%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SWW368+1 : TPTP v8.1.2. Released v5.2.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n013.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Fri Sep  1 00:20:34 EDT 2023

% Result   : Theorem 80.20s 80.50s
% Output   : Proof 80.20s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.90  % Problem    : SWW368+1 : TPTP v8.1.2. Released v5.2.0.
% 0.12/0.91  % Command    : do_cvc5 %s %d
% 0.90/1.12  % Computer : n013.cluster.edu
% 0.90/1.12  % Model    : x86_64 x86_64
% 0.90/1.12  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.90/1.12  % Memory   : 8042.1875MB
% 0.90/1.12  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.90/1.12  % CPULimit   : 300
% 0.90/1.12  % WCLimit    : 300
% 0.90/1.12  % DateTime   : Sun Aug 27 22:08:48 EDT 2023
% 0.90/1.12  % CPUTime    : 
% 1.94/2.21  %----Proving TF0_NAR, FOF, or CNF
% 80.20/80.50  ------- convert to smt2 : /export/starexec/sandbox2/tmp/tmp.4YVVLmSOGY/cvc5---1.0.5_5530.p...
% 80.20/80.50  ------- get file name : TPTP file name is SWW368+1
% 80.20/80.50  ------- cvc5-fof : /export/starexec/sandbox2/solver/bin/cvc5---1.0.5_5530.smt2...
% 80.20/80.50  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 80.20/80.50  --- Run --no-e-matching --full-saturate-quant at 5...
% 80.20/80.50  --- Run --no-e-matching --enum-inst-sum --full-saturate-quant at 5...
% 80.20/80.50  --- Run --finite-model-find --uf-ss=no-minimal at 5...
% 80.20/80.50  --- Run --multi-trigger-when-single --full-saturate-quant at 5...
% 80.20/80.50  --- Run --trigger-sel=max --full-saturate-quant at 5...
% 80.20/80.50  --- Run --multi-trigger-when-single --multi-trigger-priority --full-saturate-quant at 5...
% 80.20/80.50  --- Run --multi-trigger-cache --full-saturate-quant at 5...
% 80.20/80.50  --- Run --prenex-quant=none --full-saturate-quant at 5...
% 80.20/80.50  --- Run --enum-inst-interleave --decision=internal --full-saturate-quant at 5...
% 80.20/80.50  --- Run --relevant-triggers --full-saturate-quant at 5...
% 80.20/80.50  --- Run --finite-model-find --e-matching --sort-inference --uf-ss-fair at 5...
% 80.20/80.50  --- Run --pre-skolem-quant=on --full-saturate-quant at 10...
% 80.20/80.50  % SZS status Theorem for SWW368+1
% 80.20/80.50  % SZS output start Proof for SWW368+1
% 80.20/80.50  (
% 80.20/80.50  (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Com_Opname tptp.t_a tptp.v_mgt__call) tptp.v_U))) (let ((_let_2 (tptp.hAPP tptp.v_mgt__call tptp.v_pn))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.t_a tptp.tc_HOL_Obool)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.t_a) _let_2) tptp.v_G)) _let_1)))) (let ((_let_5 (not _let_4))) (let ((_let_6 (tptp.hAPP (tptp.c_member tptp.t_a) _let_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Com_Opname) tptp.v_pn) tptp.v_U)))) (let ((_let_8 (tptp.hAPP tptp.c_Nat_OSuc tptp.v_n))) (let ((_let_9 (tptp.c_Finite__Set_Ocard tptp.t_a))) (let ((_let_10 (tptp.hAPP _let_9 _let_1))) (let ((_let_11 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 tptp.v_G) _let_1)))) (let ((_let_12 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_13 (tptp.c_Nat_Osize__class_Osize tptp.tc_HOL_Obool))) (let ((_let_14 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (let ((_let_15 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_16 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_17 (tptp.c_Orderings_Otop__class_Otop _let_16))) (let ((_let_18 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_19 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_20 (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_21 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_22 (tptp.c_Orderings_Obot__class_Obot _let_16))) (let ((_let_23 (tptp.c_SetInterval_Oord__class_OgreaterThan tptp.tc_Nat_Onat))) (let ((_let_24 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_25 (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_26 (tptp.c_Set_OCollect _let_24))) (let ((_let_27 (tptp.tc_fun _let_24 tptp.tc_HOL_Obool))) (let ((_let_28 (tptp.c_Orderings_Otop__class_Otop _let_27))) (let ((_let_29 (tptp.c_Int_OBit1 tptp.c_Int_OPls))) (let ((_let_30 (tptp.c_Int_OBit0 _let_29))) (let ((_let_31 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_32 (tptp.hAPP _let_31 _let_30))) (let ((_let_33 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_34 (tptp.c_SetInterval_Oord__class_OatLeast tptp.tc_Nat_Onat))) (let ((_let_35 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_16) _let_17))) (let ((_let_36 (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat))) (let ((_let_37 (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat))) (let ((_let_38 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (let ((_let_39 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_40 (tptp.c_Orderings_Obot__class_Obot _let_27))) (let ((_let_41 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_42 (tptp.c_Set_Oinsert _let_24))) (let ((_let_43 (tptp.c_Relation_OImage _let_24 _let_24 tptp.c_Int_Ointrel))) (let ((_let_44 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_45 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_46 (tptp.tc_prod _let_27 _let_27))) (let ((_let_47 (tptp.tc_fun _let_46 tptp.tc_HOL_Obool))) (let ((_let_48 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_46) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_27 _let_27) _let_40) _let_40)) (tptp.c_Orderings_Obot__class_Obot _let_47)))) (let ((_let_49 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_47))) (let ((_let_50 (tptp.c_Product__Type_OPair _let_47 _let_47))) (let ((_let_51 (tptp.tc_fun tptp.tc_Int_Oint _let_38))) (let ((_let_52 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_53 (tptp.hAPP _let_52 _let_44))) (let ((_let_54 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_55 (tptp.hAPP _let_54 _let_30))) (let ((_let_56 (tptp.c_Int_OBit1 _let_29))) (let ((_let_57 (tptp.hAPP _let_54 _let_56))) (let ((_let_58 (tptp.c_Orderings_Obot__class_Obot _let_38))) (let ((_let_59 (tptp.hAPP _let_54 tptp.c_Int_OMin))) (let ((_let_60 (tptp.hAPP _let_54 tptp.c_Int_OPls))) (let ((_let_61 (tptp.hAPP _let_52 tptp.c_Int_OMin))) (let ((_let_62 (tptp.hAPP _let_52 tptp.c_Int_OPls))) (let ((_let_63 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_64 (tptp.hAPP _let_63 tptp.c_Int_OPls))) (let ((_let_65 (tptp.hAPP _let_63 tptp.c_Int_OMin))) (let ((_let_66 (tptp.hAPP tptp.c_Nat_OSuc _let_12))) (let ((_let_67 (tptp.hAPP tptp.c_Nat_OSuc _let_66))) (let ((_let_68 (tptp.hAPP tptp.c_Int_Onat _let_55))) (let ((_let_69 (tptp.hAPP _let_63 _let_44))) (let ((_let_70 (tptp.hAPP _let_31 _let_56))) (let ((_let_71 (tptp.hAPP _let_31 _let_29))) (let ((_let_72 (tptp.hAPP tptp.c_Int_Onat _let_45))) (let ((_let_73 (tptp.hAPP tptp.c_Int_Onat _let_44))) (let ((_let_74 (tptp.hAPP _let_31 tptp.c_Int_OPls))) (let ((_let_75 (tptp.hAPP _let_14 _let_19))) (let ((_let_76 (tptp.hAPP _let_14 _let_12))) (let ((_let_77 (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) V_B_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)))))))) (let ((_let_78 (forall ((T_b $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_x_2 $$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_c) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2)))))))) (let ((_let_79 (forall ((T_b $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_x_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2))))))) (let ((_let_80 (tptp.hBOOL (tptp.hAPP _let_6 _let_1)))) (let ((_let_81 (not _let_7))) (let ((_let_82 (or _let_81 _let_80))) (let ((_let_83 (EQ_RESOLVE (ASSUME :args (_let_78)) (MACRO_SR_EQ_INTRO :args (_let_78 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_84 (not _let_82))) (let ((_let_85 (and _let_80 _let_11))) (let ((_let_86 (not _let_80))) (let ((_let_87 (= _let_4 _let_85))) (let ((_let_88 (not _let_85))) (let ((_let_89 (_let_77))) (let ((_let_90 (ASSUME :args _let_89))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_83 :args (tptp.t_a tptp.v_U tptp.tc_Com_Opname tptp.v_pn tptp.v_mgt__call QUANTIFIERS_INST_CBQI_CONFLICT)) :args (_let_79))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_82)) :args ((or _let_81 _let_80 _let_84))) (ASSUME :args (_let_7)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_NEG :args (_let_85)) :args ((or (not _let_11) _let_85 _let_86))) (ASSUME :args (_let_11)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_EQUIV_POS2 :args (_let_87)) :args ((or _let_4 _let_88 (not _let_87)))) (ASSUME :args (_let_5)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_90 :args (_let_1 tptp.v_G _let_2 tptp.t_a QUANTIFIERS_INST_E_MATCHING ((tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) V_B_2)))) :args _let_89)) _let_90 :args (_let_87 false _let_77)) :args (_let_88 true _let_4 false _let_87)) :args (_let_86 false _let_11 true _let_85)) :args (_let_84 false _let_7 true _let_80)) _let_83 :args (false true _let_82 false _let_79)) :args ((forall ((V_g_2 $$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_G_2 $$unsorted) (V_ts_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.t_a tptp.tc_HOL_Obool)) V_ts_2) V_G_2)) (tptp.v_P V_G_2 V_ts_2))) (forall ((V_h_2 $$unsorted) (T_c $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_h_2) V_F_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP tptp.c_Nat_OSuc _let_2)))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (=> (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 _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)))))) _let_78 (forall ((V_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) V_A_2))) (=> (not (= V_a_2 V_b_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_a_2))) (=> (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))) (= V_a_2 V_b_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) V_B_2)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (=> (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_fun T_b 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_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_B_2)) (tptp.hAPP _let_1 V_A_2))) (= V_A_2 V_B_2)))))) (forall ((V_c_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))) (=> (= V_A_2 V_B_2) (=> (=> _let_3 (not _let_2)) (not (=> (not _let_3) _let_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b 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_A_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (=> (= 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 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (=> (= 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_x_2 $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) 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 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_b) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_b) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_y_2) V_A_2) V_x_2)) (or (= V_y_2 V_x_2) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_y_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (= (tptp.hAPP _let_1 _let_2) _let_2)))) (forall ((T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (= V_A_2 V_B_2) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_C_2))))))) (forall ((T_b $$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_b tptp.tc_HOL_Obool)) V_B_2) V_A_2)))) (forall ((T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)))) (forall ((T_b $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b 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_A_2 $$unsorted) (T_b $$unsorted)) (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_A_2))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_d T_c V_g_2) V_A_2)) (tptp.hAPP (tptp.c_Set_Oimage T_d T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c T_b T_d) V_f_2) V_g_2)) V_A_2))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2) V_A_2))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 V_A_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 V_B_2))) (= (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2)) (= V_A_2 V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_a_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) V_A_2))) (or (= V_a_2 V_b_2) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (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)) (=> (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) 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_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (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)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((T_c $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_c) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_z_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_z_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (= V_z_2 (tptp.hAPP V_f_2 B_x)))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (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_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b 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_b) V_b_2) V_B_2)))))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (= (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_c T_b V_f_2) V_A_2))) (exists ((B_AA $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) B_AA) V_A_2)) (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) B_AA)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) _let_77 (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) (tptp.hAPP V_f_2 V_x_2)) _let_1) _let_1)))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (=> (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_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_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_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_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_R_2)) (tptp.hAPP _let_3 V_S_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2))))))) (forall ((V_na_2 $$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_na_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_na_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_m))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_n)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k)) (tptp.hAPP _let_2 V_n)))))))) (forall ((V_na_2 $$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_na_2)) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_k_2)) (= V_m_2 V_na_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_k $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)) (tptp.hAPP tptp.c_Nat_OSuc V_k)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) V_k)))) (forall ((V_y $$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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Oorder T_b) (= (= V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2))))))) (forall ((V_y $$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 ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (= V_x_2 V_y_2)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))))) (forall ((V_c $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a))) (=> (tptp.class_Orderings_Oord T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP _let_1 V_c))))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a))))))) (forall ((V_y $$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_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_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_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_z))) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (tptp.hBOOL (tptp.hAPP _let_2 V_x)))))))) (forall ((V_y $$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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP tptp.c_fequal V_y_2))) (= V_x_2 V_y_2))) (forall ((V_n $$unsorted)) (not (= V_n (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_n $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_n) V_n))) (forall ((V_nat_H_2 $$unsorted) (V_nat_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Nat_OSuc V_nat_2) (tptp.hAPP tptp.c_Nat_OSuc V_nat_H_2)) (= V_nat_2 V_nat_H_2))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (= (tptp.hAPP tptp.c_Nat_OSuc V_x) (tptp.hAPP tptp.c_Nat_OSuc V_y)) (= V_x V_y))) (forall ((V_Q_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_Pa_2) V_Q_2)) (tptp.hBOOL (tptp.hAPP V_Q_2 V_x_2))))) (forall ((V_x_2 $$unsorted) (V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_Pa_2) V_Q_2)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (tptp.hBOOL (tptp.hAPP V_Q_2 V_x_2))))) (forall ((V_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_n))) (forall ((V_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 $$unsorted) (V_m $$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_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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)) (= V_m V_n))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_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_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (= (= (tptp.hAPP _let_1 V_R_2) (tptp.hAPP _let_1 V_S_2)) (= V_R_2 V_S_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Orderings_Oord T_c) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b T_c)) V_f_2) V_g_2)) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Orderings_Oord T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b T_c)) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2)))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Orderings_Oord T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b T_c)) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2)))))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_n)))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) V_m_2))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2))) (= (tptp.hBOOL (tptp.hAPP _let_2 _let_1)) (or (tptp.hBOOL (tptp.hAPP _let_2 V_na_2)) (= V_m_2 _let_1)))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.hAPP tptp.c_Nat_OSuc V_m_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_m_2))))) (forall ((V_n $$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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (=> (= (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_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_x_2) V_A_2))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) 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_na_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_na_2)) (tptp.hBOOL (tptp.c_Nat_Onat_Onat__case tptp.tc_HOL_Obool tptp.c_fFalse (tptp.hAPP _let_1 V_m_2) V_na_2))))) (forall ((V_G_2 $$unsorted) (V_c_2 $$unsorted)) (=> (tptp.v_wt V_c_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Com_Opname) B_x) tptp.v_U)) (tptp.v_P V_G_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.t_a) (tptp.hAPP tptp.v_mgt__call B_x)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.t_a tptp.tc_HOL_Obool)))))) (tptp.v_P V_G_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.t_a) (tptp.v_mgt V_c_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.t_a tptp.tc_HOL_Obool))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_B_2)) _let_1) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_F_2 V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) _let_1) _let_1)))))) (forall ((V_Y_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b (tptp.c_COMBI T_b)) V_Y_2) V_Y_2)) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ominus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_c T_b)) V_A_2) V_B_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (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) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_b V_f_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 V_x_2) _let_1) _let_1)))))) (forall ((V_pn_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Com_Opname) V_pn_2) tptp.v_U)) (tptp.v_wt (tptp.c_Option_Othe tptp.tc_Com_Ocom (tptp.c_Com_Obody