (V_m_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2))) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) (tptp.c_List_Olist_ONil T_a))) V_ys_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)) (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 V_ys_2)) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_ys_2) _let_1))))))) (forall ((V_y_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP (tptp.c_Set_Oimage T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a _let_1) V_m_2) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_y_2))) V_A_2) (tptp.hAPP (tptp.c_Set_Oimage T_a _let_1 V_m_2) V_A_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs)) (tptp.hAPP _let_1 V_xs)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Odom T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_Option_Ooption T_b) T_a) (tptp.c_Option_Ooption_OSome T_b)) V_f_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted) (V_a_2 $$unsorted) (V_m_2 $$unsorted)) (=> (= (tptp.hAPP V_m_2 V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.c_Map_Odom T_b T_a V_m_2))))) (forall ((T_b $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.c_Map_Odom T_b T_a V_f_2))) (=> (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) _let_1) _let_1)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_y_2 $$unsorted) (V_n_2 $$unsorted) (V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Option_Ooption_OSome T_b))) (let ((_let_2 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_a_2) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 V_n_2) V_a_2) (tptp.hAPP _let_1 V_y_2))) (= V_x_2 V_y_2))))) (forall ((T_b $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (V_k_2 $$unsorted) (V_t_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_x_2))) (=> (= (tptp.hAPP V_t_2 V_k_2) _let_1) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b (tptp.tc_Option_Ooption T_a)) V_t_2) V_k_2) _let_1) V_t_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Option_Ooption_OSome T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_y_2))) (let ((_let_3 (= V_x_2 V_a_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b (tptp.tc_Option_Ooption T_a)) V_m_2) V_a_2) (tptp.hAPP _let_1 V_b_2)) V_x_2) _let_2) (or (and _let_3 (= V_b_2 V_y_2)) (and (not _let_3) (= (tptp.hAPP V_m_2 V_x_2) _let_2)))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_bs_2 $$unsorted) (V_b_2 $$unsorted) (V_as_2 $$unsorted) (V_a_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a_2) V_as_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_b_2) V_bs_2)) (tptp.c_Map_Omap__upds T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)) V_m_2) V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_b_2)) V_as_2 V_bs_2))) (forall ((V_a_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.hAPP tptp.c_Com_Ocom_OBODY V_a_2))) (exists ((B_y $$unsorted)) (= (tptp.hAPP tptp.c_Com_Obody V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome tptp.tc_Com_Ocom) B_y))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_Option_Ooption T_a))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite _let_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_2 V_f_2) _let_1))) (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b _let_2) V_f_2) V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_b_2))) _let_1)))))))) (forall ((V_b $$unsorted) (V_pn $$unsorted)) (=> tptp.c_Com_OWT__bodies (=> (= (tptp.hAPP tptp.c_Com_Obody V_pn) (tptp.hAPP (tptp.c_Option_Ooption_OSome tptp.tc_Com_Ocom) V_b)) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT V_b))))) (forall ((V_bs_2 $$unsorted) (V_b_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (V_as_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_b_2))) (let ((_let_2 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_as_2)))) (= (tptp.c_Map_Omap__upds T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_a_2) _let_1) V_as_2 V_bs_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_as_2 V_bs_2)) V_a_2) _let_1)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (= (tptp.c_Map_Orestrict__map T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)) V_m_2) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_y_2)) _let_2) (tptp.c_Map_Orestrict__map T_a T_b V_m_2 _let_2))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x))) (let ((_let_2 (tptp.c_List_Olist_ONil T_a))) (and (= (tptp.hAPP (tptp.c_List_Orotate1 T_a) _let_2) _let_2) (= (tptp.hAPP (tptp.c_List_Orotate1 T_b) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs) (tptp.hAPP _let_1 (tptp.c_List_Olist_ONil T_b)))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs) V_xs))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_y_2))) (let ((_let_2 (tptp.c_Map_Omap__upds T_a T_b V_f_2 V_xs_2 V_ys_2))) (let ((_let_3 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (let ((_let_4 (tptp.c_Map_Omap__upds T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 V_f_2) V_x_2) _let_1) V_xs_2 V_ys_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) V_xs_2)))))) (and (=> _let_5 (= _let_4 _let_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_x_2) _let_1)))))))))) (forall ((V_a_2 $$unsorted) (V_fa_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Option_Ooption_Ooption__size T_a V_fa_2 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP V_fa_2 V_a_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_a_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case T_a T_b V_f1_2 V_f2_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_a_2)) (tptp.hAPP V_f2_2 V_a_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Fun_Oinj__on T_a (tptp.tc_Option_Ooption T_a) (tptp.c_Option_Ooption_OSome T_a) V_A_2)) (forall ((V_a_H_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Option_Ooption_OSome T_a))) (= (= (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_a_H_2)) (= V_a_2 V_a_H_2)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Option_Othe T_a) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_x)) V_x)) (forall ((V_P $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.hAPP tptp.c_Com_Ocom_OBODY V_P))) (not (forall ((B_y $$unsorted)) (not (= (tptp.hAPP tptp.c_Com_Obody V_P) (tptp.hAPP (tptp.c_Option_Ooption_OSome tptp.tc_Com_Ocom) B_y))))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Option_Oset T_a (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_xo_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.c_Option_Oset T_a V_xo_2))) (= V_xo_2 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_x_2)))) (forall ((V_a $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_i) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_xs)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= V_xs (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_i) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_i)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_xs)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) _let_1) _let_1))) (forall ((V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_i) V_xs)))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Odrop T_a) V_n))) (= (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs)) (tptp.c_List_Obutlast T_a (tptp.hAPP _let_1 V_xs))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Odrop T_a) V_n))) (= (tptp.c_List_Obutlast T_a (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_a V_xs))))) (forall ((V_xs $$unsorted) (V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Odrop T_a) V_m))) (= (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)) V_xs)))))) (forall ((V_xs $$unsorted) (V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Odrop T_a) V_n))) (let ((_let_2 (tptp.c_List_Otake T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m) V_xs)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP _let_1 V_xs)))))) (forall ((V_xs $$unsorted) (V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)) V_xs)))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_xs) V_xs)) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (V_n $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop (tptp.tc_prod T_a T_b)) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_n) V_ys))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) V_n)))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n_2) V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_xs)))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) V_xs)) (forall ((V_ys $$unsorted) (V_y $$unsorted) (V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y) V_ys)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_n) V_y))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs)) V_xs)) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_m_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (let ((_let_2 (tptp.c_List_Oset T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n_2) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n_2) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n_2) V_xs_2) (tptp.c_List_Olist_ONil T_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2)) V_n_2)))) (forall ((V_n $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs) (tptp.c_List_Olist_ONil T_a)))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (let ((_let_3 (tptp.c_List_Oappend T_a))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_xs) V_ys)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) V_ys))))))) (forall ((V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2) V_zs_2) (and (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) _let_1) V_zs_2)) (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) _let_1) V_zs_2)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_n_2) _let_3) (tptp.c_Nat_Onat_Onat__case _let_2 _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) _let_1) V_xs_2) V_n_2)))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_1 V_n) _let_2))) (let ((_let_4 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_xs))))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_i))) (let ((_let_2 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) _let_1) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs)) V_i) (tptp.hAPP (tptp.hAPP _let_2 V_xs) _let_1)))))) (forall ((V_ys_092_060_094isub_0622_2 $$unsorted) (V_ys_092_060_094isub_0621_2 $$unsorted) (V_xs_092_060_094isub_0622_2 $$unsorted) (V_xs_092_060_094isub_0621_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (let ((_let_2 (tptp.hAPP _let_1 V_ys_092_060_094isub_0621_2))) (let ((_let_3 (tptp.c_List_Odrop T_a))) (let ((_let_4 (tptp.c_List_Oappend T_a))) (let ((_let_5 (tptp.c_List_Otake T_a))) (let ((_let_6 (tptp.hAPP _let_1 V_xs_092_060_094isub_0621_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) _let_6) _let_2)))) (= (= (tptp.hAPP (tptp.hAPP _let_4 V_xs_092_060_094isub_0621_2) V_xs_092_060_094isub_0622_2) (tptp.hAPP (tptp.hAPP _let_4 V_ys_092_060_094isub_0621_2) V_ys_092_060_094isub_0622_2)) (and (=> _let_7 (and (= V_xs_092_060_094isub_0621_2 (tptp.hAPP (tptp.hAPP _let_5 _let_6) V_ys_092_060_094isub_0621_2)) (= V_xs_092_060_094isub_0622_2 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 _let_6) V_ys_092_060_094isub_0621_2)) V_ys_092_060_094isub_0622_2)))) (=> (not _let_7) (and (= (tptp.hAPP (tptp.hAPP _let_5 _let_2) V_xs_092_060_094isub_0621_2) V_ys_092_060_094isub_0621_2) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_xs_092_060_094isub_0621_2)) V_xs_092_060_094isub_0622_2) V_ys_092_060_094isub_0622_2))))))))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_i)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 V_i) V_xs))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v))) (let ((_let_2 (tptp.c_List_Odrop T_a))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_3))) (let ((_let_5 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_5 (= _let_4 _let_3)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_xs)))))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_j))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) _let_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP _let_1 V_i) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 V_j) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_i) V_xs)))))))) (forall ((V_zs $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys))) (let ((_let_2 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_ys) V_zs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) _let_1) V_xs)) V_ys)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) _let_1) V_xs)) V_zs)))))) (forall ((V_zs $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (let ((_let_2 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) V_zs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP _let_2 V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) _let_1) V_zs))) (tptp.hAPP (tptp.hAPP _let_2 V_ys) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) _let_1) V_zs))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.c_List_Ohd T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs))) (tptp.c_List_Olist_ONil T_a))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_xs))))) (forall ((V_n $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.c_List_Ohd T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_n))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ohd T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) V_x)) (forall ((V_ys $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Ohd T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)))) (let ((_let_3 (= V_xs (tptp.c_List_Olist_ONil T_a)))) (and (=> _let_3 (= _let_2 (tptp.hAPP _let_1 V_ys))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 V_xs)))))))) (forall ((V_ys $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Ohd T_a))) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.hAPP _let_1 V_xs))))) (forall ((T_a $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs_2)) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (forall ((T_a $$unsorted) (V_xs $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (=> (not (= V_xs _let_1)) (= (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.c_List_Otl T_a) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs)) _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_Option_Ooption T_a)) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otl T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Odrop T_a) V_n))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Odrop T_a) V_n))) (let ((_let_2 (tptp.c_List_Otl T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Otl T_a) _let_1) _let_1))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Otl T_a) V_xs)))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Otl T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) V_xs)) (forall ((V_ys $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Otl T_a))) (let ((_let_2 (tptp.c_List_Oappend T_a))) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_xs) V_ys)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)) V_ys)))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.c_List_Otl T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_xs)))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_xs) (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.c_List_Otl T_a) V_xs))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_List_Otl T_a))) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2)) (tptp.hAPP (tptp.c_List_Olist_Olist__case _let_2 T_a (tptp.hAPP _let_3 V_ys_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_2 _let_2) T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 _let_2) _let_1) V_ys_2))) V_xs_2)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Otl T_a) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otl T_a))) (let ((_let_2 (tptp.c_List_Otake T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_n) V_xs)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.hAPP _let_1 V_xs)))))) (forall ((V_n $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.c_List_Otl T_a) V_xs))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_a _let_1) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs_2)) (tptp.hAPP (tptp.c_List_Otl T_a) V_xs_2))))))) (forall ((V_n $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Ohd T_a) (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_n (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs)) (tptp.hAPP _let_1 V_xs)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ofoldl T_a T_b V_f_2 V_a_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b V_f_2 (tptp.hAPP _let_1 V_xs_2)) V_ys_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b V_f_2 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_x_2)) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_a V_n_2) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_ns_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofoldl tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n_2) V_ns_2)))))) (forall ((T_a $$unsorted)) (= (tptp.c_List_Orotate T_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Fun_Oid (tptp.tc_List_Olist T_a)))) (forall ((V_zs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_zs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 V_x_2) V_zs_2))))) (forall ((V_zs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (=> (tptp.class_Groups_Osemigroup__add T_a) (= (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 (tptp.hAPP _let_2 V_y_2)) V_zs_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 V_y_2) V_zs_2))))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_m) (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs)) (tptp.hAPP (tptp.c_List_Orotate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) V_xs))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Orotate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_n_2)) (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 _let_1 _let_1 (tptp.c_List_Orotate T_a V_m_2)) (tptp.c_List_Orotate T_a V_n_2))))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_a V_n_2) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n_2) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b V_f_2 V_a_2) (tptp.c_List_Olist_ONil T_b)) V_a_2)) _let_73 (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orotate1 T_a))) (let ((_let_2 (tptp.c_List_Orotate T_a V_n))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs) (tptp.hAPP (tptp.c_List_Orotate T_a (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_n (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) V_xs))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Orotate T_a (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_xs) (tptp.hAPP (tptp.c_List_Orotate1 T_a) (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs)))) (forall ((T_a $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.c_List_Ofoldl T_a T_a (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_ns_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.c_List_Ofoldl tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_ns_2) _let_1) (and (= V_m_2 _let_1) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) V_ns_2))) (= B_x (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (=> (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_n (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs) V_xs))) (forall ((V_n $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs) V_xs))) (forall ((V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_a T_b V_f_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_a (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) V_f_2) V_y_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_a (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.c_Orderings_Obot__class_Obot T_a)) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_a (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) (tptp.c_Orderings_Otop__class_Otop T_a)) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_a (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_a (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_a (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a (tptp.c_Orderings_Oord__class_Omin T_a) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_a (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a (tptp.c_Orderings_Oord__class_Omax T_a) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_a _let_1 (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 V_x_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_a _let_1 (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 V_x_2) V_xs_2))))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_a) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) V_f_2) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b (tptp.hAPP (tptp.c_COMBC T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a)) V_f_2)) (tptp.c_Orderings_Obot__class_Obot T_a)) V_xs_2)))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_a) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) V_f_2) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b (tptp.hAPP (tptp.c_COMBC T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a)) V_f_2)) (tptp.c_Orderings_Otop__class_Otop T_a)) V_xs_2)))) _let_72 (forall ((V_ns_2 $$unsorted) (V_n_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) V_ns_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_List_Ofoldl tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_ns_2))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_n (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)))) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) _let_1) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) _let_1) V_xs))))) _let_71 (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (exists ((B_b $$unsorted) (B_z $$unsorted)) (= V_y_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) B_b) B_z))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_x_2) V_y_2))) (tptp.c_List_Olexord T_a V_r_2)))))) (forall ((T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted) (V_a_2 $$unsorted) (V_m_2 $$unsorted)) (=> (= (tptp.hAPP V_m_2 V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_b_2) (tptp.c_Map_Oran T_b T_a V_m_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_a (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a (tptp.c_COMBI T_a)) V_x_2) (tptp.c_List_Olinorder__class_Osorted__list__of__set T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.c_List_Olist_ONil T_a)) V_y_2)) (tptp.c_List_Olexord T_a V_r_2))) (exists ((B_a $$unsorted) (B_x $$unsorted)) (= V_y_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) B_a) B_x)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_x_2) V_xs_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_x_2) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) _let_1))))) (forall ((V_ys_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_a V_f_2) V_x_2))) (let ((_let_2 (tptp.c_List_Olist_OCons T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_y_2))) (let ((_let_4 (tptp.hAPP _let_3 V_ys_2))) (let ((_let_5 (tptp.hAPP _let_1 _let_4))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))))) (=> (tptp.class_Orderings_Olinorder T_a) (and (=> _let_6 (= _let_5 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_4))) (=> (not _let_6) (= _let_5 (tptp.hAPP _let_3 (tptp.hAPP _let_1 V_ys_2))))))))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olinorder__class_Oinsort__key T_b T_a V_f_2))) (let ((_let_2 (tptp.hAPP _let_1 V_y_2))) (let ((_let_3 (tptp.hAPP _let_1 V_x_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_xs_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a (tptp.c_COMBI T_a)))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_y_2))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_xs_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs_2)))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_Finite__Set_Ofun__left__comm T_a (tptp.tc_List_Olist T_a) (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a (tptp.c_COMBI T_a))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Oremove1 T_a V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a (tptp.c_COMBI T_a)) V_x_2) V_xs_2)) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_a_2) V_xs_2) (tptp.c_List_Olist_ONil T_a))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_x_2) V_xs_2))) (and (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_a V_f_2 V_x_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_a V_f_2) V_x_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_COMBI T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_a T_a _let_1 V_x_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a _let_1) V_x_2) V_xs_2)))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Oinsort__insert__key T_a T_b V_f_2 V_x_2 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_a T_a (tptp.c_COMBI T_a) V_x_2 V_xs_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Oinsort__insert__key T_a T_a (tptp.c_COMBI T_a) V_x_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_a V_f_2 V_x_2 V_xs_2) V_xs_2)))) (forall ((V_x $$unsorted) (V_k $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_k) V_x)))) (let ((_let_2 (= V_k (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (and (=> _let_2 (= _let_1 V_x)) (=> (not _let_2) (= _let_1 (tptp.c_List_Olast T_a V_xs)))))))) (forall ((T_a $$unsorted) (V_xs $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Olast T_a V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_ys $$unsorted)) (=> (= V_ys (tptp.c_List_Olist_ONil T_a)) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.c_List_Olast T_a V_xs)))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_ys $$unsorted)) (=> (not (= V_ys (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.c_List_Olast T_a V_ys)))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_ys $$unsorted)) (let ((_let_1 (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)))) (let ((_let_2 (= V_ys (tptp.c_List_Olist_ONil T_a)))) (and (=> _let_2 (= _let_1 (tptp.c_List_Olast T_a V_xs))) (=> (not _let_2) (= _let_1 (tptp.c_List_Olast T_a V_ys))))))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (=> (= V_xs (tptp.c_List_Olist_ONil T_a)) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) V_x))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.c_List_Olast T_a V_xs)))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)))) (let ((_let_2 (= V_xs (tptp.c_List_Olist_ONil T_a)))) (and (=> _let_2 (= _let_1 V_x)) (=> (not _let_2) (= _let_1 (tptp.c_List_Olast T_a V_xs))))))) (forall ((T_a $$unsorted) (V_as_2 $$unsorted)) (=> (not (= V_as_2 (tptp.c_List_Olist_ONil T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.c_List_Olast T_a V_as_2)) (tptp.hAPP (tptp.c_List_Oset T_a) V_as_2))))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) (tptp.c_List_Olist_ONil T_a)))) V_x)) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs)) (tptp.c_List_Olast T_a V_xs)))) (forall ((T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (=> (not (= V_xs _let_1)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_Obutlast T_a V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.c_List_Olast T_a V_xs)) _let_1)) V_xs)))) (forall ((V_ys_2 $$unsorted) (V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) _let_1)) V_ys_2) (and (not (= V_ys_2 _let_1)) (= (tptp.c_List_Obutlast T_a V_ys_2) V_xs_2) (= (tptp.c_List_Olast T_a V_ys_2) V_x_2))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_a)) V_a_2) (tptp.c_List_Olists T_a V_A_2))) (or (= V_a_2 (tptp.c_List_Olist_ONil T_a)) (exists ((B_a $$unsorted) (B_l $$unsorted)) (and (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) B_a) B_l)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_a)) B_l) (tptp.c_List_Olists T_a V_A_2)))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_zs_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_ys_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ozip T_b T_a) V_xs_2))) (let ((_let_2 (tptp.c_Map_Omap__of T_b T_a))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_zs_2)))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_ys_2)))) (let ((_let_5 (tptp.c_Option_Ooption_OSome T_a))) (let ((_let_6 (tptp.c_Fun_Ofun__upd T_b (tptp.tc_Option_Ooption T_a)))) (let ((_let_7 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (let ((_let_8 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (= (tptp.hAPP _let_8 V_ys_2) _let_7) (=> (= (tptp.hAPP _let_8 V_zs_2) _let_7) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_6 _let_4) V_x_2) (tptp.hAPP _let_5 V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_6 _let_3) V_x_2) (tptp.hAPP _let_5 V_z_2))) (= _let_4 _let_3)))))))))))))) (forall ((V_l_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) (tptp.c_Map_Odom T_a T_b (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_l_2))))) (forall ((V_zs_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_ys_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ozip T_b T_a) V_xs_2))) (let ((_let_2 (tptp.c_Map_Omap__of T_b T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (let ((_let_4 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (= (tptp.hAPP _let_4 V_ys_2) _let_3) (=> (= (tptp.hAPP _let_4 V_zs_2) _let_3) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (=> (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_ys_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_zs_2))) (= V_ys_2 V_zs_2)))))))))) _let_70 (forall ((V_ps $$unsorted) (V_v $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_k $$unsorted) (V_l $$unsorted)) (let ((_let_1 (tptp.c_Map_Omap__of T_b T_c))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_l) V_v)) V_ps)) V_k))) (let ((_let_3 (= V_l V_k))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_c) V_v))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ps) V_k)))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.c_List_Oset _let_1))) (let ((_let_3 (tptp.c_Set_Oimage _let_1 T_a (tptp.c_Product__Type_Ofst T_a T_b)))) (let ((_let_4 (tptp.c_Map_Omap__of T_a T_b))) (=> (= (tptp.hAPP _let_4 V_xs_2) (tptp.hAPP _let_4 V_ys_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ys_2))))))))) (forall ((V_xys_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_b _let_1 (tptp.hAPP (tptp.c_Map_Omap__of T_b T_a) V_xys_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_xys_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_a))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_b T_a) V_xys_2) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xys_2)))))) (forall ((V_y_2 $$unsorted) (V_k_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_a))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_b T_a) V_xs_2) V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_k_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xs_2)))))) (forall ((V_xys_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.c_Map_Odom T_a T_b (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_xys_2)) (tptp.hAPP (tptp.c_Set_Oimage _let_1 T_a (tptp.c_Product__Type_Ofst T_a T_b)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xys_2))))) (forall ((V_x_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (exists ((B_y $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) B_y)))))) (forall ((V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (= (tptp.c_Map_Odom T_a T_b (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2))) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (forall ((V_ps_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Map_Omap__of T_a T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_prod T_a T_b)) V_p_2) V_ps_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)) (tptp.hAPP _let_1 V_ps_2)) (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) V_p_2)) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) (tptp.hAPP (tptp.c_Product__Type_Osnd T_a T_b) V_p_2)))))) (forall ((V_al_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_al_2))) (= (tptp.c_Map_Oran T_a T_b (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_al_2)) (tptp.hAPP (tptp.c_Set_Oimage _let_1 T_b (tptp.c_Product__Type_Osnd T_a T_b)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_al_2)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_1))) (let ((_let_5 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_5 T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_xs_2))) (= (tptp.hAPP (tptp.c_List_Oset _let_5) V_xs_2) (tptp.hAPP (tptp.c_Set_OCollect _let_5) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_a) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 T_a) tptp.c_fequal) (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_xs_2)))) (tptp.c_Option_Ooption_OSome T_b)))))))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_b) (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_a) _let_1) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (V_g_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c T_a))) (let ((_let_2 (tptp.tc_fun T_c T_b))) (= (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b V_g_2 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_a) (tptp.tc_fun _let_2 _let_1) T_a) (tptp.c_COMBB T_b T_a T_c)) V_g_2)) V_f_2) V_a_2) V_xs_2))))) (forall ((V_ys_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_c T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_a T_c) _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_c _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) (tptp.tc_fun _let_3 _let_2) T_a) (tptp.c_COMBB T_b _let_1 T_c)) (tptp.c_Product__Type_OPair T_a T_b))) V_f_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_c) V_xs_2) V_ys_2))))))) (forall ((V_ys_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun T_d _let_1))) (let ((_let_3 (tptp.tc_fun T_d T_c))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_g_2) V_ys_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_b T_d) T_a) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_d T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun T_d T_a) T_b) (tptp.hAPP (tptp.c_COMBB _let_1 T_a T_d) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_COMBB T_c _let_1 T_d)) (tptp.c_Product__Type_OPair T_b T_c))) V_g_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_d) V_xs_2) V_ys_2))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun T_c _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) V_g_2) V_xs_2)) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_d T_c) T_a) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_c T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun T_c T_a) T_d) (tptp.hAPP (tptp.c_COMBB _let_1 T_a T_c) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_2 T_d) (tptp.c_Product__Type_OPair T_b T_c)) V_g_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_d T_c) V_xs_2) V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_a) V_f_2) V_xs_2)) V_ys_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_c T_b) _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b _let_1) T_c) (tptp.c_Product__Type_OPair T_a T_b)) V_f_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_c T_b) V_xs_2) V_ys_2))))) (forall ((V_ys_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_d _let_1))) (let ((_let_4 (tptp.tc_fun T_d T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_a) V_f_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) V_g_2) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_c T_d) _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_c) (tptp.c_COMBB T_b _let_1 T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a _let_2 T_c) (tptp.c_Product__Type_OPair T_a T_b)) V_f_2))) V_g_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_c T_d) V_xs_2) V_ys_2)))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_List_Olist T_c) (tptp.tc_List_Olist T_b) (tptp.tc_List_Olist T_a) (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2)) (tptp.hAPP (tptp.c_List_Omap T_a T_c) V_g_2)) (tptp.hAPP (tptp.c_List_Omap T_a T_b) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_f_2) V_g_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_List_Olist T_c) (tptp.tc_List_Olist T_b) (tptp.tc_List_Olist T_a) (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2)) (tptp.hAPP (tptp.c_List_Omap T_a T_c) V_g_2)) (tptp.hAPP (tptp.c_List_Omap T_a T_b) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_f_2) V_g_2)))) (forall ((V_list_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_g_2) V_list_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_a) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c V_f_2) V_g_2)) V_list_2))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_g_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_a) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c V_f_2) V_g_2)) V_xs_2))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Omap T_a T_a) (tptp.c_COMBI T_a)) (tptp.c_Fun_Oid (tptp.tc_List_Olist T_a)))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Omap T_a T_a) (tptp.c_Fun_Oid T_a)) (tptp.c_Fun_Oid (tptp.tc_List_Olist T_a)))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_b $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist_Olist__size T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_g_2) V_xs_2)) (tptp.c_List_Olist_Olist__size T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_a tptp.tc_Nat_Onat T_b V_f_2) V_g_2) V_xs_2))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_n_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) V_xs_2)) V_n_2) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_n_2))))) (forall ((V_v_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_a T_b))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a T_b) V_f_2) V_y_2) V_v_2)) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Otl T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Otl T_a) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (forall ((V_ys_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_a) V_g_2) V_ys_2)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_ys_2)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_a V_f_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 V_ys_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_y_2 $$unsorted) (V_k_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_k_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_1 V_xs_2)) V_k_2) (tptp.hAPP V_f_2 V_y_2))))) (forall ((V_g_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_b T_a))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_xs_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) (tptp.hAPP (tptp.c_COMBK T_a T_b) _let_1)) V_xs_2)) _let_1)))) (forall ((V_xs_2 $$unsorted) (V_c_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_b T_a))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Osemiring__0 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_3) V_f_2)) V_c_2)) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))) V_c_2))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_b T_a))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_c_2))) (=> (tptp.class_Rings_Osemiring__0 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a T_a T_b) _let_3) V_f_2)) V_xs_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_b T_a))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_3) V_f_2)) V_g_2)) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_b T_a))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_a T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a T_a) T_b) _let_3) V_f_2)) V_g_2)) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_a) (tptp.c_COMBI T_a)) V_xs_2) V_xs_2)) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_n_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_b V_xs_2)) (tptp.c_List_Obutlast T_a (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_Olist_ONil T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2)) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) (tptp.c_List_Olist_ONil T_b)) (tptp.c_List_Olist_ONil T_a))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2) (tptp.c_List_Olist_ONil T_a)) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= V_xs_2 V_ys_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (=> (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (=> (tptp.c_Fun_Oinj__on T_b T_a V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= V_xs_2 V_ys_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Ohd T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) V_xs_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_n_2) V_xs_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (=> (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)) (=> (tptp.c_Fun_Oinj__on T_b T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 V_ys_2))) (= V_xs_2 V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2))) (let ((_let_2 (tptp.c_List_Oset T_a))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 V_ys_2))) (= (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= V_xs_2 V_ys_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_b V_n_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (tptp.c_Fun_Oinj__on T_b T_a V_f_2 (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Olast T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) V_xs_2)) (tptp.hAPP V_f_2 (tptp.c_List_Olast T_a V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_List_Omap T_b T_a))) (let ((_let_3 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_xs_2))) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_a T_a T_b _let_1) V_f_2)) V_xs_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 (tptp.hAPP (tptp.c_Set_Oimage _let_1 _let_2 (tptp.c_List_Oset T_a)) V_A_2))) (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_b) (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) V_A_2))))) (forall ((V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_a T_b) T_a) (tptp.c_Product__Type_Ofst T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)) V_xs_2))) (forall ((V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_a T_b) T_b) (tptp.c_Product__Type_Osnd T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)) V_ys_2))) (forall ((V_zs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_zs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.c_Product__Type_Osnd T_a T_b)) V_zs_2)) V_zs_2))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_b) (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_b) (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) V_i_2) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_Ofst tptp.tc_Code__Numeral_Ocode__numeral T_a)) V_xs_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Random_Opick T_a V_xs_2) V_i_2)) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Osnd tptp.tc_Code__Numeral_Ocode__numeral T_a)) V_xs_2))))))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_a T_b) V_f_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))))) (forall ((V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2) V_zs_2) (and (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_zs_2) V_xs_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.c_Product__Type_Osnd T_a T_b)) V_zs_2) V_ys_2)))))) (forall ((V_xs_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) (tptp.hAPP (tptp.c_COMBK T_a T_b) V_r_2)) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) V_r_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.c_List_Oset _let_1))) (let ((_let_3 (tptp.c_Map_Omap__of T_a T_b))) (let ((_let_4 (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a T_b)))) (let ((_let_5 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_ys_2))) (= (= (tptp.hAPP _let_3 V_xs_2) (tptp.hAPP _let_3 V_ys_2)) (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2))))))))))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_x_2 V_xs_2)))))))))) (forall ((V_s_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_Ofst tptp.tc_Code__Numeral_Ocode__numeral T_a)) V_xs_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) (tptp.hAPP (tptp.c_Product__Type_Ofst T_a (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP (tptp.c_Random_Oselect__weight T_a V_xs_2) V_s_2))) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Osnd tptp.tc_Code__Numeral_Ocode__numeral T_a)) V_xs_2))))))) _let_68 (forall ((V_x_2 $$unsorted) (V_k_2 $$unsorted) (V_t_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_c) V_x_2))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_c) V_t_2) V_k_2) _let_1) (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_a T_c) _let_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_c _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c _let_2) T_a) (tptp.c_Product__Type_OPair T_b T_c)) V_f_2))) V_t_2)) (tptp.hAPP V_f_2 V_k_2)) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_xys_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_xys_2))) (= (= (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_y_2) (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_xys_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xys_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_xys_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_xys_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_xys_2) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xys_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_xys_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_xys_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xys_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_xys_2) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_y_2)))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Osetsum_H T_a T_b V_f_2 V_A_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (let ((_let_4 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) (tptp.c_Hilbert__Choice_OEps _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_3 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_3) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 _let_4 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_4 tptp.tc_HOL_Obool) _let_3) tptp.c_fequal) (tptp.c_List_Oset T_a))) V_A_2))) (tptp.c_List_Odistinct T_a))))))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b)))))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_a T_b))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (=> (and (tptp.class_Groups_Omonoid__add T_b) (tptp.class_Groups_Oordered__ab__semigroup__add T_b)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_a) V_xs_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_a _let_1) (tptp.c_Product__Type_OPair T_a T_a)) (tptp.c_COMBI T_a))) V_xs_2)))) _let_67 (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oupto V_k_2 V_l_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Int_Oint T_a) V_f_2) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_a) V_f_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) _let_1)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oupto V_i_2 V_j_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_a) V_f_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) _let_1)) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Int_Oint T_a) V_f_2) _let_1)))))) (forall ((V_ks_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_1) (tptp.c_Product__Type_OPair T_a T_b)) V_f_2)) V_ks_2)) (tptp.c_Map_Orestrict__map T_a T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_b (tptp.tc_Option_Ooption T_b) T_a (tptp.c_Option_Ooption_OSome T_b)) V_f_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_ks_2))))) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (= (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2) (tptp.c_Map_Odom T_a T_b V_m_2)) (= (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b _let_1) (tptp.c_Product__Type_OPair T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_Option_Ooption T_b) T_b T_a) (tptp.c_Option_Othe T_b)) V_m_2))) V_xs_2)) V_m_2)))) (forall ((T_a $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_Random_Oselect__weight T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a)) (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) V_xs_2)) (tptp.c_Random_Oselect T_a V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_l_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_l_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= (tptp.hAPP (tptp.c_Random_Opick T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a)) (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral T_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) V_xs_2)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_l_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_l_2)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_c T_b))) (= (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_a T_c) _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_c _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) (tptp.tc_fun _let_3 _let_2) T_a) (tptp.c_COMBB T_b _let_1 T_c)) (tptp.c_Product__Type_OPair T_a T_b))) V_f_2))) V_xs_2)) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_Option_Ooption T_c) (tptp.tc_Option_Ooption T_b) T_a (tptp.hAPP (tptp.c_Option_Omap T_c T_b) V_f_2)) (tptp.hAPP (tptp.c_Map_Omap__of T_a T_c) V_xs_2))))))) (forall ((V_opt_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_b T_a) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_c T_b) V_g_2) V_opt_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_c T_a) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c V_f_2) V_g_2)) V_opt_2))) (forall ((V_option_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_b T_a) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_c T_b) V_g_2) V_option_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_c T_a) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_a T_c V_f_2) V_g_2)) V_option_2))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_Option_Ooption T_c) (tptp.tc_Option_Ooption T_b) (tptp.tc_Option_Ooption T_a) (tptp.hAPP (tptp.c_Option_Omap T_c T_b) V_f_2)) (tptp.hAPP (tptp.c_Option_Omap T_a T_c) V_g_2)) (tptp.hAPP (tptp.c_Option_Omap T_a T_b) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_a V_f_2) V_g_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (let ((_let_2 (tptp.tc_Option_Ooption T_c))) (let ((_let_3 (tptp.c_Fun_Ocomp _let_2 _let_1 T_a (tptp.hAPP (tptp.c_Option_Omap T_c T_b) V_f_2)))) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a _let_2) V_m_2) V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_c) V_b_2))) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a _let_1) (tptp.hAPP _let_3 V_m_2)) V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) (tptp.hAPP V_f_2 V_b_2)))))))) (= _let_66 (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_Nat_Onat tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Int_Oint tptp.c_Code__Numeral_Oof__nat) tptp.c_Int_Onat)) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Option_Omap T_a T_a) (tptp.c_COMBI T_a)) (tptp.c_Fun_Oid (tptp.tc_Option_Ooption T_a)))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Option_Omap T_a T_a) (tptp.c_Fun_Oid T_a)) (tptp.c_Fun_Oid (tptp.tc_Option_Ooption T_a)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_member tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1)) (= (= (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_x_2) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_y_2)) (= V_x_2 V_y_2))))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_n_2)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) V_m_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_b T_a) V_f_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_x_2)) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) (tptp.hAPP V_f_2 V_x_2)))) (forall ((V_y_2 $$unsorted) (V_xo_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_b T_a) V_f_2) V_xo_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_y_2)) (exists ((B_z $$unsorted)) (and (= V_xo_2 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) B_z)) (= (tptp.hAPP V_f_2 B_z) V_y_2))))) (forall ((V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (let ((_let_2 (tptp.tc_Option_Ooption T_c))) (= (tptp.c_Map_Odom T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_b) (tptp.tc_fun _let_2 _let_1) T_a) (tptp.c_Option_Omap T_c T_b)) V_f_2)) V_m_2)) (tptp.c_Map_Odom T_a T_c V_m_2))))) _let_65 (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_n)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) V_m)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_n_2) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m_2)) (= V_n_2 V_m_2))) (= _let_64 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat _let_31)) (= _let_63 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat _let_62)) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_n)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_n_2)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n_2) V_m_2)))) (forall ((V_i_2 $$unsorted)) (= (not (= V_i_2 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)) V_i_2)))) _let_61 (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP tptp.c_Code__Numeral_Osubtract__code__numeral (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_n)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) V_m)))) (tptp.c_Typedef_Otype__definition tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Nat_Onat tptp.c_Code__Numeral_Onat__of tptp.c_Code__Numeral_Oof__nat _let_60) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m))))) (forall ((V_x $$unsorted)) (= (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_x)) V_x)) (forall ((V_k $$unsorted)) (= (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_k)) V_k)) (forall ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_n)) V_n)) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_k_2) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_l_2)) (= V_k_2 V_l_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_x_2) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_y_2)) (= V_x_2 V_y_2))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) V_n_2) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n_2)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m_2))))) (forall ((V_x_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_x_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (forall ((V_k $$unsorted)) (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral) V_k)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_k))) _let_58 (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) V_n_2) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n_2)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m_2))))) (forall ((V_i $$unsorted)) (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_i) (tptp.c_Code__Numeral_Onat__of__aux V_i (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) _let_57 _let_56 (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m))))) (forall ((V_y_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_y_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_y_2)) V_y_2))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m))))) _let_55 (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m) (tptp.hAPP (tptp.hAPP tptp.c_Code__Numeral_Osubtract__code__numeral V_n) V_m))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Code__Numeral_Ocode__numeral V_n V_m) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m))))) (forall ((V_k $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_k))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not (= V_k (tptp.hAPP tptp.c_Code__Numeral_Oof__nat _let_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.hAPP tptp.c_Nat_OSuc _let_2))) _let_1)))))) (forall ((V_i_2 $$unsorted) (V_xq_2 $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Opos__not__seq V_xq_2) V_i_2) (tptp.c_Lazy__Sequence_Ohb__not__seq (tptp.hAPP V_xq_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_i_2))))) (forall ((V_l_2 $$unsorted) (V_x_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_k_2) V_x_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_l_2))) (exists ((B_x $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_l_2) V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) B_x)))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) V_xs_2))) V_x_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) (tptp.hAPP V_f_2 V_x_2)))) (=> (not _let_2) (= _let_1 (tptp.c_Option_Ooption_ONone T_b))))))) (forall ((V_no_2 $$unsorted) (V_yes_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (=> (= (tptp.c_List_Opartition T_a V_P_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) V_yes_2) V_no_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_yes_2)) (tptp.hAPP _let_1 V_no_2)) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 (tptp.tc_Option_Ooption T_a) T_c (tptp.hAPP (tptp.c_Option_Omap T_b T_a) V_f_2)) (tptp.hAPP (tptp.c_COMBK _let_1 T_c) (tptp.c_Option_Ooption_ONone T_b))) V_x_2) (tptp.c_Option_Ooption_ONone T_a)))) (forall ((V_opt_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_b T_a) V_f_2) V_opt_2) (tptp.c_Option_Ooption_ONone T_a)) (= V_opt_2 (tptp.c_Option_Ooption_ONone T_b)))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Option_Omap T_b T_a) V_f_2) (tptp.c_Option_Ooption_ONone T_b)) (tptp.c_Option_Ooption_ONone T_a))) (forall ((V_k $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_b T_a) (tptp.c_List_Olist_ONil (tptp.tc_prod T_b T_a))) V_k) (tptp.c_Option_Ooption_ONone T_a))) (forall ((V_xo_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Option_Oset T_a V_xo_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_xo_2 (tptp.c_Option_Ooption_ONone T_a)))) (forall ((T_a $$unsorted)) (= (tptp.c_Option_Oset T_a (tptp.c_Option_Ooption_ONone T_a)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_xa_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Option_Ooption_ONone T_a))) (let ((_let_2 (tptp.tc_Option_Ooption T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b _let_2) (tptp.hAPP (tptp.c_COMBK _let_2 T_b) _let_1)) V_x_2) _let_1) V_xa_2) _let_1)))) _let_53 (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.c_Map_Odom T_a T_b V_m_2))) (not (= (tptp.hAPP V_m_2 V_a_2) (tptp.c_Option_Ooption_ONone T_b))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Odom T_a T_b (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_b) T_a) (tptp.c_Option_Ooption_ONone T_b))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Map_Odom T_a T_b V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_f_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_b) T_a) (tptp.c_Option_Ooption_ONone T_b))))) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_A_2) V_x_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2)))) (and (=> _let_2 (= _let_1 (tptp.hAPP V_m_2 V_x_2))) (=> (not _let_2) (= _let_1 (tptp.c_Option_Ooption_ONone T_b))))))) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_A_2))) (= (tptp.hAPP (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_A_2) V_x_2) (tptp.c_Option_Ooption_ONone T_b)))) (forall ((V_x_2 $$unsorted) (V_D_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Option_Ooption_ONone T_a))) (= (tptp.hAPP (tptp.c_Map_Orestrict__map T_b T_a (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_a) T_b) _let_1) V_D_2) V_x_2) _let_1))) (forall ((V_x_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Map_Orestrict__map T_b T_a V_m_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_x_2) (tptp.c_Option_Ooption_ONone T_a))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case T_a T_b V_f1_2 V_f2_2) (tptp.c_Option_Ooption_ONone T_b)) V_f1_2)) (forall ((V_x_2 $$unsorted) (V_k_2 $$unsorted) (V_t_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (not (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a _let_1) V_t_2) V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_x_2)) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) (tptp.c_Option_Ooption_ONone T_b)))))) (forall ((T_a $$unsorted) (V_x_2 $$unsorted)) (= (not (= V_x_2 (tptp.c_Option_Ooption_ONone T_a))) (exists ((B_y $$unsorted)) (= V_x_2 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) B_y))))) (forall ((T_a $$unsorted) (V_x_2 $$unsorted)) (= (forall ((B_y $$unsorted)) (not (= V_x_2 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) B_y)))) (= V_x_2 (tptp.c_Option_Ooption_ONone T_a)))) (forall ((V_a_H $$unsorted) (T_a $$unsorted)) (not (= (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_a_H) (tptp.c_Option_Ooption_ONone T_a)))) (forall ((V_a_H $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Option_Ooption_ONone T_a) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_a_H)))) (forall ((T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Map_Oran T_b T_a (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_a) T_b) (tptp.c_Option_Ooption_ONone T_a))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_x_1 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.c_List_Olist_ONil (tptp.tc_prod T_a T_b))) V_x_1) (tptp.c_Option_Ooption_ONone T_b))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (= (tptp.hAPP (tptp.c_Option_Omap T_a T_b) V_f_2) (tptp.c_Option_Ooption_Ooption__case _let_1 T_a (tptp.c_Option_Ooption_ONone T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_1 T_a) (tptp.c_Option_Ooption_OSome T_b)) V_f_2))))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_Option_Ooption T_a)) (tptp.c_Option_Ooption_ONone T_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_fa_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Option_Ooption_Ooption__size T_a V_fa_2 (tptp.c_Option_Ooption_ONone T_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_Map_Odom T_b T_a V_f_2)))) (=> (= (tptp.hAPP V_f_2 V_x_2) (tptp.c_Option_Ooption_ONone T_a)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP _let_1 V_A_2))))) (forall ((V_pn $$unsorted)) (=> (not (= (tptp.hAPP tptp.c_Com_Obody V_pn) (tptp.c_Option_Ooption_ONone tptp.tc_Com_Ocom))) (tptp.hBOOL (tptp.hAPP tptp.c_Com_OWT (tptp.hAPP tptp.c_Com_Ocom_OBODY V_pn))))) (forall ((V_no_2 $$unsorted) (V_yes_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (= (tptp.c_List_Opartition T_a V_P_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_yes_2) V_no_2)) (and (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_yes_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_no_2))) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Opartition T_a V_P_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_1) _let_1))))) (forall ((V_b_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_a_2 $$unsorted) (V_m_2 $$unsorted)) (=> (= (tptp.hAPP V_m_2 V_a_2) (tptp.c_Option_Ooption_ONone T_a)) (= (tptp.c_Map_Oran T_b T_a (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b (tptp.tc_Option_Ooption T_a)) V_m_2) V_a_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_b_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_b_2) (tptp.c_Map_Oran T_b T_a V_m_2))))) (forall ((V_x_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)) V_x_2) (tptp.c_Option_Ooption_ONone T_b)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.tc_prod _let_2 _let_2))) (let ((_let_5 (tptp.tc_fun _let_2 _let_4))) (let ((_let_6 (tptp.tc_fun _let_2 _let_2))) (let ((_let_7 (tptp.tc_fun _let_4 _let_4))) (let ((_let_8 (tptp.tc_fun _let_2 _let_7))) (= (tptp.c_List_Opartition T_a V_P_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_2 _let_2 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_2 _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_5 _let_5) _let_2) (tptp.c_COMBS _let_2 _let_4 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 _let_2) (tptp.hAPP (tptp.c_COMBB _let_4 _let_7 _let_2) (tptp.hAPP (tptp.c_If _let_4) (tptp.hAPP V_P_2 V_x_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_2) _let_3) _let_1)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_6 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_6 _let_5) _let_2) (tptp.c_COMBB _let_2 _let_4 _let_2)) _let_3)) _let_1))) (tptp.c_List_Opartition T_a V_P_2 V_xs_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.c_Map_Odom T_b T_a V_f_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_3 (tptp.c_Map_Odom T_b T_a (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b (tptp.tc_Option_Ooption T_a)) V_f_2) V_x_2) V_y_2)))) (let ((_let_4 (= V_y_2 (tptp.c_Option_Ooption_ONone T_a)))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_5) _let_1) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_5))))) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 _let_1)))))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Map_Orestrict__map T_a T_b V_f_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)) V_f_2) V_x_2) (tptp.c_Option_Ooption_ONone T_b))))) (forall ((V_x_2 $$unsorted) (V_xys_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_a))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_b T_a) V_xys_2) V_x_2) (tptp.c_Option_Ooption_ONone T_a)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_Set_Oimage _let_1 T_b (tptp.c_Product__Type_Ofst T_b T_a)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xys_2)))))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (= (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_Option_Ooption_ONone T_a)) (tptp.hAPP (tptp.c_Set_Oimage T_a _let_1 (tptp.c_Option_Ooption_OSome T_a)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Map_Odom T_a T_b V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (exists ((B_v $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (= V_f_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a _let_1) (tptp.hAPP (tptp.c_COMBK _let_1 T_a) (tptp.c_Option_Ooption_ONone T_b))) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) B_v))))))) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (V_D_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 V_D_2))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)) _let_1) V_x_2) (tptp.c_Option_Ooption_ONone T_b)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) V_D_2)))) (let ((_let_4 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (and (=> _let_3 (= _let_2 (tptp.c_Map_Orestrict__map T_a T_b V_m_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_4) V_D_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_4)))))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)))) (let ((_let_4 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.hAPP (tptp.c_COMBC T_a _let_4 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (= (tptp.c_List_Olistsp T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_4) (tptp.hAPP _let_5 V_A_2)) (tptp.hAPP _let_5 V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) (tptp.hAPP _let_3 (tptp.c_List_Olists T_a V_A_2))) (tptp.hAPP _let_3 (tptp.c_List_Olists T_a V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_l_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_a _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_a)))) (let ((_let_3 (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_a)) V_l_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olists T_a V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olists T_a V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) V_l_2)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a V_A_2) (tptp.c_List_Olist_ONil T_a)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistsp T_a V_A_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_ys_2)))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olistsp T_a V_A_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a1_2) V_a2_2))) (and (tptp.hBOOL (tptp.hAPP V_A_2 V_a1_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_a2_2)))))) (forall ((V_xs_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a V_A_2) V_xs_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_A_2 B_x)))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) V_A_2)) V_B_2)) V_x_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a V_A_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a V_B_2) V_x_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olistsp T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_List_Olist T_a) tptp.tc_HOL_Obool)) (tptp.c_List_Olistsp T_a V_A_2)) (tptp.c_List_Olistsp T_a V_B_2)))) (forall ((V_B_2 $$unsorted) (V_l_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a V_A_2) V_l_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a V_B_2) V_l_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) V_l_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a V_A_2) (tptp.c_List_Olist_ONil T_a)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_List_Olist T_a) tptp.tc_HOL_Obool)) (tptp.c_List_Olistsp T_a V_A_2)) (tptp.c_List_Olistsp T_a V_B_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) V_A_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_a)) V_x_2) (tptp.c_List_Olists T_a V_A_2))))) _let_52 (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.c_Enum_Oproduct T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2) V_ys_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b _let_1) (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x_2)) V_ys_2)) (tptp.c_Enum_Oproduct T_a T_b V_xs_2 V_ys_2))))) (forall ((V_uu $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Enum_Oproduct T_a T_b (tptp.c_List_Olist_ONil T_a) V_uu) (tptp.c_List_Olist_ONil (tptp.tc_prod T_a T_b)))) (forall ((V_ys $$unsorted) (T_b $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_ys)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_prod T_a T_b)) (tptp.c_Enum_Oproduct T_a T_b V_xs V_ys)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset (tptp.tc_prod T_a T_b)) (tptp.c_Enum_Oproduct T_a T_b V_xs_2 V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_a T_b) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) (tptp.hAPP (tptp.c_List_Oset T_b) V_ys_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_Set_Oimage _let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool) _let_1) (tptp.hAPP (tptp.c_List_Oset _let_2) (tptp.c_Enum_Osublists T_a V_xs_2))) (tptp.hAPP (tptp.c_Set_OPow T_a) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) (tptp.c_Enum_Osublists T_a V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) V_xs))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_Enum_Osublists T_a _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons _let_1) _let_2) (tptp.c_List_Olist_ONil _let_1)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_List_Olist _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (= (tptp.c_Enum_Osublists T_a (tptp.hAPP _let_3 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet _let_2 _let_2) (tptp.c_Enum_Osublists T_a V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_2 _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) _let_2) (tptp.c_List_Oappend _let_1)) (tptp.hAPP (tptp.c_List_Omap _let_1 _let_1) _let_3))) (tptp.c_COMBI _let_2)))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_List_Olist T_a) _let_1) (tptp.c_List_Oset T_a)) (tptp.c_Enum_Osublists T_a V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_List_Oset _let_3) (tptp.c_Enum_On__lists T_a V_n_2 V_xs_2)) (tptp.hAPP (tptp.c_Set_OCollect _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_3 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_3) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_3) tptp.c_fequal) (tptp.c_Nat_Osize__class_Osize _let_3))) V_n_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_3) (tptp.c_Orderings_Oord__class_Oless__eq _let_2)) _let_1)) (tptp.hAPP _let_1 V_xs_2))))))))) _let_51 (forall ((V_n $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_List_Olist T_a)) (tptp.c_Enum_On__lists T_a V_n V_xs))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_Enum_On__lists T_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons _let_1) (tptp.c_List_Olist_ONil T_a)) (tptp.c_List_Olist_ONil _let_1))))) (forall ((T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_ONil _let_1))) (let ((_let_3 (tptp.c_List_Olist_ONil T_a))) (let ((_let_4 (tptp.c_Enum_On__lists T_a V_n _let_3))) (let ((_let_5 (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons _let_1) _let_3) _let_2))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) (tptp.c_Enum_On__lists T_a V_n V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) V_xs)) V_n)))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (V_ys_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_ys_2) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.c_Enum_On__lists T_a V_n_2 V_xs_2)))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) V_ys_2) V_n_2)))) (forall ((V_d_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Code__Numeral_Oint__of V_d_2))) (= (tptp.c_Smallcheck_Ofull__small__class_Ofull__small tptp.tc_Int_Oint V_f_2 V_d_2) (tptp.c_Smallcheck_Ofull__small_H V_f_2 _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) _let_1))))) (forall ((V_f_2 $$unsorted) (V_i_2 $$unsorted) (V_d_2 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist tptp.tc_Code__Evaluation_Oterm))) (let ((_let_2 (tptp.c_Smallcheck_Osmall_H V_f_2 V_d_2 V_i_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_d_2) V_i_2)))) (and (=> _let_3 (= _let_2 (tptp.c_Option_Ooption_ONone _let_1))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case (tptp.tc_Option_Ooption _let_1) _let_1 (tptp.c_Smallcheck_Osmall_H V_f_2 V_d_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (tptp.c_Option_Ooption_OSome _let_1)) (tptp.hAPP V_f_2 V_i_2))))))))) (forall ((V_i_2 $$unsorted) (V_d_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist tptp.tc_Code__Evaluation_Oterm))) (let ((_let_2 (tptp.tc_Option_Ooption _let_1))) (let ((_let_3 (tptp.c_Smallcheck_Osmall_H V_f_2 V_d_2 V_i_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_d_2) V_i_2)))) (let ((_let_5 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_6 (tptp.tc_fun tptp.tc_Int_Oint _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod _let_6 _let_5) tptp.c_Smallcheck_Osmall_H__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_6 _let_5) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_d_2) V_i_2)))) (and (=> _let_4 (= _let_3 (tptp.c_Option_Ooption_ONone _let_1))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case _let_2 _let_1 (tptp.c_Smallcheck_Osmall_H V_f_2 V_d_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (tptp.c_Option_Ooption_OSome _let_1)) (tptp.hAPP V_f_2 V_i_2))))))))))))) (forall ((V_d_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Code__Numeral_Oint__of