V_pn_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (not (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (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_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_G_2 $$unsorted) (V_pn_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.t_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert tptp.t_a))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP tptp.v_mgt__call V_pn_2)))) (=> (tptp.v_P (tptp.hAPP _let_3 V_G_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.v_mgt (tptp.c_Option_Othe tptp.tc_Com_Ocom (tptp.c_Com_Obody V_pn_2)))) _let_1)) (tptp.v_P V_G_2 (tptp.hAPP _let_3 _let_1))))))) (forall ((V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Obot T_b) (= (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c T_b)) V_x_2) (tptp.c_Orderings_Obot__class_Obot T_b)))) (forall ((V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Obot T_b) (= (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c T_b)) V_x_2) (tptp.c_Orderings_Obot__class_Obot T_b)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Fun_Oinj__on T_b T_b (tptp.c_COMBI T_b) V_A_2)) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (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_b) B_xa) V_A_2)) (=> (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_f_2 B_xa)) (= B_x B_xa)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)))) (forall ((V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (= (tptp.hAPP V_F_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_z_2))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) _let_2) V_A_2) _let_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (= V_x_2 V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (= (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (not (= V_x_2 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (not (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_4 (tptp.hAPP _let_3 V_A_2))) (= (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_4 V_B_2)) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_4 (tptp.hAPP _let_3 V_A_2))) (= (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_4 (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$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))) (= (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_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_B_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_b V_f_2 V_F_2) (= (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_x_2) V_x_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_C_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_b T_c 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_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (= (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.hAPP _let_2 V_A_2)) (and (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_x_2))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$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_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_A_2))))) (forall ((V_N_2 $$unsorted)) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc V_N_2)) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) 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_c) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (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))) (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (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.hBOOL (tptp.hAPP _let_1 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (= (tptp.hBOOL (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))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (= (tptp.hAPP (tptp.hAPP _let_2 V_B_2) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2)) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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.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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) V_A_2)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))))) (forall ((V_c_2 $$unsorted) (T_b $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_c_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2))) (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_b))) (=> (= (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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_b))) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_a_2) (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_c_2) (tptp.hAPP (tptp.hAPP _let_2 V_d_2) _let_1))) (or (and (= V_a_2 V_c_2) (= V_b_2 V_d_2)) (and (= V_a_2 V_d_2) (= V_b_2 V_c_2))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (not (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) _let_2)) (= V_A_2 _let_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2)) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_F_2 V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_2 (tptp.tc_fun T_b 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_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) V_x_2) V_A_2)))) (let ((_let_5 (tptp.hAPP (tptp.c_Set_Oinsert T_b) 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_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) _let_1))))) (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_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2))) (=> (tptp.c_Fun_Oinj__on T_c T_b V_f_2 V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_3 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (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_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_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_2))))) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b 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_b) 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_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_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_b_2 V_a_2))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (= V_b_2 V_a_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) 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_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_2))))) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_nat_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Nat_Onat_Onat__case T_b V_f1_2 V_f2_2 (tptp.hAPP tptp.c_Nat_OSuc V_nat_2)) (tptp.hAPP V_f2_2 V_nat_2))) (forall ((V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c (tptp.hAPP (tptp.c_COMBK T_c