V_d_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Smallcheck_Osmall__class_Osmall tptp.tc_Int_Oint) V_f_2) V_d_2) (tptp.c_Smallcheck_Osmall_H V_f_2 _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) _let_1))))) (forall ((V_d_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption (tptp.tc_List_Olist tptp.tc_Code__Evaluation_Oterm)))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (=> (and (tptp.class_Smallcheck_Osmall T_b) (tptp.class_Smallcheck_Osmall T_a)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Smallcheck_Osmall__class_Osmall _let_2) V_f_2) V_d_2) (tptp.hAPP (tptp.hAPP (tptp.c_Smallcheck_Osmall__class_Osmall T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a tptp.tc_Code__Numeral_Ocode__numeral _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_1) T_a) (tptp.c_Smallcheck_Osmall__class_Osmall T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_2) _let_3 T_a) (tptp.hAPP (tptp.c_COMBB _let_2 _let_1 T_b) V_f_2)) (tptp.c_Product__Type_OPair T_a T_b)))) V_d_2)) V_d_2))))))) (forall ((V_P_2 $$unsorted) (V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_a0_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_2 (tptp.tc_fun tptp.tc_Int_Oint (tptp.tc_Option_Ooption (tptp.tc_List_Olist tptp.tc_Code__Evaluation_Oterm))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod _let_2 _let_1) tptp.c_Smallcheck_Osmall_H__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_1) V_a0_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_a1_2) V_a2_2)))) (=> (forall ((B_f $$unsorted) (B_d $$unsorted) (B_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP V_P_2 B_f) B_d))) (let ((_let_2 (tptp.tc_List_Olist tptp.tc_Code__Evaluation_Oterm))) (let ((_let_3 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_4 (tptp.tc_fun tptp.tc_Int_Oint (tptp.tc_Option_Ooption _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod _let_4 _let_3) tptp.c_Smallcheck_Osmall_H__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_4 _let_3) B_f) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) B_d) B_i)))) (=> (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_d) B_i))) (=> (= (tptp.hAPP B_f B_i) (tptp.c_Option_Ooption_ONone _let_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))) (tptp.hBOOL (tptp.hAPP _let_1 B_i))))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_P_2 V_a0_2) V_a1_2) V_a2_2))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Enum_Oenum__class_Oenum T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Ozip T_b T_a) _let_1))) (let ((_let_3 (tptp.c_Map_Omap__of T_b T_a))) (let ((_let_4 (tptp.c_Fun_Ocomp (tptp.tc_Option_Ooption T_a) T_a T_b (tptp.c_Option_Othe T_a)))) (let ((_let_5 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) _let_1))) (let ((_let_6 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (and (tptp.class_Enum_Oenum T_b) (tptp.class_Enum_Oenum T_a)) (=> (= (tptp.hAPP _let_6 V_xs_2) _let_5) (=> (= (tptp.hAPP _let_6 V_ys_2) _let_5) (=> (= (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs_2))) (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ys_2)))) (= V_xs_2 V_ys_2)))))))))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Enum_Oenum T_a) (tptp.class_Enum_Oenum T_b)) (= (tptp.c_Enum_Oenum__class_Oenum (tptp.tc_prod T_a T_b)) (tptp.c_Enum_Oproduct T_a T_b (tptp.c_Enum_Oenum__class_Oenum T_a) (tptp.c_Enum_Oenum__class_Oenum T_b))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_Enum_Oenum__class_Oenum T_a))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_Enum_Oenum__class_Oenum T_a)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_Enum_Oenum__class_Oenum T_a)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.c_Enum_Oenum__class_Oenum _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons _let_1) (tptp.c_Option_Ooption_ONone T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a _let_1) (tptp.c_Option_Ooption_OSome T_a)) (tptp.c_Enum_Oenum__class_Oenum T_a))))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.c_Enum_Oenum__class_Oenum T_a))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Enum_Oenum__class_Oenum T_a))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.tc_Option_Ooption T_b))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a T_b))) (=> (and (tptp.class_Enum_Oenum T_a) (tptp.class_HOL_Oequal T_a) (tptp.class_Enum_Oenum T_b)) (= (tptp.c_Enum_Oenum__class_Oenum _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 _let_2) (tptp.c_Fun_Ocomp _let_3 T_b T_a (tptp.c_Option_Othe T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_List_Olist (tptp.tc_prod T_a T_b)) _let_4 _let_2) (tptp.c_Map_Omap__of T_a T_b)) (tptp.hAPP (tptp.c_List_Ozip T_a T_b) _let_1)))) (tptp.c_Enum_On__lists T_b (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_1) (tptp.c_Enum_Oenum__class_Oenum T_b))))))))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Enum_Oenum__class_Oenum T_a))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.tc_Option_Ooption T_b))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a T_b))) (=> (and (tptp.class_Enum_Oenum T_a) (tptp.class_Enum_Oenum T_b)) (= (tptp.c_Enum_Oenum__class_Oenum _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 _let_2) (tptp.c_Fun_Ocomp _let_3 T_b T_a (tptp.c_Option_Othe T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_List_Olist (tptp.tc_prod T_a T_b)) _let_4 _let_2) (tptp.c_Map_Omap__of T_a T_b)) (tptp.hAPP (tptp.c_List_Ozip T_a T_b) _let_1)))) (tptp.c_Enum_On__lists T_b (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_1) (tptp.c_Enum_Oenum__class_Oenum T_b))))))))))) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.c_Enum_Oall__n__lists T_a V_P_2 V_n_2) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.c_Enum_On__lists T_a V_n_2 (tptp.c_Enum_Oenum__class_Oenum T_a))))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))))) (forall ((V_n_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.c_Enum_Oex__n__lists T_a V_P_2 V_n_2) (exists ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.c_Enum_On__lists T_a V_n_2 (tptp.c_Enum_Oenum__class_Oenum T_a))))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Enum_Oenum__class_Oenum T_a))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.tc_Option_Ooption T_b))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a T_b))) (=> (and (tptp.class_Enum_Oenum T_b) (tptp.class_Enum_Oenum T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all _let_5) V_P_2)) (tptp.c_Enum_Oall__n__lists T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 tptp.tc_HOL_Obool _let_2) V_P_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 _let_2) (tptp.c_Fun_Ocomp _let_3 T_b T_a (tptp.c_Option_Othe T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_List_Olist (tptp.tc_prod T_a T_b)) _let_4 _let_2) (tptp.c_Map_Omap__of T_a T_b)) (tptp.hAPP (tptp.c_List_Ozip T_a T_b) _let_1)))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_1)))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Enum_Oenum__class_Oenum T_a))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.tc_Option_Ooption T_b))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a T_b))) (=> (and (tptp.class_Enum_Oenum T_b) (tptp.class_Enum_Oenum T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex _let_5) V_P_2)) (tptp.c_Enum_Oex__n__lists T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 tptp.tc_HOL_Obool _let_2) V_P_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 _let_2) (tptp.c_Fun_Ocomp _let_3 T_b T_a (tptp.c_Option_Othe T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_List_Olist (tptp.tc_prod T_a T_b)) _let_4 _let_2) (tptp.c_Map_Omap__of T_a T_b)) (tptp.hAPP (tptp.c_List_Ozip T_a T_b) _let_1)))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_1)))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (and (tptp.class_Enum_Oenum T_b) (tptp.class_Enum_Oenum T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all _let_1) V_P_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_a) (tptp.c_Enum_Oenum__class_Oenum__all T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_2 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) V_P_2)) (tptp.c_Product__Type_OPair T_a T_b)))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (and (tptp.class_Enum_Oenum T_b) (tptp.class_Enum_Oenum T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex _let_1) V_P_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_a) (tptp.c_Enum_Oenum__class_Oenum__ex T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_2 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) V_P_2)) (tptp.c_Product__Type_OPair T_a T_b)))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_a) V_P_2))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_a) V_P_2))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_a) V_P_2)) (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1)))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_a) V_P_2)) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1)))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all _let_1) V_P_2)) (and (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Option_Ooption_ONone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) V_P_2) (tptp.c_Option_Ooption_OSome T_a))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (=> (tptp.class_Enum_Oenum T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex _let_1) V_P_2)) (or (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Option_Ooption_ONone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_a) V_P_2) (tptp.c_Option_Ooption_OSome T_a))))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Orderings_Oorder T_b) (tptp.class_Enum_Oenum T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b)) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_b)) V_f_2)) V_g_2)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a T_b))) (=> (and (tptp.class_Orderings_Oorder T_b) (tptp.class_Enum_Oenum T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_f_2) V_g_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_a) tptp.c_fequal) V_f_2)) V_g_2))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Enum_Oenum__class_Oenum T_a))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.tc_Option_Ooption T_b))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a T_b))) (=> (and (tptp.class_Enum_Oenum T_b) (tptp.class_Enum_Oenum T_a) (tptp.class_HOL_Oequal T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex _let_5) V_P_2)) (tptp.c_Enum_Oex__n__lists T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 tptp.tc_HOL_Obool _let_2) V_P_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 _let_2) (tptp.c_Fun_Ocomp _let_3 T_b T_a (tptp.c_Option_Othe T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_List_Olist (tptp.tc_prod T_a T_b)) _let_4 _let_2) (tptp.c_Map_Omap__of T_a T_b)) (tptp.hAPP (tptp.c_List_Ozip T_a T_b) _let_1)))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_1)))))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Enum_Oenum__class_Oenum T_a))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.tc_Option_Ooption T_b))) (let ((_let_4 (tptp.tc_fun T_a _let_3))) (let ((_let_5 (tptp.tc_fun T_a T_b))) (=> (and (tptp.class_Enum_Oenum T_b) (tptp.class_Enum_Oenum T_a) (tptp.class_HOL_Oequal T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all _let_5) V_P_2)) (tptp.c_Enum_Oall__n__lists T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 tptp.tc_HOL_Obool _let_2) V_P_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 _let_2) (tptp.c_Fun_Ocomp _let_3 T_b T_a (tptp.c_Option_Othe T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_List_Olist (tptp.tc_prod T_a T_b)) _let_4 _let_2) (tptp.c_Map_Omap__of T_a T_b)) (tptp.hAPP (tptp.c_List_Ozip T_a T_b) _let_1)))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_1)))))))))) (forall ((V_x $$unsorted) (V_ys $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.c_Enum_Oenum__class_Oenum T_a))) (exists ((B_y $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) (tptp.c_Enum_Oenum__class_Oenum T_a)) V_ys)) V_x) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) B_y)))))) _let_50 (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)))) (forall ((V_n_2 $$unsorted) (V_bs_2 $$unsorted) (V_as_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_as_2 V_bs_2) (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n_2) V_as_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_n_2) V_bs_2)))) (forall ((V_n_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_n_2) V_ys_2)))) (forall ((T_a $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted)) (= (= V_xs_2 V_ys_2) (tptp.c_List_Olist__all2 T_a T_a tptp.c_fequal V_xs_2 V_ys_2))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_a_2 V_b_2) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_c_2 V_d_2) (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_b_2) V_d_2))))) (forall ((V_ys_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.c_List_Olist_ONil T_a) V_ys_2) (= V_ys_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 (tptp.c_List_Olist_ONil T_b)) (= V_xs_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((V_vs_2 $$unsorted) (V_us_2 $$unsorted) (V_P_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_us_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_ys_2) V_vs_2)) (and (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_us_2 V_vs_2))))) (forall ((V_bs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_as_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c T_b))) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_as_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_bs_2)) (tptp.c_List_Olist__all2 T_a T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_2 _let_1) T_a) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_c)) V_P_2)) V_f_2) V_as_2 V_bs_2))))) (forall ((V_bs_2 $$unsorted) (V_as_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_a) V_f_2) V_as_2) V_bs_2) (tptp.c_List_Olist__all2 T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_c) V_P_2) V_f_2) V_as_2 V_bs_2))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_y_2) V_ys_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_x_2) V_y_2)) (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2)))) (forall ((V_p_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_p_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_p_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_ys_2) V_p_2)))))) (forall ((V_p_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_p_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_p_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_ys_2) V_p_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (and (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2)) (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_i)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_ys_2) B_i)))))))) (forall ((V_i_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_x_2) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys_2))) (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_i_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_ys_2) V_i_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_ys_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_i_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (=> (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_x_2) V_y_2)) (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs_2) V_i_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_ys_2) V_i_2) V_y_2)))))) (forall ((V_P_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_a_2) V_b_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b tptp.tc_HOL_Obool) V_P_2) B_x))))) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_a_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_b_2)) (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_a_2 V_b_2)))) (forall ((V_x_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_i))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))))) (forall ((V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_a T_b V_f_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_a T_b V_f_2) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_a (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) V_f_2) V_y_2) (tptp.c_List_Oremdups T_a V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x_2) V_xs_2)) (tptp.hAPP (tptp.c_List_OremoveAll T_b (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.c_List_Oremdups T_a V_xs)))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_List_Oremdups T_a V_xs))) (tptp.hAPP _let_1 V_xs))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (= (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x_2) V_xs_2) V_xs_2))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_Oremdups T_a V_xs_2) V_xs_2) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x) V_xs)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (= (tptp.c_List_Oremdups T_a V_xs) V_xs))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (= (tptp.c_List_Oremove1 T_a V_x (tptp.c_List_Oremdups T_a V_xs)) (tptp.c_List_Oremdups T_a (tptp.c_List_Oremove1 T_a V_x V_xs))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oremdups T_a V_xs_2))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (= (tptp.hAPP _let_2 _let_1) (tptp.hAPP _let_2 V_xs_2)) (= _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.c_List_Oremdups T_a V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oremdups T_a V_xs))) (= (tptp.c_List_Oremdups T_a _let_1) _let_1))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_OremoveAll T_a V_x))) (let ((_let_2 (tptp.c_List_Oappend T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_xs) V_ys)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 V_ys)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Oremdups T_a _let_1) _let_1))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= _let_1 (tptp.c_List_Oremdups T_a V_x_2)) (= V_x_2 _let_1)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x) _let_1) _let_1))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.c_List_Oremdups T_a V_x_2) _let_1) (= V_x_2 _let_1)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.c_List_Oremdups T_a (tptp.hAPP _let_1 (tptp.c_List_Oremdups T_b V_xs_2))) (tptp.c_List_Oremdups T_a (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_List_OremoveAll T_a V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_xs))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_xs)))) (let ((_let_5 (= V_x V_y))) (and (=> _let_5 (= _let_4 _let_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 _let_2)))))))))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (= (tptp.c_List_Oremove1 T_a V_x V_xs) (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x) V_xs)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oremdups T_a V_xs_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (let ((_let_3 (tptp.c_List_Oremdups T_a (tptp.hAPP _let_2 V_xs_2)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (and (=> _let_4 (= _let_3 _let_1)) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 _let_1))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.c_List_Oremdups T_a V_xs_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2))) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x_2) V_xs_2)) (tptp.hAPP (tptp.c_List_OremoveAll T_b (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))) (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) (tptp.c_List_Oremdups T_a (tptp.c_List_Oconcat T_a V_xss_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xss_2)) (tptp.c_List_Oset T_a)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_a (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)) (tptp.c_List_Olinorder__class_Osort__key T_a T_a (tptp.c_COMBI T_a) (tptp.c_List_Oremdups T_a V_xs_2))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_List_Olist T_a)) V_x) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_x) (tptp.c_List_Oconcat T_a V_xs)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP _let_1 (tptp.c_List_Oconcat T_b V_xs_2)) (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_List_Olist T_b) (tptp.tc_List_Olist T_a)) _let_1) V_xs_2))))) (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_Olist_ONil T_a) (tptp.c_List_Oconcat T_a V_xss_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xss_2))) (= B_x (tptp.c_List_Olist_ONil T_a))))))) (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_Oconcat T_a V_xss_2) (tptp.c_List_Olist_ONil T_a)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xss_2))) (= B_x (tptp.c_List_Olist_ONil T_a))))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 _let_1) _let_1)))) (forall ((T_a $$unsorted)) (= (tptp.c_List_Oconcat T_a (tptp.c_List_Olist_ONil (tptp.tc_List_Olist T_a))) (tptp.c_List_Olist_ONil T_a))) _let_49 (forall ((V_xss_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_List_Ofoldl _let_2 _let_2 _let_1 V_xs_2) V_xss_2) (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) (tptp.c_List_Oconcat T_a V_xss_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 V_xs_2)) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 V_xs_2)) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (= (tptp.hAPP _let_2 (tptp.c_List_Oconcat T_a V_xss_2)) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat) _let_2) V_xss_2)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP _let_1 (tptp.c_List_Oconcat T_a V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_List_Oset _let_2) V_xs_2)) _let_1))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_x_2) (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 V_xs_2))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend (tptp.tc_List_Olist T_a)) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_Oconcat T_a V_xs)) (tptp.c_List_Oconcat T_a V_ys)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_COMBI T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osort__key T_a T_a _let_1 V_xs_2) (tptp.hAPP (tptp.c_List_Ofoldl _let_2 T_a (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a _let_1)) (tptp.c_List_Olist_ONil T_a)) V_xs_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist (tptp.tc_List_Olist T_a))))) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun _let_1 tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.tc_prod _let_1 _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_6) B_x) (tptp.hAPP (tptp.c_List_Oset _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip _let_1 _let_1) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_5 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_5 _let_4) _let_1) (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_3 _let_1) tptp.c_fequal) _let_2))) _let_2)) B_x)))))))))) (=> (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= (= (tptp.c_List_Oconcat T_a V_xs_2) (tptp.c_List_Oconcat T_a V_ys_2)) (= V_xs_2 V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist (tptp.tc_List_Olist T_a))))) (=> (= (tptp.c_List_Oconcat T_a V_xs_2) (tptp.c_List_Oconcat T_a V_ys_2)) (=> (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun _let_1 tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.tc_prod _let_1 _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_6) B_x) (tptp.hAPP (tptp.c_List_Oset _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip _let_1 _let_1) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_5 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_5 _let_4) _let_1) (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_3 _let_1) tptp.c_fequal) _let_2))) _let_2)) B_x)))))))))) (= V_xs_2 V_ys_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_List_Olist_OCons T_a)) V_f_2)) (tptp.c_List_Olist_ONil T_a))) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.tc_List_Olist _let_1))) (= (tptp.c_Enum_On__lists T_a (tptp.hAPP tptp.c_Nat_OSuc V_n_2) V_xs_2) (tptp.c_List_Oconcat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_a _let_1) (tptp.tc_fun _let_1 _let_2) _let_1) (tptp.c_List_Omap T_a _let_1)) (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_List_Olist_OCons T_a)))) V_xs_2)) (tptp.c_Enum_On__lists T_a V_n_2 V_xs_2))))))) (forall ((V_xss_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_ONil _let_1))) (let ((_let_3 (tptp.c_List_Olist_OCons _let_1))) (let ((_let_4 (tptp.tc_List_Olist _let_1))) (let ((_let_5 (tptp.c_List_Olist_ONil T_a))) (let ((_let_6 (tptp.c_List_Olist_OCons T_a))) (let ((_let_7 (tptp.hAPP _let_6 V_x_2))) (= (tptp.c_List_Otranspose T_a (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_7 V_xs_2)) V_xss_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_7 (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_1) (tptp.c_List_Olist_Olist__case _let_1 T_a _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBK _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) _let_6) _let_5)))) V_xss_2)))) (tptp.c_List_Otranspose T_a (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) (tptp.c_List_Oconcat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_4) (tptp.c_List_Olist_Olist__case _let_4 T_a _let_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 _let_4) T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_4 _let_4) _let_3) _let_2)))) V_xss_2)))))))))))))) (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 _let_2) (tptp.c_List_Olist_Olist__case _let_2 T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_a) (tptp.c_COMBK _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.c_List_Olist_OCons T_a)) _let_1)))) V_xss_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 T_a) (tptp.c_List_Ohd T_a)) (tptp.hAPP (tptp.c_List_Ofilter _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_2) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 tptp.tc_HOL_Obool) tptp.c_fequal) _let_1))) V_xss_2)))))) (forall ((V_xs_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_a V_p_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP _let_1 (tptp.c_List_Oconcat T_a V_xs_2)) (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 _let_2) _let_1) V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_a V_P_2))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 V_xs_2)) (tptp.c_List_Olinorder__class_Osort__key T_a T_b V_f_2 (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xss $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Otranspose T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_List_Olist T_a)) (tptp.c_List_Olist_ONil T_a)) V_xss)) (tptp.c_List_Otranspose T_a V_xss))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_Product__Type_Osnd _let_1 _let_1) (tptp.c_List_Opartition T_a V_P_2 V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a tptp.c_fNot) V_P_2)) V_xs_2)))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) V_P_2)) V_xs_2))) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2) V_xs_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_a V_P_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_x_2) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))) (= (tptp.c_List_Oremove1 T_a V_x_2 _let_1) _let_1)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_Q_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_a V_Q_2))) (= (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_a V_x_2 V_xs_2)) (tptp.c_List_Oremove1 T_a V_x_2 (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) (tptp.hAPP (tptp.c_List_Ofilter T_a V_Q_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) V_Q_2)) V_P_2)) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) (tptp.c_List_Opartition T_a V_P_2 V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_a V_P_2))) (let ((_let_2 (tptp.c_List_Oappend T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) V_ys_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 V_ys_2)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil (tptp.tc_List_Olist T_a)))) (= (tptp.c_List_Otranspose T_a _let_1) _let_1))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) _let_1) _let_1))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2) (tptp.c_List_Olist_ONil T_a)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_a V_P_2))) (let ((_let_2 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_xs_2)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP _let_3 _let_2))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_a tptp.tc_HOL_Obool T_b V_P_2) V_f_2)) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))) (= (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x_2) _let_1) _let_1)))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_a V_P_2))) (= (tptp.c_List_Oremdups T_a (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_Oremdups T_a V_xs_2))))) _let_48 (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2))) (let ((_let_3 (tptp.c_List_Otranspose T_a V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) _let_3))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_1) _let_3) V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat T_a) (tptp.c_List_Onth T_a)) V_i_2)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1) _let_2) (tptp.c_Nat_Osize__class_Osize _let_1))) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap (tptp.tc_List_Olist T_b) (tptp.tc_List_Olist T_a)) (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2)))) (= (tptp.c_List_Otranspose T_a (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_Otranspose T_b V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2)) (tptp.hAPP (tptp.c_Set_OCollect T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) (tptp.hAPP _let_1 V_xs_2)))) V_P_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_a tptp.tc_HOL_Obool T_b V_P_2) V_f_2)) V_xs_2)))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))))) _let_47 (forall ((V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_y_2) (tptp.c_Orderings_Obot__class_Obot _let_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_y_2))) V_xs_2)))))) (forall ((V_s_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Osublist T_a V_xs_2 V_s_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (= (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_a)) (tptp.hAPP (tptp.c_List_Oset T_a) _let_1))) V_xs_2) _let_1)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_Otranspose T_a V_xs_2) (tptp.c_List_Olist_ONil (tptp.tc_List_Olist T_a))) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xs_2))) (= B_x (tptp.c_List_Olist_ONil T_a))))))) (forall ((V_xs_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_p_2) V_xs_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP _let_1 V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a tptp.tc_HOL_Obool tptp.tc_Nat_Onat) V_p_2) (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2)))))))) (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_ONil _let_1))) (let ((_let_3 (tptp.tc_List_Olist _let_1))) (= (tptp.c_List_Oconcat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_3) (tptp.c_List_Olist_Olist__case _let_3 T_a _let_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 _let_3) T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_3 _let_3) (tptp.c_List_Olist_OCons _let_1)) _let_2)))) V_xss_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_1) (tptp.c_List_Otl T_a)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_1) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olist_ONil T_a)))) V_xss_2))))))) (forall ((V_xss_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_ONil _let_1))) (let ((_let_3 (tptp.c_List_Olist_OCons _let_1))) (let ((_let_4 (tptp.tc_List_Olist _let_1))) (let ((_let_5 (tptp.c_List_Olist_ONil T_a))) (let ((_let_6 (tptp.c_List_Olist_OCons T_a))) (let ((_let_7 (tptp.hAPP _let_6 V_x_2))) (let ((_let_8 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_7 V_xs_2)) V_xss_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp _let_4 (tptp.c_List_Otranspose__rel T_a)) _let_8)) (= (tptp.c_List_Otranspose T_a _let_8) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_7 (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_1) (tptp.c_List_Olist_Olist__case _let_1 T_a _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_a) (tptp.c_COMBK _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) _let_6) _let_5)))) V_xss_2)))) (tptp.c_List_Otranspose T_a (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) (tptp.c_List_Oconcat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_4) (tptp.c_List_Olist_Olist__case _let_4 T_a _let_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 _let_4) T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_4 _let_4) _let_3) _let_2)))) V_xss_2)))))))))))))))) _let_46 (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Omap__filter T_b T_a V_f_2 (tptp.c_List_Olist_ONil T_b)) (tptp.c_List_Olist_ONil T_a))) (forall ((V_P_2 $$unsorted) (V_z_2 $$unsorted) (V_k_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_z_2))) (let ((_let_2 (tptp.c_Map_Omap__of T_b T_a))) (=> (= (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) V_k_2) _let_1) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_k_2) V_z_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_Ofilter (tptp.tc_prod T_b T_a) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_a tptp.tc_HOL_Obool) V_P_2)) V_xs_2)) V_k_2) _let_1)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap__filter T_b T_a V_f_2 V_xs_2))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Omap__filter T_b T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case _let_2 T_a _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_2) (tptp.c_List_Olist_OCons T_a)) _let_1)) (tptp.hAPP V_f_2 V_x_2)))))) (forall ((V_is_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ozip T_a tptp.tc_Nat_Onat) V_xs_2))) (let ((_let_2 (tptp.c_Product__Type_Osnd T_a tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_prod T_a tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_3) V_P_2))) (let ((_let_5 (tptp.hAPP (tptp.c_List_Omap _let_3 T_a) (tptp.c_Product__Type_Ofst T_a tptp.tc_Nat_Onat)))) (= (tptp.hAPP _let_5 (tptp.hAPP (tptp.c_List_Ofilter _let_3 (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_3) tptp.c_Nat_OSuc) _let_2))) (tptp.hAPP _let_1 V_is_2))) (tptp.hAPP _let_5 (tptp.hAPP (tptp.c_List_Ofilter _let_3 (tptp.hAPP _let_4 _let_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) V_is_2))))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Random_Oselect__weight T_a (tptp.hAPP (tptp.c_List_Ofilter (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Code__Numeral_Ocode__numeral T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_COMBK tptp.tc_HOL_Obool T_a)) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))))) V_xs_2)) (tptp.c_Random_Oselect__weight T_a V_xs_2))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Random_Opick T_a (tptp.hAPP (tptp.c_List_Ofilter (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_a) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Code__Numeral_Ocode__numeral T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_a tptp.tc_HOL_Obool) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_COMBK tptp.tc_HOL_Obool T_a)) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))))) V_xs_2)) (tptp.c_Random_Opick T_a V_xs_2))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Olist_ONil _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_List_Olist _let_1) (tptp.c_List_Otranspose__rel T_a)) _let_2)) (= (tptp.c_List_Otranspose T_a _let_2) _let_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_a))) (let ((_let_2 (tptp.tc_fun _let_1 _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) (tptp.hAPP (tptp.c_List_Ofilter T_b V_P_2) V_xs_2)) (tptp.c_List_Omap__filter T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_1 _let_2) T_b) (tptp.c_If _let_1)) V_P_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a _let_1 T_b) (tptp.c_Option_Ooption_OSome T_a)) V_f_2))) (tptp.c_Option_Ooption_ONone T_a)) V_xs_2))))) (forall ((V_xss_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons _let_1) (tptp.c_List_Olist_ONil T_a)) V_xss_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_List_Olist _let_1) (tptp.c_List_Otranspose__rel T_a)) _let_2)) (= (tptp.c_List_Otranspose T_a _let_2) (tptp.c_List_Otranspose T_a V_xss_2)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.c_List_Ofoldr _let_1 tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat)) (tptp.c_Nat_Osize__class_Osize _let_1)) (tptp.c_List_Otranspose T_a V_xs_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_1) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olist_ONil T_a)))) V_xs_2))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize _let_2))) (let ((_let_4 (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.hAPP _let_4 _let_5))) (let ((_let_7 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP tptp.c_Nat_OSuc _let_7)) (tptp.c_List_Ofoldr _let_2 tptp.tc_Nat_Onat (tptp.hAPP _let_6 _let_3) V_xss_2 _let_1)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_5 _let_7) (tptp.c_List_Ofoldr _let_2 tptp.tc_Nat_Onat (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat)) _let_3)) (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (tptp.hAPP (tptp.c_List_Ofilter _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_2) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olist_ONil T_b)))) V_xss_2) _let_1)))))))))))) (forall ((V_a_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Ofoldr T_b T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2) V_a_2) (tptp.c_List_Ofoldr T_b T_a V_f_2 V_xs_2 (tptp.c_List_Ofoldr T_b T_a V_f_2 V_ys_2 V_a_2)))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Ofoldr T_b T_a V_f_2 (tptp.c_List_Olist_ONil T_b) V_a_2) V_a_2)) (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Ofoldr T_b T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2) V_a_2) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.c_List_Ofoldr T_b T_a V_f_2 V_xs_2 V_a_2)))) (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oab__semigroup__add T_a) (= (tptp.c_List_Ofoldr T_a T_a _let_1 V_xs_2 V_a_2) (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 V_a_2) V_xs_2))))) (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_g_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Ofoldr T_b T_a V_g_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2) V_a_2) (tptp.c_List_Ofoldr T_c T_a (tptp.hAPP (tptp.c_Fun_Ocomp T_b (tptp.tc_fun T_a T_a) T_c V_g_2) V_f_2) V_xs_2 V_a_2))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_2 _let_1) V_xs_2) (tptp.c_List_Ofoldr T_a T_a _let_2 V_xs_2 _let_1)))))) (forall ((V_xs_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Omonoid__add T_a) (= (tptp.hAPP (tptp.c_List_Ofoldl T_a T_a _let_1 V_a_2) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.c_List_Ofoldr T_a T_a _let_1 V_xs_2 (tptp.c_Groups_Ozero__class_Ozero T_a))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) (tptp.c_List_Otranspose T_a V_xs_2)) (tptp.c_List_Ofoldr _let_1 tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat)) (tptp.c_Nat_Osize__class_Osize _let_1)) V_xs_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) _let_45 (forall ((V_i_2 $$unsorted) (V_xs_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Ozip T_a tptp.tc_Nat_Onat) V_xs_2))) (let ((_let_3 (tptp.c_Product__Type_Osnd T_a tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.tc_prod T_a tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_6 tptp.tc_HOL_Obool) _let_5) (tptp.c_member tptp.tc_Nat_Onat)))) (let ((_let_8 (tptp.c_COMBC _let_5 _let_6 tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.hAPP (tptp.c_List_Omap _let_5 T_a) (tptp.c_Product__Type_Ofst T_a tptp.tc_Nat_Onat)))) (= (tptp.hAPP _let_9 (tptp.hAPP (tptp.c_List_Ofilter _let_5 (tptp.hAPP (tptp.hAPP _let_8 (tptp.hAPP _let_7 _let_3)) V_A_2)) (tptp.hAPP _let_2 (tptp.c_List_Oupt V_i_2 (tptp.hAPP (tptp.hAPP _let_4 V_i_2) _let_1))))) (tptp.hAPP _let_9 (tptp.hAPP (tptp.c_List_Ofilter _let_5 (tptp.hAPP (tptp.hAPP _let_8 (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_5 tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_5) _let_4) _let_3)) V_i_2))) V_A_2)) (tptp.hAPP _let_2 (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) _let_1))))))))))))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_i_2) V_j_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i_2 V_j_2)))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i_2 V_j_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_i_2) V_j_2))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist tptp.tc_Nat_Onat)) (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j) V_i))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.c_List_Oupt V_i (tptp.hAPP tptp.c_Nat_OSuc V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) V_j) (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat)))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_List_Oupt V_i (tptp.hAPP tptp.c_Nat_OSuc V_j)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) V_j) _let_1)))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Oupt V_i V_j))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_j)))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) V_i) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_i) V_j)))) (=> (not _let_2) (= _let_1 (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat))))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.c_List_Oupt V_i V_j) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) V_i) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_i) V_j))))) (forall ((V_i $$unsorted) (V_j $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j) V_i)) (= (tptp.c_List_Oupt V_i V_j) (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat)))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (= (tptp.c_List_Oupt V_i_2 V_j_2) (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat)) (or (= V_j_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j_2) V_i_2))))) (forall ((V_i $$unsorted)) (= (tptp.c_List_Oupt V_i (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.c_List_Oupt V_i _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)) (tptp.c_List_Oupt V_j _let_1)))))) _let_44 (forall ((V_n $$unsorted) (V_m $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_m))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) _let_1) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake tptp.tc_Nat_Onat) V_m) (tptp.c_List_Oupt V_i V_n)) (tptp.c_List_Oupt V_i _let_1))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.hAPP (tptp.c_List_Ohd tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)) V_i))) (forall ((V_j $$unsorted) (V_i $$unsorted) (V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop tptp.tc_Nat_Onat) V_m) (tptp.c_List_Oupt V_i V_j)) (tptp.c_List_Oupt (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_m) V_j))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_n))) (let ((_let_3 (tptp.hAPP _let_1 V_m))) (let ((_let_4 (tptp.c_List_Oupt _let_3 _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_3) _let_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) _let_3) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc _let_3) _let_2)))) (=> (not _let_5) (= _let_4 (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat)))))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (= (= (tptp.c_List_Oupt V_i_2 V_j_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) V_x_2) V_xs_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) V_j_2)) (= V_i_2 V_x_2) (= (tptp.c_List_Oupt (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_j_2) V_xs_2)))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.c_List_Olast tptp.tc_Nat_Onat (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.c_List_Oupt V_m_2 V_n_2)) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_n_2)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_n_2 V_m_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_n_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2))))) (forall ((V_n_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_n_2)))) (forall ((V_j $$unsorted) (V_k $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_j)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)) V_k) _let_1)))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_n_2 V_m_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_n_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2))))) (forall ((V_m_2 $$unsorted) (V_n_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Nat_Onat V_n_2 V_m_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_n_2) V_m_2)))) (forall ((V_n_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_a) (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2)) (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) V_xs_2)) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oupt V_m_2 V_n_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_a) V_f_2) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) _let_1)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oupt V_m_2 V_n_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_a) V_f_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) _let_1)) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_a) V_f_2) _let_1)))))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (V_m_2 $$unsorted) (V_n_2 $$unsorted) (V_i_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n_2) V_m_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_a) V_f_2) (tptp.c_List_Oupt V_m_2 V_n_2))) V_i_2) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_i_2))))) (forall ((V_n_2 $$unsorted) (T_a $$unsorted) (V_xs_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat T_a))) (let ((_let_4 (tptp.tc_List_Olist tptp.tc_Nat_Onat))) (=> (=> (= V_xs_2 (tptp.c_List_Olist_ONil _let_2)) (= V_n_2 _let_1)) (=> (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) V_xs_2))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_1) V_xs_2) B_i)) V_n_2)))) (= (tptp.c_List_Otranspose T_a V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_4 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_2) tptp.tc_Nat_Onat) (tptp.c_List_Omap tptp.tc_Nat_Onat T_a)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_3 tptp.tc_Nat_Onat) (tptp.c_List_Onth T_a)) (tptp.hAPP (tptp.c_List_Onth _let_2) V_xs_2))))) (tptp.c_List_Oupt _let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_2)) V_xs_2)))) (tptp.c_List_Oupt _let_1 V_n_2)))))))))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2) V_xs_2)) (=> (= (tptp.hAPP (tptp.c_List_Ohd T_a) (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_a) (tptp.hAPP tptp.c_fequal (tptp.hAPP V_f_2 V_a_2))) V_f_2)) V_xs_2)) V_a_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_a_2) (tptp.c_List_Oremove1 T_a V_a_2 V_xs_2)) V_xs_2)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Olist_ONil T_a)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) (tptp.c_List_Olist_ONil T_a))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.c_List_Oupt V_i V_j))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) (and (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) B_x)))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Olinorder__class_Osort__key T_a T_a (tptp.c_COMBI T_a) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a (tptp.c_COMBI T_a)) V_x_2) V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2)))) (forall ((V_n $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs))))) (forall ((V_a $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Oremove1 T_a V_a V_xs))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (tptp.c_List_Olinorder__class_Osorted tptp.tc_Int_Oint (tptp.c_List_Oupto V_i V_j))) (forall ((V_n $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.c_List_Otl T_a) V_xs))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Olinorder__class_Oinsort__insert__key T_a T_a (tptp.c_COMBI T_a) V_x_2 V_xs_2))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Obutlast T_a V_xs)))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Olist_ONil T_a)))) (forall ((V_zs $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_zs))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (=> (tptp.c_List_Olinorder__class_Osorted T_a _let_2) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) _let_2)))))))) (forall ((V_zs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_zs_2))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP _let_1 V_x_2) _let_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) V_y_2)) (tptp.c_List_Olinorder__class_Osorted T_a _let_2))))))) (forall ((V_l $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_l) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Oremdups T_a V_l))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.c_List_Odistinct T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_xs_2)) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_ys_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_ys_2)) (=> (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= V_xs_2 V_ys_2)))))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) (tptp.c_List_Olinorder__class_Osort__key T_b T_a V_f_2 V_xs_2))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_a V_f_2 V_x_2 V_xs_2))))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_b V_x_2 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_a V_f_2) V_x_2) V_xs_2))) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_a tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a (tptp.tc_fun T_a tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_f_2)) (tptp.hAPP V_g_2 V_xs_2))) V_xs_2))))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_P_2) V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.c_List_Ofilter