T_b) V_c_2)) V_A_2))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_c_2) _let_1)))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_f_2) V_A_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)) _let_1) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_b T_b V_f_2 V_A_2) (= _let_1 V_A_2)))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_f_2) V_A_2))) (tptp.c_Fun_Oinj__on T_b T_b V_f_2 V_A_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2)) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_Groups_Oab__group__add T_b) (=> (= (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_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ominus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_c T_b)) V_A_2) V_B_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP V_A_2 V_x_2)) (tptp.hAPP V_B_2 V_x_2))))) (forall ((V_c_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c (tptp.hAPP (tptp.c_COMBK T_c T_b) V_c_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_c_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_B_2)) (=> (tptp.c_Fun_Oinj__on T_c T_b V_g_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 (tptp.c_Set_Oimage T_c T_b V_g_2) V_B_2)) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2))))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (=> (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.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_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)) (= (exists ((B_f $$unsorted)) (and (tptp.c_Fun_Oinj__on T_b T_c B_f V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c B_f) V_A_2)) V_B_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2))))))) (forall ((V_A_H_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (exists ((B_f $$unsorted)) (and (tptp.c_Fun_Oinj__on T_b T_c B_f V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c B_f) V_A_2)) V_A_H_2)))) (exists ((B_g $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b B_g) V_A_H_2) V_A_2))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Set_Othe__elem T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_2)) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) 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_b V_f_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)) (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_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_F_2 V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_x_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_b T_c V_A_2 V_f_2) V_x_2)) V_B_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (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 _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_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (=> (not _let_5) (= _let_4 _let_2)))))))))) (forall ((V_Pa_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_F_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (forall ((B_a $$unsorted) (B_F $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) B_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) B_F)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 B_F))) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_F)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) B_a) B_F))))))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_F_2)))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_b 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_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b 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_b T_b V_h_2) V_N_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.hAPP V_F_2 (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.c_Finite__Set_Ofolding__one T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_F_2 _let_4)))))))))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Groups_Oone T_b) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b 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_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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_b T_c 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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_b) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_b T_c V_A_2 V_f_2) V_y_2) V_x_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_x) B_m))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (tptp.c_Fun_Oinj__on T_c T_b (tptp.c_Fun_Othe__inv__into T_b T_c V_A_2 V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.c_Fun_Othe__inv__into T_b T_c V_A_2 V_f_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_A_2))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_b V_f_2 V_F_2) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_y_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (= (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_b T_c V_A_2 V_f_2) V_y_2)) V_y_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_member T_b) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b) 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_k_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OatMost T_b 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 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OatLeast T_b 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 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OlessThan T_b V_less_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_i_2) V_k_2)))) (forall ((V_k_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OgreaterThan T_b V_less_2 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less_2 V_k_2) V_i_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OatLeastAtMost T_b V_less__eq_2 V_l_2 V_u_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_l_2) V_i_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_less__eq_2 V_i_2) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (V_less_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OgreaterThanLessThan T_b 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_2 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OatLeastLessThan T_b 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 $$unsorted) (V_less__eq_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.c_SetInterval_Oord_OgreaterThanAtMost T_b 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_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP V_f_2 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_F_2 V_A_2)) V_A_2))))))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_x_2 V_y_2)) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)))) (let ((_let_4 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_y_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_4) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_4))))))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 