T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.hAPP V_g_2 V_xs_2))) V_xs_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (and (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_a V_ys_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) (tptp.hAPP (tptp.c_List_Oset T_a) V_ys_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) B_x) B_xa)))))))))) (forall ((V_x_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_P_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_a V_f_2) V_x_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))))) (forall ((V_a1_2 $$unsorted) (V_a2_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a2_2) V_a1_2)) (and (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_a1_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a2_2) B_x)))) (tptp.c_List_Olinorder__class_Osorted T_a V_a1_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olinorder__class_Osorted__list__of__set T_a V_A_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (and (= (tptp.hAPP (tptp.c_List_Oset T_a) _let_1) V_A_2) (tptp.c_List_Olinorder__class_Osorted T_a _let_1) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) _let_1))))))) (forall ((V_xs_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_a_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_a (tptp.c_COMBI T_a)) V_a_2) (tptp.c_List_Oremove1 T_a V_a_2 V_xs_2)) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (forall ((B_j $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_i) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 B_i)) (tptp.hAPP _let_1 B_j))))))))))) (forall ((V_j $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_a) V_xs))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_j) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_ys_2))) (let ((_let_3 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_4 (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2))) (let ((_let_5 (tptp.hAPP _let_4 V_ys_2))) (let ((_let_6 (tptp.c_List_Odistinct T_b))) (let ((_let_7 (tptp.hAPP _let_4 V_xs_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_Fun_Oinj__on T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) _let_3) _let_2)) (=> (tptp.c_List_Olinorder__class_Osorted T_b _let_7) (=> (tptp.hBOOL (tptp.hAPP _let_6 _let_7)) (=> (tptp.c_List_Olinorder__class_Osorted T_b _let_5) (=> (tptp.hBOOL (tptp.hAPP _let_6 _let_5)) (=> (= _let_3 _let_2) (= V_xs_2 V_ys_2)))))))))))))))) (forall ((V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2))) (=> (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_List_Orev tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat) _let_2) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat T_a) (tptp.c_List_Onth T_a)) V_i_2)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1) _let_3) _let_2)) (tptp.c_List_Otranspose T_a V_xs_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_1) V_xs_2) V_i_2)))))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Onth _let_1))) (let ((_let_3 (tptp.c_List_Onth T_a))) (let ((_let_4 (tptp.c_List_Otranspose T_a V_xs_2))) (let ((_let_5 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_7 (tptp.hAPP _let_6 V_i_2))) (let ((_let_8 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)))) (=> (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_List_Orev tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat) _let_5) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_7 (tptp.hAPP _let_8 _let_4))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_6 V_j_2) (tptp.hAPP _let_8 (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1) _let_7) _let_5)) V_xs_2)))) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 _let_4) V_i_2)) V_j_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) V_j_2)) V_i_2)))))))))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (tptp.c_Fun_Oinj__on _let_1 _let_1 (tptp.c_List_Orev T_a) V_A_2))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (let ((_let_2 (tptp.c_List_Oappend T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_xs) V_ys)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_ys)) (tptp.hAPP _let_1 V_xs)))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Orev T_a) _let_1) _let_1))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2)) (= V_xs_2 _let_1)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (= (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) (tptp.c_List_Olist_ONil T_a)))) (= (= _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2)) (= V_xs_2 _let_1)))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) (tptp.c_List_Olist_ONil T_a)))) (= (= (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (let ((_let_2 (tptp.c_List_Orev T_a))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_1 (tptp.c_List_Olist_ONil T_a))))))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y_2))) (let ((_let_2 (tptp.c_List_Orev T_a))) (= (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP _let_2 V_ys_2)) (tptp.hAPP _let_1 (tptp.c_List_Olist_ONil T_a)))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_List_Orev T_a))) (= (tptp.hAPP _let_2 (tptp.c_List_Oconcat T_a V_xs_2)) (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_1) _let_2) (tptp.hAPP (tptp.c_List_Orev _let_1) V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.hAPP (tptp.c_List_Orev T_a) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (let ((_let_2 (tptp.c_List_Ofilter T_a V_P_2))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_ys $$unsorted) (T_b $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) (tptp.hAPP (tptp.c_List_Orev T_b) V_ys)) (tptp.hAPP (tptp.c_List_Orev (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2) V_ys_2) (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 (tptp.hAPP (tptp.c_List_Orev T_b) V_ys_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all2 T_a T_b V_P_2 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2) (tptp.hAPP (tptp.c_List_Orev T_b) V_ys_2)) (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_xs_2 V_ys_2))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) (tptp.hAPP _let_1 V_xs))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (= (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= V_xs_2 V_ys_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (= (= (tptp.hAPP _let_1 V_xs_2) V_ys_2) (= V_xs_2 (tptp.hAPP _let_1 V_ys_2))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_xs)) V_xs))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) (tptp.hAPP _let_1 V_xs)))) (forall ((V_xs_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofoldl T_a T_b V_f_2 V_a_2) V_xs_2) (tptp.c_List_Ofoldr T_b T_a (tptp.hAPP (tptp.c_COMBC T_a T_b T_a) V_f_2) (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2) V_a_2))) _let_43 (forall ((T_a $$unsorted) (V_xs $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs)))) (forall ((T_a $$unsorted) (V_xs $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Ohd T_a) (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) (tptp.c_List_Olast T_a V_xs)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_List_Orev tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat) (tptp.c_Nat_Osize__class_Osize _let_1)) (tptp.c_List_Otranspose T_a V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (= (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2) (tptp.hAPP (tptp.c_List_Ofoldl _let_1 T_a (tptp.hAPP (tptp.c_COMBC T_a _let_1 _let_1) (tptp.c_List_Olist_OCons T_a)) (tptp.c_List_Olist_ONil T_a)) V_xs_2)))) (forall ((V_xs $$unsorted) (V_i $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_i) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_i)) (tptp.hAPP _let_1 V_xs))))) (forall ((V_xs $$unsorted) (V_i $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_i) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_i)) (tptp.hAPP _let_1 V_xs))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (let ((_let_2 (tptp.c_List_Orev T_a))) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n) (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_Orotate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_n _let_1))) V_xs)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (let ((_let_2 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) _let_1)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) V_n) (tptp.hAPP (tptp.hAPP _let_2 V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.hAPP tptp.c_Nat_OSuc V_n)))))))) (forall ((V_y $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_List_Orev T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_k) _let_1)) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_k) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_3 V_xs)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_k)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_y))))))) (forall ((V_j $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_a) V_xs))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_j) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 V_j)) (tptp.hAPP _let_1 V_i))))))))) (forall ((V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_a))) (let ((_let_2 (tptp.c_List_Ofoldr T_a T_a _let_1 V_xs_2 V_y_2))) (let ((_let_3 (= V_xs_2 (tptp.c_List_Olist_ONil T_a)))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2)) (and (=> _let_3 (= _let_2 V_y_2)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_y_2)))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist T_a))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize _let_2))) (let ((_let_4 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_2)) (tptp.c_List_Otranspose T_a V_xs_2)))) (let ((_let_5 (= V_xs_2 (tptp.c_List_Olist_ONil _let_2)))) (=> (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_List_Orev tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 tptp.tc_Nat_Onat) _let_3) V_xs_2))) (and (=> _let_5 (= _let_4 _let_1)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_2) V_xs_2) _let_1)))))))))))) (forall ((V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2))) (let ((_let_4 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)))) (=> (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_List_Orev tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat) _let_2) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP _let_4 V_xs_2))) (= (tptp.hAPP _let_4 (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1) _let_3) _let_2)) (tptp.c_List_Otranspose T_a V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_1) V_xs_2) V_i_2)))))))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_List_Orev tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat) (tptp.c_Nat_Osize__class_Osize _let_1)) V_xs_2))) (= (tptp.c_List_Otranspose T_a (tptp.c_List_Otranspose T_a V_xs_2)) (tptp.c_List_OtakeWhile _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_1) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olist_ONil T_a))) V_xs_2))))) (= (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat) (tptp.c_List_Oreturn__list (tptp.c_List_Olist_ONil tptp.tc_Int_Oint))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2) V_xs_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_a) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)))) (and (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)) V_ys_2) (tptp.c_List_OtakeWhile _let_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_a tptp.tc_HOL_Obool _let_2 V_P_2) (tptp.c_Product__Type_Ofst T_a T_b)) (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2))) (let ((_let_2 (tptp.tc_prod T_a T_b))) (= (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_P_2 V_ys_2)) (tptp.c_List_OtakeWhile _let_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b tptp.tc_HOL_Obool _let_2 V_P_2) (tptp.c_Product__Type_Osnd T_a T_b)) (tptp.hAPP _let_1 V_ys_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.c_List_OtakeWhile T_a V_P_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_a tptp.tc_HOL_Obool T_b V_P_2) V_f_2) V_xs_2))))) (forall ((V_l_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))) (= (tptp.c_List_OtakeWhile T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_l_2))) (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2))) (let ((_let_2 (tptp.c_List_OtakeWhile T_a V_P_2 (tptp.hAPP _let_1 V_xs_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)))) (=> (not _let_3) (= _let_2 (tptp.c_List_Olist_ONil T_a)))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_OtakeWhile T_a V_P_2 _let_1) _let_1))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2))) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_1)) V_xs_2)))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)))))) _let_42 (forall ((V_ys_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))) (= (tptp.c_List_OtakeWhile T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_2) (tptp.hAPP _let_1 V_xs_2))) (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) _let_2)))))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (let ((_let_2 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_j_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_j_2) (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_j_2)))))) (forall ((T_a $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (= (tptp.c_List_OtakeWhile T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olast T_a V_xs_2))) V_xs_2) (tptp.c_List_Obutlast T_a V_xs_2))))) (forall ((V_t_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a tptp.tc_HOL_Obool T_b) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_t_2)) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.c_List_Orev T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2) V_xs_2))) (= (tptp.hAPP (tptp.c_List_Ofilter T_b _let_1) V_xs_2) (tptp.c_List_OtakeWhile T_b _let_1 V_xs_2)))))) (forall ((V_l $$unsorted)) (=> (tptp.c_List_Onat__list V_l) (= (tptp.c_List_Oembed__list (tptp.c_List_Oreturn__list V_l)) V_l))) (forall ((V_m $$unsorted) (V_l $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Nat_Onat) V_l) V_m) (tptp.c_List_Oreturn__list (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Int_Oint) (tptp.c_List_Oembed__list V_l)) (tptp.c_List_Oembed__list V_m))))) _let_41 (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_x_2)))) (let ((_let_2 (tptp.c_List_Orev T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= (tptp.c_List_OtakeWhile T_a _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_Otl T_a) (tptp.c_List_OdropWhile T_a _let_1 V_xs_2))))))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a T_a tptp.tc_HOL_Obool) tptp.c_fequal) V_x_2)))) (let ((_let_2 (tptp.c_List_Orev T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= (tptp.c_List_OdropWhile T_a _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) (tptp.hAPP _let_2 (tptp.c_List_OtakeWhile T_a _let_1 V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)) (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2)) V_xs_2)) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_a) V_f_2))) (= (tptp.c_List_OdropWhile T_a V_P_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_a tptp.tc_HOL_Obool T_b V_P_2) V_f_2) V_xs_2))))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2) (tptp.c_List_Olist_ONil T_a)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))) (let ((_let_2 (tptp.c_List_OdropWhile T_a V_P_2 _let_1))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)))) (and (=> _let_3 (= _let_2 (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2))) (=> (not (= _let_1 (tptp.c_List_Olist_ONil T_a))) (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.c_List_Ohd T_a) _let_1))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_OdropWhile T_a V_P_2 _let_1) _let_1))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2)))))) (forall ((V_ys_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2))) (= (tptp.c_List_OdropWhile T_a V_P_2 (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2)) V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y_2) V_ys_2))) (= (= (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2) _let_1) (and (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)) _let_1)) (not (tptp.hBOOL (tptp.hAPP V_P_2 V_y_2))))))) _let_40 (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (let ((_let_2 (tptp.c_List_Onth T_a))) (let ((_let_3 (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_j_2) (tptp.hAPP _let_1 _let_3))) (= (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_j_2) (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j_2) (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2)))))))))) (forall ((V_f_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_a T_b) V_f_2))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (=> (tptp.class_Groups_Omonoid__add T_b) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (forall ((B_i $$unsorted) (B_j $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_a) V_xs))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_i) B_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 B_i)) (tptp.hAPP _let_1 B_j))))))) (tptp.c_List_Olinorder__class_Osorted T_a V_xs)))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_List_OdropWhile T_a V_P_2 V_xs_2)))) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))) (= (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2) (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_xs_2)))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_n_2 $$unsorted)) (=> (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_i))))))) (=> (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (not (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_n_2))))) (= (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted) (V_P_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j_2))) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_j_2)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) B_i))))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_xs_2))))))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.c_List_Olinorder__class_Osorted T_a V_a_2) (or (= V_a_2 (tptp.c_List_Olist_ONil T_a)) (exists ((B_xs $$unsorted) (B_x $$unsorted)) (and (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) B_x) B_xs)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_xa) (tptp.hAPP (tptp.c_List_Oset T_a) B_xs))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) B_x) B_xa)))) (tptp.c_List_Olinorder__class_Osorted T_a B_xs))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__upds T_a T_b V_m_2 V_xs_2 V_ys_2) (tptp.c_Map_Omap__add T_a T_b V_m_2 (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) (tptp.hAPP (tptp.c_List_Orev (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs_2) V_ys_2)))))) _let_39 (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Map_Omap__of T_a T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend (tptp.tc_prod T_a T_b)) V_xs_2) V_ys_2)) (tptp.c_Map_Omap__add T_a T_b (tptp.hAPP _let_1 V_ys_2) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_x_2 $$unsorted) (V_m_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Map_Omap__comp T_b T_a T_c V_m_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_b) T_c) (tptp.c_Option_Ooption_ONone T_b)) V_x_2) (tptp.c_Option_Ooption_ONone T_a))) (forall ((V_x_2 $$unsorted) (V_m_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Option_Ooption_ONone T_a))) (= (tptp.c_Map_Omap__comp T_b T_a T_c (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_a) T_b) _let_1) V_m_2 V_x_2) _let_1))) (forall ((V_m1_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (V_k_H_2 $$unsorted) (T_a $$unsorted) (V_k_2 $$unsorted) (V_m2_2 $$unsorted)) (=> (= (tptp.hAPP V_m2_2 V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_k_H_2)) (= (tptp.c_Map_Omap__comp T_a T_c T_b V_m1_2 V_m2_2 V_k_2) (tptp.hAPP V_m1_2 V_k_H_2)))) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (V_xx_2 $$unsorted) (T_a $$unsorted) (V_k_2 $$unsorted) (V_n_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_xx_2))) (=> (= (tptp.hAPP V_n_2 V_k_2) _let_1) (= (tptp.hAPP (tptp.c_Map_Omap__add T_b T_a V_m_2 V_n_2) V_k_2) _let_1)))) (forall ((V_v_2 $$unsorted) (V_k_2 $$unsorted) (V_m2_2 $$unsorted) (V_m1_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Map_Omap__comp T_b T_a T_c V_m1_2 V_m2_2 V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_v_2)) (exists ((B_k_H $$unsorted)) (and (= (tptp.hAPP V_m2_2 V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) B_k_H)) (= (tptp.hAPP V_m1_2 B_k_H) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_v_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_y_2))) (let ((_let_2 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (= (tptp.c_Map_Omap__add T_a T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 V_g_2) V_x_2) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Map_Omap__add T_a T_b V_f_2 V_g_2)) V_x_2) _let_1))))) (forall ((V_x_2 $$unsorted) (V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_x_2))) (let ((_let_2 (tptp.hAPP V_n_2 V_k_2))) (=> (= (tptp.hAPP (tptp.c_Map_Omap__add T_b T_a V_m_2 V_n_2) V_k_2) _let_1) (or (= _let_2 _let_1) (and (= _let_2 (tptp.c_Option_Ooption_ONone T_a)) (= (tptp.hAPP V_m_2 V_k_2) _let_1))))))) (forall ((V_x_2 $$unsorted) (V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_x_2))) (let ((_let_2 (tptp.hAPP V_n_2 V_k_2))) (= (= (tptp.hAPP (tptp.c_Map_Omap__add T_b T_a V_m_2 V_n_2) V_k_2) _let_1) (or (= _let_2 _let_1) (and (= _let_2 (tptp.c_Option_Ooption_ONone T_a)) (= (tptp.hAPP V_m_2 V_k_2) _let_1))))))) (forall ((V_m1_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (V_k_2 $$unsorted) (V_m2_2 $$unsorted)) (=> (= (tptp.hAPP V_m2_2 V_k_2) (tptp.c_Option_Ooption_ONone T_a)) (= (tptp.c_Map_Omap__comp T_a T_c T_b V_m1_2 V_m2_2 V_k_2) (tptp.c_Option_Ooption_ONone T_c)))) (forall ((V_k_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Option_Ooption_ONone T_a))) (= (= (tptp.hAPP (tptp.c_Map_Omap__add T_b T_a V_m_2 V_n_2) V_k_2) _let_1) (and (= (tptp.hAPP V_n_2 V_k_2) _let_1) (= (tptp.hAPP V_m_2 V_k_2) _let_1))))) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__add T_a T_b V_m_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_b) T_a) (tptp.c_Option_Ooption_ONone T_b))) V_m_2)) (forall ((V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__add T_a T_b (tptp.hAPP (tptp.c_COMBK (tptp.tc_Option_Ooption T_b) T_a) (tptp.c_Option_Ooption_ONone T_b)) V_m_2) V_m_2)) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_m2_2 $$unsorted) (V_m1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__add T_a T_b V_m1_2 (tptp.c_Map_Omap__upds T_a T_b V_m2_2 V_xs_2 V_ys_2)) (tptp.c_Map_Omap__upds T_a T_b (tptp.c_Map_Omap__add T_a T_b V_m1_2 V_m2_2) V_xs_2 V_ys_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Odom T_a T_b (tptp.c_Map_Omap__add T_a T_b V_m_2 V_n_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Map_Odom T_a T_b V_n_2)) (tptp.c_Map_Odom T_a T_b V_m_2)))) (forall ((V_l2_2 $$unsorted) (V_l1_2 $$unsorted) (T_b $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_m_2) (tptp.c_Map_Odom T_a T_b V_l1_2)))) (= (tptp.hAPP (tptp.c_Map_Omap__add T_a T_b V_l1_2 V_l2_2) V_m_2) (tptp.hAPP V_l2_2 V_m_2)))) (forall ((V_l1_2 $$unsorted) (V_l2_2 $$unsorted) (T_b $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_m_2) (tptp.c_Map_Odom T_a T_b V_l2_2)))) (= (tptp.hAPP (tptp.c_Map_Omap__add T_a T_b V_l1_2 V_l2_2) V_m_2) (tptp.hAPP V_l1_2 V_m_2)))) (forall ((V_l1_2 $$unsorted) (V_l2_2 $$unsorted) (T_b $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_m_2) (tptp.c_Map_Odom T_a T_b V_l2_2))) (= (tptp.hAPP (tptp.c_Map_Omap__add T_a T_b V_l1_2 V_l2_2) V_m_2) (tptp.hAPP V_l2_2 V_m_2)))) (forall ((V_m3_2 $$unsorted) (V_m2_2 $$unsorted) (V_m1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Map_Omap__add T_a T_b V_m1_2 (tptp.c_Map_Omap__add T_a T_b V_m2_2 V_m3_2)) (tptp.c_Map_Omap__add T_a T_b (tptp.c_Map_Omap__add T_a T_b V_m1_2 V_m2_2) V_m3_2))) (forall ((V_m_H_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Map_Odom T_a T_b V_m_H_2))) (let ((_let_2 (tptp.tc_Option_Ooption T_b))) (= (tptp.c_Fun_Oinj__on T_a _let_2 (tptp.c_Map_Omap__add T_a T_b V_m_2 V_m_H_2) _let_1) (tptp.c_Fun_Oinj__on T_a _let_2 V_m_H_2 _let_1))))) (forall ((V_x_2 $$unsorted) (V_m2_2 $$unsorted) (V_m1_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Map_Omap__add T_b T_a V_m1_2 V_m2_2) V_x_2) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case (tptp.tc_Option_Ooption T_a) T_a (tptp.hAPP V_m1_2 V_x_2) (tptp.c_Option_Ooption_OSome T_a)) (tptp.hAPP V_m2_2 V_x_2)))) (forall ((V_m2_2 $$unsorted) (V_m1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_Map_Odom T_a T_b V_m1_2)) (tptp.c_Map_Odom T_a T_b V_m2_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)) (= (tptp.c_Map_Omap__add T_a T_b V_m1_2 V_m2_2) (tptp.c_Map_Omap__add T_a T_b V_m2_2 V_m1_2))))) (forall ((V_k_2 $$unsorted) (V_m2_2 $$unsorted) (V_m1_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Map_Omap__comp T_b T_a T_c V_m1_2 V_m2_2 V_k_2) (tptp.c_Option_Ooption_ONone T_a)) (or (= (tptp.hAPP V_m2_2 V_k_2) (tptp.c_Option_Ooption_ONone T_b)) (exists ((B_k_H $$unsorted)) (and (= (tptp.hAPP V_m2_2 V_k_2) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) B_k_H)) (= (tptp.hAPP V_m1_2 B_k_H) (tptp.c_Option_Ooption_ONone T_a))))))) (forall ((V_ps_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (let ((_let_2 (tptp.tc_fun T_a _let_1))) (let ((_let_3 (tptp.tc_fun _let_1 _let_2))) (let ((_let_4 (tptp.tc_fun _let_2 _let_2))) (let ((_let_5 (tptp.tc_fun _let_1 _let_4))) (let ((_let_6 (tptp.tc_fun T_b _let_4))) (let ((_let_7 (tptp.tc_fun T_b _let_1))) (= (tptp.c_Map_Omap__add T_a T_b V_m_2 (tptp.hAPP (tptp.c_Map_Omap__of T_a T_b) V_ps_2)) (tptp.c_List_Ofoldr (tptp.tc_prod T_a T_b) _let_2 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a T_b _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_7 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_7 _let_6) T_a) (tptp.c_COMBB _let_1 _let_4 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_2 _let_3) _let_5 T_a) (tptp.c_COMBC _let_2 _let_1 _let_2)) (tptp.hAPP (tptp.c_COMBC _let_2 T_a _let_3) (tptp.c_Fun_Ofun__upd T_a _let_1))))) (tptp.c_Option_Ooption_OSome T_b))) V_ps_2 V_m_2)))))))))) (forall ((V_u1_2 $$unsorted) (V_e1_2 $$unsorted) (V_e2_2 $$unsorted) (T_b $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_u1_2))) (let ((_let_2 (tptp.c_Fun_Ofun__upd T_a (tptp.tc_Option_Ooption T_b)))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_m_2) (tptp.c_Map_Odom T_a T_b V_e2_2)))) (= (tptp.c_Map_Omap__add T_a T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 V_e1_2) V_m_2) _let_1) V_e2_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Map_Omap__add T_a T_b V_e1_2 V_e2_2)) V_m_2) _let_1)))))) (forall ((V_l_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_Option_Ooption T_a))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite _let_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_2 V_f_2) _let_1))) (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_Set_Oimage T_b _let_2 (tptp.c_Map_Omap__add T_b T_a V_f_2 (tptp.hAPP (tptp.c_Map_Omap__of T_b T_a) V_l_2))) _let_1)))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oinsert T_a V_x_2 V_xs_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))))) (and (=> _let_2 (= _let_1 V_xs_2)) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2)))) (= (tptp.c_List_Oinsert T_a V_x_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Oinsert T_a V_x V_xs)))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oinsert T_a V_x (tptp.c_List_Oremdups T_a V_xs)) (tptp.c_List_Oremdups T_a (tptp.c_List_Oinsert T_a V_x V_xs)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= (tptp.c_List_Oinsert T_a V_x_2 V_xs_2) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_a))) (= (tptp.hAPP _let_1 (tptp.c_List_Oinsert T_a V_x_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_a) V_x_2) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Oinsert T_a V_x _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) _let_1)))) _let_38 (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Omaps T_b T_a V_f_2 V_xs_2) (tptp.c_List_Oconcat T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b (tptp.tc_List_Olist T_a)) V_f_2) V_xs_2)))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Omaps T_b T_a V_f_2 (tptp.c_List_Olist_ONil T_b)) (tptp.c_List_Olist_ONil T_a))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Omaps T_b T_a V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP V_f_2 V_x_2)) (tptp.c_List_Omaps T_b T_a V_f_2 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_List_Olist T_a)) V_xs_2)) (=> (forall ((B_ys $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_ys) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) B_ys))))) (=> (forall ((B_ys $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_ys) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xs_2))) (forall ((B_zs $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_a))) (let ((_let_3 (tptp.tc_List_Olist T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_3) B_zs) (tptp.hAPP (tptp.c_List_Oset _let_3) V_xs_2))) (=> (not (= B_ys B_zs)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 B_ys)) (tptp.hAPP _let_2 B_zs)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) (tptp.c_List_Oconcat T_a V_xs_2))))))) (forall ((V_A_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_y_2) (tptp.c_Map_Oran T_b T_a (tptp.c_Map_Orestrict__map T_b T_a V_m_2 V_A_2)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_m_2 B_x) (tptp.hAPP (tptp.c_Option_Ooption_OSome T_a) V_y_2)))))) (forall ((V_fs_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Omeasures T_a V_fs_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Omeasures T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_a tptp.tc_Nat_Onat)) V_f_2) V_fs_2)))))))) (forall ((V_fs_2 $$unsorted) (V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)))) (let ((_let_2 (tptp.hAPP V_f_2 V_y_2))) (let ((_let_3 (tptp.hAPP V_f_2 V_x_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Omeasures T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_a tptp.tc_Nat_Onat)) V_f_2) V_fs_2)))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_3) _let_2)) (and (= _let_3 _let_2) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Omeasures T_a V_fs_2)))))))))) (forall ((V_fs_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Wellfounded_Owf T_a (tptp.c_List_Omeasures T_a V_fs_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_List_Omeasures T_a (tptp.c_List_Olist_ONil (tptp.tc_fun T_a tptp.tc_Nat_Onat))))))) _let_37 (forall ((V_fs_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_a T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_a) V_x_2) V_y_2)) (tptp.c_List_Omeasures T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_a tptp.tc_Nat_Onat)) V_f_2) V_fs_2)))))) (forall ((V_s_2 $$unsorted) (V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_b (tptp.hAPP V_f_2 B_x)) V_h_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_b V_h_2) (tptp.hAPP V_g_2 B_x))))) (= (tptp.hAPP (tptp.c_List_Ofoldl T_c T_a (tptp.hAPP (tptp.c_COMBC T_a T_c T_c) V_f_2) (tptp.hAPP V_h_2 V_s_2)) V_xs_2) (tptp.hAPP V_h_2 (tptp.hAPP (tptp.c_List_Ofoldl T_b T_a (tptp.hAPP (tptp.c_COMBC T_a T_b T_b) V_g_2) V_s_2) V_xs_2))))) (forall ((V_P_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) (tptp.hAPP _let_3 V_xs_2)) (tptp.hAPP _let_2 V_ys_2)))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) _let_4) V_ys_2))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) _let_4) V_xs_2))) (=> (=> (= (tptp.hAPP _let_3 _let_6) (tptp.hAPP _let_2 _let_5)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP _let_1 _let_6) _let_5)))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2))))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Map_Omap__of T_a T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_a T_b) T_a) (tptp.c_Product__Type_Ofst T_a T_b)))) (let ((_let_3 (tptp.c_List_Oset T_a))) (=> (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ys_2))) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.c_Map_Omap__of T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_a T_b) T_a) (tptp.c_Product__Type_Ofst T_a T_b)) V_xs_2)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) B_x) (tptp.hAPP (tptp.hAPP _let_1 V_ys_2) B_x))))) (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)))))))) _let_36 (forall ((V_ys_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (= (tptp.c_List_OdropWhile T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (tptp.c_List_OdropWhile T_a V_P_2 V_ys_2)))) (forall ((V_P_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_a_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_b_2)) (=> (forall ((B_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_a_2) B_n)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_b_2) B_n))))) (tptp.c_List_Olist__all2 T_a T_b V_P_2 V_a_2 V_b_2)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Map_Odom T_a T_b V_f_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (exists ((B_x $$unsorted)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Option_Ooption_ONone T_b))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_a) V_A_2)) (exists ((B_x $$unsorted)) (and (= (tptp.hAPP (tptp.c_List_Oset T_a) B_x) V_A_2) (tptp.c_List_Olinorder__class_Osorted T_a B_x) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) B_x)) (forall ((B_y $$unsorted)) (=> (and (= (tptp.hAPP (tptp.c_List_Oset T_a) B_y) V_A_2) (tptp.c_List_Olinorder__class_Osorted T_a B_y) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) B_y))) (= B_y B_x)))))))) (forall ((V_ys_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (= (tptp.c_List_OtakeWhile T_a V_P_2 (tptp.hAPP _let_1 V_ys_2)) (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_a V_P_2 V_ys_2)))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP V_f_2 B_x))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_a T_b V_f_2) V_a_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_a_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_ys_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_ys_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) (exists ((B_us $$unsorted) (B_vs $$unsorted)) (and (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) B_us) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) B_vs))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) B_us))) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))) (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (= V_xs_2 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) B_vs)))))) (forall ((V_ys_2 $$unsorted) (V_P_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2) (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) V_ys_2)) (exists ((B_us $$unsorted) (B_vs $$unsorted)) (and (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) B_us) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) B_vs))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) B_us))) (not (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))) (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (= V_xs_2 (tptp.hAPP (tptp.c_List_Ofilter T_a V_P_2) B_vs)))))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted) (V_n_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs_2))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs_2) V_n_2)))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x_2) B_x)))) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)))))) _let_35 (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (exists ((B_ys $$unsorted)) (and (exists ((B_zs $$unsorted)) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) B_ys) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) B_zs)))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) B_ys)))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (not (tptp.c_List_Olist__ex T_a V_P_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__ex T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (or (tptp.c_List_Olist__ex T_a V_P_2 V_xs_2) (tptp.c_List_Olist__ex T_a V_P_2 V_ys_2)))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__ex T_a V_P_2 V_xs_2) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__ex T_a V_P_2 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2)) (tptp.c_List_Olist__ex T_a V_P_2 V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__ex T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) (or (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (tptp.c_List_Olist__ex T_a V_P_2 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (exists ((B_ys $$unsorted) (B_zs $$unsorted)) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) B_ys) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) B_zs)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (exists ((B_ys $$unsorted) (B_zs $$unsorted)) (and (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) B_ys) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) B_zs))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) B_zs)))))))) (forall ((V_a_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Option_Ooption_Ooption__rec T_a T_b V_f1_2 V_f2_2 (tptp.hAPP (tptp.c_Option_Ooption_OSome T_b) V_a_2)) (tptp.hAPP V_f2_2 V_a_2))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Option_Ooption_Ooption__rec T_a T_b V_f1_2 V_f2_2 (tptp.c_Option_Ooption_ONone T_b)) V_f1_2)) _let_34 (tptp.c_FunDef_Ois__measure tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Int_Oint tptp.c_Int_Onat) (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) (and (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (tptp.c_List_Olist__all T_a V_P_2 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all T_a V_P_2 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs_2)) (tptp.c_List_Olist__all T_a V_P_2 V_xs_2))) (forall ((V_P_2 $$unsorted) (V_xs_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (tptp.c_List_Olist__all T_a V_P_2 V_xs_2))) (forall ((T_b $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.c_FunDef_Ois__measure T_a V_f_2) (tptp.c_FunDef_Ois__measure _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a tptp.tc_Nat_Onat _let_1) V_f_2) (tptp.c_Product__Type_Ofst T_a T_b)))))) (forall ((T_b $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_a))) (=> (tptp.c_FunDef_Ois__measure T_a V_f_2) (tptp.c_FunDef_Ois__measure _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_a tptp.tc_Nat_Onat _let_1) V_f_2) (tptp.c_Product__Type_Osnd T_b T_a)))))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all T_a V_P_2 V_xs_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) _let_33 (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.c_FunDef_Ois__measure T_a V_a_2)) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (tptp.c_FunDef_Ois__measure T_a V_f_2)) (forall ((T_a $$unsorted)) (=> (tptp.class_Nat_Osize T_a) (tptp.c_FunDef_Ois__measure T_a (tptp.c_Nat_Osize__class_Osize T_a)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs_2) V_ys_2)) (and (tptp.c_List_Olist__all T_a V_P_2 V_xs_2) (tptp.c_List_Olist__all T_a V_P_2 V_ys_2)))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (tptp.c_List_Olist__all T_a V_P_2 (tptp.c_List_Olist_ONil T_a))) (forall ((V_xs_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Olist__all T_a V_P_2 V_xs_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)))) (= (tptp.c_List_Olist__ex1 T_a V_P_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x_2) V_xs_2)) (and (=> _let_1 (tptp.c_List_Olist__all T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_a tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_a) tptp.c_fdisj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot) V_P_2))) (tptp.hAPP tptp.c_fequal V_x_2)) V_xs_2)) (=> (not _let_1) (tptp.c_List_Olist__ex1 T_a V_P_2 V_xs_2)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_HOL_Oequal T_b) (tptp.class_Enum_Oenum T_a)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal (tptp.tc_fun T_a T_b)) V_f_2) V_g_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_Enum_Oenum__class_Oenum T_a)))) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))))))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (not (tptp.c_List_Olist__ex1 T_a V_P_2 (tptp.c_List_Olist_ONil T_a)))) (forall ((T_a $$unsorted)) (=> (tptp.class_HOL_Oequal T_a) (= (tptp.c_HOL_Oequal__class_Oequal T_a) tptp.c_fequal))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_HOL_Oequal T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal T_a) V_x) V_x)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_HOL_Oequal T_a) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal T_a) V_x_2) V_y_2)) (= V_x_2 V_y_2)))) (forall ((T_a $$unsorted)) (=> (tptp.class_HOL_Oequal T_a) (= tptp.c_fequal (tptp.c_HOL_Oequal__class_Oequal T_a)))) (forall ((V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Enum_Oenum T_a) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_y)) (= B_y B_x))))) (tptp.c_List_Olist__ex1 T_a V_P_2 (tptp.c_Enum_Oenum__class_Oenum T_a))))) (= (tptp.hAPP _let_32 tptp.c_fTrue) _let_31) (= (tptp.hAPP _let_32 tptp.c_fFalse) _let_31) _let_30 (forall ((V_xq_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Lazy__Sequence_Olazy__sequence T_a))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (= (tptp.hAPP _let_2 V_xq_2) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case tptp.tc_Nat_Onat (tptp.tc_prod T_a _let_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a _let_1 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 tptp.tc_Nat_Onat) T_a) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) _let_2)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (tptp.hAPP (tptp.c_Lazy__Sequence_Oyield T_a) V_xq_2)))))) (tptp.c_Nat__Transfer_Otransfer__morphism tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_29 (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fTrue)) (forall ((V_k2 $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) tptp.c_Int_OMin) (tptp.c_Int_OBit0 V_k2))))) (forall ((V_k1 $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) (tptp.c_Int_OBit0 V_k1)) tptp.c_Int_OMin)))) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k1_2)) (tptp.c_Int_OBit0 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (tptp.hBOOL (tptp.hAPP _let_27 tptp.c_Int_OPls)) (forall ((V_k2_2 $$unsorted) (V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k1_2)) (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) V_k2_2))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_k_2)) (tptp.hAPP _let_2 V_l_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2)))))) (tptp.hBOOL (tptp.hAPP _let_28 tptp.c_Int_OMin)) (forall ((V_k2 $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) tptp.c_Int_OPls) (tptp.c_Int_OBit1 V_k2))))) (forall ((V_k1 $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) (tptp.c_Int_OBit1 V_k1)) tptp.c_Int_OPls)))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) V_k_2) V_l_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k_2) V_l_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (forall ((V_k2 $$unsorted) (V_k1 $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) (tptp.c_Int_OBit0 V_k1)) (tptp.c_Int_OBit1 V_k2))))) (forall ((V_k2 $$unsorted) (V_k1 $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) (tptp.c_Int_OBit1 V_k1)) (tptp.c_Int_OBit0 V_k2))))) (forall ((V_k2_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) tptp.c_Int_OPls))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k2_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k2_2))))) (forall ((V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k1_2)) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) tptp.c_Int_OPls))))) (forall ((V_k2_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k2_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k2_2))))) (forall ((V_k1_2 $$unsorted)) (let ((_let_1 (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k1_2)) tptp.c_Int_OMin)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k1_2) tptp.c_Int_OMin))))) (not (tptp.hBOOL (tptp.hAPP _let_28 tptp.c_Int_OPls))) (not (tptp.hBOOL (tptp.hAPP _let_27 tptp.c_Int_OMin))) (forall ((V_xq_2 $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size T_a V_s_2))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.tc_Lazy__Sequence_Olazy__sequence T_a))) (let ((_let_5 (tptp.tc_fun _let_4 tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.tc_fun _let_4 _let_3))) (= (tptp.hAPP _let_1 V_xq_2) (tptp.hAPP (tptp.c_Option_Ooption_Ooption__case tptp.tc_Nat_Onat (tptp.tc_prod T_a _let_4) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_a _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a tptp.tc_Nat_Onat _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun tptp.tc_Nat_Onat _let_5) T_a) (tptp.c_COMBC _let_4 tptp.tc_Nat_Onat tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_6 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_3 _let_4) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_5 _let_5) T_a) (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_3 T_a) _let_2) V_s_2))) _let_1)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (tptp.hAPP (tptp.c_Lazy__Sequence_Oyield T_a) V_xq_2)))))))))) _let_25 (forall ((V_k $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_Oequal__class_Oequal tptp.tc_Int_Oint) V_k) V_k))) _let_24 (forall ((V_lazy__sequence_2 $$unsorted) (V_a_2 $$unsorted) (V_fa_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size T_a V_fa_2))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_a_2 V_lazy__sequence_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP V_fa_2 V_a_2)) (tptp.hAPP _let_1 V_lazy__sequence_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_lazy__sequence_2 $$unsorted) (V_a_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case T_a T_b V_f1_2 V_f2_2 (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_b V_a_2 V_lazy__sequence_2)) (tptp.hAPP (tptp.hAPP V_f2_2 V_a_2) V_lazy__sequence_2))) (forall ((V_lazy__sequence_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_lazy__sequence_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_a_2 V_lazy__sequence_2) (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_a_H_2 V_lazy__sequence_H_2)) (and (= V_a_2 V_a_H_2) (= V_lazy__sequence_2 V_lazy__sequence_H_2)))) (forall ((V_xq $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_Lazy__Sequence_Olazy__sequence T_a))) (= (tptp.hAPP (tptp.c_Lazy__Sequence_Oyield T_a) (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_x V_xq)) (tptp.hAPP (tptp.c_Option_Ooption_OSome (tptp.tc_prod T_a _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a _let_1) V_x) V_xq))))) (forall ((V_lazy__sequence $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_Lazy__Sequence_Olazy__sequence T_a)))) (= (tptp.hAPP _let_1 (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_a V_lazy__sequence)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_lazy__sequence)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_New__DSequence_Opos__decr__bind T_b T_a V_x_2 V_f_2) V_i_2))) (let ((_let_2 (= V_i_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (and (=> _let_2 (= _let_1 (tptp.c_Lazy__Sequence_Oempty T_a))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.c_Lazy__Sequence_Obind T_b T_a (tptp.hAPP V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_a)) V_f_2) V_i_2)))))))) (forall ((V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_New__DSequence_Oneg__decr__bind T_b T_a V_x_2 V_f_2) V_i_2))) (let ((_let_2 (= V_i_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (and (=> _let_2 (= _let_1 (tptp.c_Lazy__Sequence_Ohit__bound T_a))) (=> (not _let_2) (= _let_1 (tptp.c_Lazy__Sequence_Ohb__bind T_b T_a (tptp.hAPP V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_a))) V_f_2) V_i_2)))))))) _let_23 _let_22 (forall ((V_i_2 $$unsorted) (V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Oneg__bind T_b T_a V_x_2 V_f_2) V_i_2) (tptp.c_Lazy__Sequence_Ohb__bind T_b T_a (tptp.hAPP V_x_2 V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_a))) V_f_2) V_i_2)))) (forall ((V_i_1 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Opos__empty T_a) V_i_1) (tptp.c_Lazy__Sequence_Oempty T_a))) _let_21 (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_a))))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_New__Random__Sequence_Oneg__bind T_b T_a V_R_2 V_f_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Oneg__bind T_b T_a (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_nrandom_2) V_size_2) V_seed_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) V_f_2) V_nrandom_2)) V_size_2)) V_seed_2))))))) (forall ((V_i_2 $$unsorted) (V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Opos__bind T_b T_a V_x_2 V_f_2) V_i_2) (tptp.hAPP (tptp.c_Lazy__Sequence_Obind T_b T_a (tptp.hAPP V_x_2 V_i_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_a)) V_f_2) V_i_2)))) _let_20 (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_New__Random__Sequence_Oneg__single T_a) V_x_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Oneg__single T_a V_x_2))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_a)))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_New__Random__Sequence_Opos__bind T_b T_a V_R_2 V_f_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Opos__bind T_b T_a (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_nrandom_2) V_size_2) V_seed_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) V_f_2) V_nrandom_2)) V_size_2)) V_seed_2))))))) (forall ((V_i_1 $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Oneg__single T_a V_x) V_i_1) (tptp.c_Lazy__Sequence_Ohb__single T_a V_x))) _let_19 (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_New__Random__Sequence_Opos__single T_a) V_x_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Opos__single T_a V_x_2))) (forall ((V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.c_List_Osplice T_a V_xs V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 V_ys))))) (forall ((V_ys $$unsorted) (V_y $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (let ((_let_3 (tptp.hAPP _let_1 V_x))) (= (tptp.c_List_Osplice T_a (tptp.hAPP _let_3 V_xs) (tptp.hAPP _let_2 V_ys)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.c_List_Osplice T_a V_xs V_ys)))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Osplice T_a V_xs (tptp.c_List_Olist_ONil T_a)) V_xs)) (forall ((V_ys $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Osplice T_a (tptp.c_List_Olist_ONil T_a) V_ys) V_ys)) (forall ((V_va $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_v) V_va))) (= (tptp.c_List_Osplice T_a _let_1 (tptp.c_List_Olist_ONil T_a)) _let_1))) (forall ((V_i_1 $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Opos__single T_a V_x) V_i_1) (tptp.hAPP (tptp.c_Lazy__Sequence_Osingle T_a) V_x))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_Oprod_Oprod__case T_a T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_OCollect (tptp.tc_prod T_a T_a)))) (= (tptp.c_Wellfounded_Oacyclic T_a (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Predicate_Oconversep T_a T_a V_r_2)))) (tptp.c_Wellfounded_Oacyclic T_a (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_r_2))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Oconversep T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Predicate_Oconversep T_b T_a V_r_2)) (tptp.c_Predicate_Oconversep T_b T_a V_s_2)))) (forall ((T_a $$unsorted)) (= (tptp.c_Predicate_Oconversep T_a T_a tptp.c_fequal) tptp.c_fequal)) (forall ((V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Oconversep T_b T_a (tptp.c_Predicate_Oconversep T_a T_b V_r_2)) V_r_2)) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_a T_b V_r_2) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_b_2) V_a_2)))) (forall ((V_a1_2 $$unsorted) (V_a2_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_a T_b V_r_2) V_a2_2) V_a1_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a1_2) V_a2_2)))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_a T_b V_r_2) V_b_2) V_a_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_a T_b V_r_2) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_b_2) V_a_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_a T_a (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_a) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_a) tptp.c_fNot)) tptp.c_fequal)) V_x_2) V_y_2)) (not (= V_x_2 V_y_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Oconversep T_b T_a (tptp.c_Predicate_Opred__comp T_b T_c T_a V_r_2 V_s_2)) (tptp.c_Predicate_Opred__comp T_a T_c T_b (tptp.c_Predicate_Oconversep T_c T_a V_s_2) (tptp.c_Predicate_Oconversep T_b T_c V_r_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Oconversep T_b T_a (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b (tptp.tc_fun T_a tptp.tc_HOL_Obool))) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Predicate_Oconversep T_b T_a V_r_2)) (tptp.c_Predicate_Oconversep T_b T_a V_s_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_a T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_a _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_a) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_a) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_a T_b)))) V_r_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_a) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_a T_b) V_r_2)))))))))) _let_18 (forall ((V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (=> (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_List_Otl T_a) (tptp.c_List_Oreplicate T_a V_n V_x)) (tptp.c_List_Oreplicate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_x)))) (forall ((V_y $$unsorted) (V_j $$unsorted) (V_x $$unsorted) (V_i $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) (tptp.c_List_Oreplicate T_a V_i V_x)) (tptp.c_List_Oreplicate T_b V_j V_y)) (tptp.c_List_Oreplicate (tptp.tc_prod T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) V_i) V_j) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x) V_y)))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_Oreplicate T_a V_n V_x)))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oreplicate T_a (tptp.hAPP tptp.c_Nat_OSuc V_n) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) (tptp.c_List_Oreplicate T_a V_n V_x)))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oreplicate T_a V_n V_x))) (= (tptp.hAPP (tptp.c_List_Orev T_a) _let_1) _let_1))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (=> (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.c_List_Olast T_a (tptp.c_List_Oreplicate T_a V_n V_x)) V_x))) (forall ((V_x $$unsorted) (V_k $$unsorted) (V_i $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_i) (tptp.c_List_Oreplicate T_a V_k V_x)) (tptp.c_List_Oreplicate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_k) V_i) V_x))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted)) (=> (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_List_Ohd T_a) (tptp.c_List_Oreplicate T_a V_n V_x)) V_x))) (forall ((V_x $$unsorted) (V_k $$unsorted) (V_i $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_i) (tptp.c_List_Oreplicate T_a V_k V_x)) (tptp.c_List_Oreplicate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat) V_i) V_k) V_x))) (forall ((V_P_2 $$unsorted) (V_a_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_List_Oreplicate T_a V_n_2 V_a_2)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (or (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)) (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_P_2 $$unsorted) (V_a_2 $$unsorted) (V_n_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) B_x) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_List_Oreplicate T_a V_n_2 V_a_2)))) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (and (tptp.hBOOL (tptp.hAPP V_P_2 V_a_2)) (not (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_y_2 $$unsorted) (V_n_2 $$unsorted) (V_x_2 $$unsorted) (V_m_2 $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_List_Oreplicate T_a V_m_2 V_x_2) (tptp.c_List_Oreplicate T_a V_n_2 V_y_2)) (and (= V_m_2 V_n_2) (=> (not (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= V_x_2 V_y_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) (tptp.c_List_Oreplicate T_a V_n V_x)) V_i) V_x))) (forall ((V_y_2 $$unsorted) (V_n_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_a) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_a) (tptp.c_List_Oreplicate T_a V_n_2 V_y_2)))) (and (= V_x_2 V_y_2) (not (= V_n_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_x $$unsorted) (V_m $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oreplicate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_Oreplicate T_a V_n V_x)) (tptp.c_List_Oreplicate T_a V_m V_x)))) (forall ((V_k $$unsorted) (V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oreplicate T_a V_n V_x))) (let ((_let_2 (tptp.c_List_Oreplicate T_a V_k V_x))) (let ((_let_3 (tptp.c_List_Oappend T_a))) (= (tptp.hAPP (tptp.hAPP _let_3 _let_1) _let_2) (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)))))) (tptp.class_Lattices_Obounded__lattice tptp.tc_HOL_Obool) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice T_1) (tptp.class_Lattices_Obounded__lattice (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_1) (tptp.class_Complete__Lattice_Ocomplete__lattice (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice T_1) (tptp.class_Lattices_Obounded__lattice__top (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice T_1) (tptp.class_Lattices_Obounded__lattice__bot (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Olattice T_1) (tptp.class_Lattices_Osemilattice__sup (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Olattice T_1) (tptp.class_Lattices_Osemilattice__inf (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Odistrib__lattice T_1) (tptp.class_Lattices_Odistrib__lattice (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_1) (tptp.class_Lattices_Oboolean__algebra (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_1) (tptp.class_Orderings_Opreorder (tptp.tc_fun T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Finite__Set_Ofinite T_2) (tptp.class_Finite__Set_Ofinite T_1)) (tptp.class_Finite__Set_Ofinite (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Lattices_Olattice T_1) (tptp.class_Lattices_Olattice (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Oorder T_1) (tptp.class_Orderings_Oorder (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Otop T_1) (tptp.class_Orderings_Otop (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Oord T_1) (tptp.class_Orderings_Oord (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Orderings_Obot T_1) (tptp.class_Orderings_Obot (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Groups_Ouminus T_1) (tptp.class_Groups_Ouminus (tptp.tc_fun T_2 T_1)))) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (tptp.class_Groups_Ominus T_1) (tptp.class_Groups_Ominus (tptp.tc_fun T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Enum_Oenum T_2) (tptp.class_HOL_Oequal T_1)) (tptp.class_HOL_Oequal (tptp.tc_fun T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Enum_Oenum T_2) (tptp.class_Enum_Oenum T_1)) (tptp.class_Enum_Oenum (tptp.tc_fun T_2 T_1)))) (tptp.class_HOL_Oequal tptp.tc_Com_Ocom) (tptp.class_Nat_Osize tptp.tc_Com_Ocom) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__comm__semiring__strict tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semiring__1__strict tptp.tc_Int_Oint) (tptp.class_Groups_Olinordered__ab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semiring__strict tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__group__add__abs tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__comm__monoid__add tptp.tc_Int_Oint) (tptp.class_Groups_Olinordered__ab__group__add tptp.tc_Int_Oint) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Rings_Oring__1__no__zero__divisors tptp.tc_Int_Oint) (tptp.class_Rings_Oordered__cancel__semiring tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__ring__strict tptp.tc_Int_Oint) (tptp.class_Rings_Oring__no__zero__divisors tptp.tc_Int_Oint) (tptp.class_Rings_Oordered__comm__semiring tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semiring__1 tptp.tc_Int_Oint) (tptp.class_Groups_Oordered__ab__group__add tptp.tc_Int_Oint) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semiring tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__semidom tptp.tc_Int_Oint) (tptp.class_Lattices_Osemilattice__sup tptp.tc_Int_Oint) (tptp.class_Lattices_Osemilattice__inf tptp.tc_Int_Oint) (tptp.class_Lattices_Odistrib__lattice tptp.tc_Int_Oint) (tptp.class_Groups_Oab__semigroup__mult tptp.tc_Int_Oint) (tptp.class_Groups_Ocomm__monoid__mult tptp.tc_Int_Oint) (tptp.class_Groups_Oab__semigroup__add tptp.tc_Int_Oint) (tptp.class_Rings_Oordered__semiring tptp.tc_Int_Oint) (tptp.class_Rings_Oordered__ring__abs tptp.tc_Int_Oint) (tptp.class_Rings_Ono__zero__divisors tptp.tc_Int_Oint) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__ring tptp.tc_Int_Oint) (tptp.class_Rings_Olinordered__idom tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_Int_Oint) (tptp.class_Groups_Osemigroup__add tptp.tc_Int_Oint) (tptp.class_Divides_Osemiring__div tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__semiring tptp.tc_Int_Oint) (tptp.class_Nat_Osemiring__char__0 tptp.tc_Int_Oint) (tptp.class_Groups_Oab__group__add tptp.tc_Int_Oint) (tptp.class_Rings_Ozero__neq__one tptp.tc_Int_Oint) (tptp.class_Rings_Oordered__ring tptp.tc_Int_Oint) (tptp.class_Orderings_Opreorder tptp.tc_Int_Oint) (tptp.class_Orderings_Olinorder tptp.tc_Int_Oint) (tptp.class_Groups_Omonoid__mult tptp.tc_Int_Oint) (tptp.class_Rings_Ocomm__ring__1 tptp.tc_Int_Oint) (tptp.class_Groups_Omonoid__add tptp.tc_Int_Oint) (tptp.class_Smallcheck_Osmall tptp.tc_Int_Oint) (tptp.class_Rings_Osemiring__1 tptp.tc_Int_Oint) (tptp.class_Rings_Osemiring__0 tptp.tc_Int_Oint) (tptp.class_Lattices_Olattice tptp.tc_Int_Oint) (tptp.class_Groups_Ogroup__add tptp.tc_Int_Oint) (tptp.class_Divides_Oring__div tptp.tc_Int_Oint) (tptp.class_Rings_Omult__zero tptp.tc_Int_Oint) (tptp.class_Orderings_Oorder tptp.tc_Int_Oint) (tptp.class_Int_Oring__char__0 tptp.tc_Int_Oint) (tptp.class_Int_Onumber__ring tptp.tc_Int_Oint) (tptp.class_Rings_Osemiring tptp.tc_Int_Oint) (tptp.class_Orderings_Oord tptp.tc_Int_Oint) (tptp.class_Groups_Ouminus tptp.tc_Int_Oint) (tptp.class_Groups_Osgn__if tptp.tc_Int_Oint) (tptp.class_Groups_Oabs__if tptp.tc_Int_Oint) (tptp.class_Rings_Oring__1 tptp.tc_Int_Oint) (tptp.class_Groups_Ominus tptp.tc_Int_Oint) (tptp.class_Power_Opower tptp.tc_Int_Oint) (tptp.class_Groups_Ozero tptp.tc_Int_Oint) (tptp.class_Rings_Oring tptp.tc_Int_Oint) (tptp.class_Rings_Oidom tptp.tc_Int_Oint) (tptp.class_Int_Onumber tptp.tc_Int_Oint) (tptp.class_Groups_Oone tptp.tc_Int_Oint) (tptp.class_HOL_Oequal tptp.tc_Int_Oint) (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__comm__semiring__strict tptp.tc_Nat_Onat) (tptp.class_Groups_Olinordered__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__semiring__strict tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Groups_Oordered__comm__monoid__add tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Oordered__cancel__semiring tptp.tc_Nat_Onat) (tptp.class_Rings_Oordered__comm__semiring tptp.tc_Nat_Onat) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__semiring tptp.tc_Nat_Onat) (tptp.class_Rings_Olinordered__semidom tptp.tc_Nat_Onat) (tptp.class_Lattices_Osemilattice__sup tptp.tc_Nat_Onat) (tptp.class_Lattices_Osemilattice__inf tptp.tc_Nat_Onat) (tptp.class_Lattices_Odistrib__lattice tptp.tc_Nat_Onat) (tptp.class_Groups_Oab__semigroup__mult tptp.tc_Nat_Onat) (tptp.class_Groups_Ocomm__monoid__mult tptp.tc_Nat_Onat) (tptp.class_Groups_Oab__semigroup__add tptp.tc_Nat_Onat) (tptp.class_Rings_Oordered__semiring tptp.tc_Nat_Onat) (tptp.class_Rings_Ono__zero__divisors tptp.tc_Nat_Onat) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_Nat_Onat) (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_Nat_Onat) (tptp.class_Groups_Osemigroup__add tptp.tc_Nat_Onat) (tptp.class_Divides_Osemiring__div tptp.tc_Nat_Onat) (tptp.class_Rings_Ocomm__semiring tptp.tc_Nat_Onat) (tptp.class_Orderings_Owellorder tptp.tc_Nat_Onat) (tptp.class_Nat_Osemiring__char__0 tptp.tc_Nat_Onat) (tptp.class_Rings_Ozero__neq__one tptp.tc_Nat_Onat) (tptp.class_Orderings_Opreorder tptp.tc_Nat_Onat) (tptp.class_Orderings_Olinorder tptp.tc_Nat_Onat) (tptp.class_Groups_Omonoid__mult tptp.tc_Nat_Onat) (tptp.class_Groups_Omonoid__add tptp.tc_Nat_Onat) (tptp.class_Rings_Osemiring__1 tptp.tc_Nat_Onat) (tptp.class_Rings_Osemiring__0 tptp.tc_Nat_Onat) (tptp.class_Lattices_Olattice tptp.tc_Nat_Onat) (tptp.class_Rings_Omult__zero tptp.tc_Nat_Onat) (tptp.class_Orderings_Oorder tptp.tc_Nat_Onat) (tptp.class_Rings_Osemiring tptp.tc_Nat_Onat) (tptp.class_Orderings_Oord tptp.tc_Nat_Onat) (tptp.class_Orderings_Obot tptp.tc_Nat_Onat) (tptp.class_Groups_Ominus tptp.tc_Nat_Onat) (tptp.class_Power_Opower tptp.tc_Nat_Onat) (tptp.class_Groups_Ozero tptp.tc_Nat_Onat) (tptp.class_Int_Onumber tptp.tc_Nat_Onat) (tptp.class_Groups_Oone tptp.tc_Nat_Onat) (tptp.class_HOL_Oequal tptp.tc_Nat_Onat) (tptp.class_Nat_Osize tptp.tc_Nat_Onat) (tptp.class_Complete__Lattice_Ocomplete__lattice tptp.tc_HOL_Obool) (tptp.class_Lattices_Obounded__lattice__top tptp.tc_HOL_Obool) (tptp.class_Lattices_Obounded__lattice__bot tptp.tc_HOL_Obool) (tptp.class_Lattices_Osemilattice__sup tptp.tc_HOL_Obool) (tptp.class_Lattices_Osemilattice__inf tptp.tc_HOL_Obool) (tptp.class_Lattices_Odistrib__lattice tptp.tc_HOL_Obool) (tptp.class_Lattices_Oboolean__algebra tptp.tc_HOL_Obool) (tptp.class_Orderings_Opreorder tptp.tc_HOL_Obool) (tptp.class_Finite__Set_Ofinite tptp.tc_HOL_Obool) (tptp.class_Lattices_Olattice tptp.tc_HOL_Obool) (tptp.class_Orderings_Oorder tptp.tc_HOL_Obool) (tptp.class_Orderings_Otop tptp.tc_HOL_Obool) (tptp.class_Orderings_Oord tptp.tc_HOL_Obool) (tptp.class_Orderings_Obot tptp.tc_HOL_Obool) (tptp.class_Groups_Ouminus tptp.tc_HOL_Obool) (tptp.class_Groups_Ominus tptp.tc_HOL_Obool) (tptp.class_HOL_Oequal tptp.tc_HOL_Obool) (tptp.class_Enum_Oenum tptp.tc_HOL_Obool) (tptp.class_Nat_Osize tptp.tc_HOL_Obool) (tptp.class_HOL_Oequal tptp.tc_Com_Ostate) (tptp.class_Nat_Osize tptp.tc_Com_Ostate) (forall ((T_1 $$unsorted)) (tptp.class_HOL_Oequal (tptp.tc_List_Olist T_1))) (forall ((T_1 $$unsorted)) (tptp.class_Nat_Osize (tptp.tc_List_Olist T_1))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Finite__Set_Ofinite T_2) (tptp.class_Finite__Set_Ofinite T_1)) (tptp.class_Finite__Set_Ofinite (tptp.tc_sum T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (tptp.class_HOL_Oequal (tptp.tc_sum T_2 T_1))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Enum_Oenum T_2) (tptp.class_Enum_Oenum T_1)) (tptp.class_Enum_Oenum (tptp.tc_sum T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (tptp.class_Nat_Osize (tptp.tc_sum T_2 T_1))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_1) (tptp.class_Finite__Set_Ofinite (tptp.tc_Option_Ooption T_1)))) (forall ((T_1 $$unsorted)) (tptp.class_HOL_Oequal (tptp.tc_Option_Ooption T_1))) (forall ((T_1 $$unsorted)) (=> (tptp.class_Enum_Oenum T_1) (tptp.class_Enum_Oenum (tptp.tc_Option_Ooption T_1)))) (forall ((T_1 $$unsorted)) (tptp.class_Nat_Osize (tptp.tc_Option_Ooption T_1))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Finite__Set_Ofinite T_2) (tptp.class_Finite__Set_Ofinite T_1)) (tptp.class_Finite__Set_Ofinite (tptp.tc_prod T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Smallcheck_Osmall T_2) (tptp.class_Smallcheck_Osmall T_1)) (tptp.class_Smallcheck_Osmall (tptp.tc_prod T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (tptp.class_HOL_Oequal (tptp.tc_prod T_2 T_1))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Enum_Oenum T_2) (tptp.class_Enum_Oenum T_1)) (tptp.class_Enum_Oenum (tptp.tc_prod T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (tptp.class_Nat_Osize (tptp.tc_prod T_2 T_1))) (tptp.class_Finite__Set_Ofinite tptp.tc_Product__Type_Ounit) (tptp.class_Smallcheck_Osmall tptp.tc_Product__Type_Ounit) (tptp.class_HOL_Oequal tptp.tc_Product__Type_Ounit) (tptp.class_Enum_Oenum tptp.tc_Product__Type_Ounit) (tptp.class_Nat_Osize tptp.tc_Product__Type_Ounit) (tptp.class_HOL_Oequal tptp.tc_Code__Evaluation_Oterm) (tptp.class_Nat_Osize tptp.tc_Code__Evaluation_Oterm) (forall ((T_1 $$unsorted)) (tptp.class_HOL_Oequal (tptp.tc_Hoare__Mirabelle_Otriple T_1))) (forall ((T_1 $$unsorted)) (tptp.class_Nat_Osize (tptp.tc_Hoare__Mirabelle_Otriple T_1))) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Olinordered__comm__semiring__strict tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Olinordered__ab__semigroup__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Olinordered__semiring__strict tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Oordered__ab__semigroup__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Oordered__comm__monoid__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Ocancel__ab__semigroup__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Oordered__cancel__semiring tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Oordered__comm__semiring tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Ocancel__semigroup__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Olinordered__semiring tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Olinordered__semidom tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Oab__semigroup__mult tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Ocomm__monoid__mult tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Oab__semigroup__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Oordered__semiring tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Ono__zero__divisors tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Ocomm__monoid__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Ocomm__semiring__1 tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Osemigroup__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Divides_Osemiring__div tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Ocomm__semiring tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Nat_Osemiring__char__0 tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Ozero__neq__one tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Orderings_Opreorder tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Orderings_Olinorder tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Omonoid__mult tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Omonoid__add tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Osemiring__1 tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Osemiring__0 tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Omult__zero tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Orderings_Oorder tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Rings_Osemiring tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Orderings_Oord tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Ominus tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Power_Opower tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Ozero tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Int_Onumber tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Groups_Oone tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_HOL_Oequal tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Nat_Osize tptp.tc_Code__Numeral_Ocode__numeral) (forall ((T_1 $$unsorted)) (tptp.class_HOL_Oequal (tptp.tc_Lazy__Sequence_Olazy__sequence T_1))) (forall ((T_1 $$unsorted)) (tptp.class_Nat_Osize (tptp.tc_Lazy__Sequence_Olazy__sequence T_1))) (forall ((V_P $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_COMBI T_a) V_P) V_P)) (forall ((V_Q $$unsorted) (V_P $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_COMBK T_a T_b) V_P) V_Q) V_P)) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_c $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_a T_c) V_P_2) V_Q_2) V_R_2) (tptp.hAPP V_P_2 (tptp.hAPP V_Q_2 V_R_2)))) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_a) V_P_2) V_Q_2) V_R_2) (tptp.hAPP (tptp.hAPP V_P_2 V_R_2) V_Q_2))) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c T_a) V_P_2) V_Q_2) V_R_2) (tptp.hAPP (tptp.hAPP V_P_2 V_R_2) (tptp.hAPP V_Q_2 V_R_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fequal V_x_2) V_y_2))) (= V_x_2 V_y_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (= V_x_2 V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fequal V_x_2) V_y_2)))) (not (tptp.hBOOL tptp.c_fFalse)) (tptp.hBOOL tptp.c_fTrue) (forall ((V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP tptp.c_fNot V_P_2))) (not (tptp.hBOOL V_P_2)))) (forall ((V_P_2 $$unsorted)) (or (not (not (tptp.hBOOL V_P_2))) (tptp.hBOOL (tptp.hAPP tptp.c_fNot V_P_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL V_P_2)) (not (tptp.hBOOL V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_P_2) V_Q_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_P_2) V_Q_2))) (tptp.hBOOL V_P_2))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_P_2) V_Q_2))) (tptp.hBOOL V_Q_2))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL V_P_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fdisj V_P_2) V_Q_2)))) (forall ((V_P_2 $$unsorted) (V_Q_2 $$unsorted)) (or (not (tptp.hBOOL V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fdisj V_P_2) V_Q_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fdisj V_P_2) V_Q_2))) (tptp.hBOOL V_P_2) (tptp.hBOOL V_Q_2))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (tptp.hBOOL V_P_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fimplies V_P_2) V_Q_2)))) (forall ((V_P_2 $$unsorted) (V_Q_2 $$unsorted)) (or (not (tptp.hBOOL V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fimplies V_P_2) V_Q_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fimplies V_P_2) V_Q_2))) (not (tptp.hBOOL V_P_2)) (tptp.hBOOL V_Q_2))) (tptp.c_Hoare__Mirabelle_Ohoare__derivs tptp.tc_Com_Ostate _let_14 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname _let_1 (tptp.hAPP (tptp.hAPP _let_13 (tptp.hAPP _let_12 _let_17)) (tptp.hAPP _let_11 _let_17))) tptp.v_Procs)) _let_16 _let_15 true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 123.70/124.44  )
% 123.70/124.44  % SZS output end Proof for SWW379+1
% 123.70/124.44  % cvc5---1.0.5 exiting
% 123.70/124.45  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------