V_A_2))))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_x_2 (tptp.hAPP V_f_2 V_x_2)) V_f_2)) (forall ((V_na_2 $$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_na_2) V_k_2)) (= V_m_2 V_na_2)))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_na_2)) (= V_m_2 V_na_2)))) (forall ((V_k $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a)) (= V_b V_c))))) (forall ((V_c $$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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_a) (=> (= (tptp.hAPP _let_1 V_b) (tptp.hAPP _let_1 V_c)) (= V_b V_c))))) (forall ((T_b $$unsorted) (T_c $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (=> (= (tptp.hAPP V_f_2 V_x_2) V_y_2) (= (tptp.c_Fun_Ofun__upd T_c T_b V_f_2 V_x_2 V_y_2) V_f_2))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted)) (=> (not (= V_z_2 V_x_2)) (= (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b T_c 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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted)) (=> (not (= V_a_2 V_c_2)) (= (tptp.c_Fun_Ofun__upd T_b T_c (tptp.c_Fun_Ofun__upd T_b T_c V_m_2 V_a_2 V_b_2) V_c_2 V_d_2) (tptp.c_Fun_Ofun__upd T_b T_c (tptp.c_Fun_Ofun__upd T_b T_c V_m_2 V_c_2 V_d_2) V_a_2 V_b_2)))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_x_2 V_y_2) V_z_2))) (let ((_let_2 (= V_z_2 V_x_2))) (and (=> _let_2 (= _let_1 V_y_2)) (=> (not _let_2) (= _let_1 (tptp.hAPP V_f_2 V_z_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ofun__upd T_c T_b 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_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Fun_Ofun__upd T_b T_c (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_x_2 V_y_2) V_x_2 V_z_2) (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_x_2 V_z_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_x_2 V_y_2) V_f_2) (= (tptp.hAPP V_f_2 V_x_2) V_y_2))) (forall ((V_c_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_b) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_a_2)) (= V_b_2 V_c_2))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2))) (=> (tptp.class_Groups_Ocancel__semigroup__add T_b) (= (= (tptp.hAPP _let_1 V_b_2) (tptp.hAPP _let_1 V_c_2)) (= V_b_2 V_c_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Groups_Oab__semigroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_b_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_a_2 V_b_2) V_x_2))) (let ((_let_2 (= V_x_2 V_a_2))) (and (=> _let_2 (= _let_1 V_b_2)) (=> (not _let_2) (= _let_1 (tptp.hAPP V_f_2 V_x_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_d $$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_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_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_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_c_2))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_b) (= (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 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_b) (= (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 $$unsorted) (V_a $$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 $$unsorted) (V_m $$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_n $$unsorted) (V_k $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_k)) V_n)) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k) V_n)))))))) (forall ((V_n $$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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_k)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k) V_n))))) (forall ((V_l $$unsorted) (V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k) V_l)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k)) (tptp.hAPP (tptp.hAPP _let_1 V_j) V_l)))))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.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_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_j)))))) (forall ((V_m $$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_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)) (exists ((B_k $$unsorted)) (= V_na_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) B_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) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) V_m)))) (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_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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) V_n) V_m)) (forall ((V_i $$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 ((V_n $$unsorted)) (= (tptp.hAPP tptp.c_Nat_OSuc V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n))) (forall ((V_n $$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 $$unsorted) (V_k_2 $$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_m $$unsorted) (V_j $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_j) V_k))) V_m) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k) V_m)))))) (forall ((V_m $$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_x_2 $$unsorted) (V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_y_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)))) (tptp.c_Fun_Oinj__on T_b T_c (tptp.c_Fun_Ofun__upd T_b T_c V_f_2 V_x_2 V_y_2) V_A_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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))) (let ((_let_3 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_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_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Set_Oinsert T_b))) (let ((_let_5 (tptp.c_Groups_Ominus__class_Ominus _let_2))) (let ((_let_6 (tptp.c_member T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_2))))) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_Pa_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_A_2)) (=> (forall ((B_a $$unsorted) (B_A $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) B_A)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_a) B_A)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_A)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) B_A) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) B_a) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (forall ((V_X_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Set_Othe__elem T_b V_X_2) (tptp.c_HOL_OThe T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.hAPP tptp.c_fequal V_X_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 ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Finite__Set_Ofun__left__comm__idem T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_1) (tptp.c_Groups_Ominus__class_Ominus _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_k_2 $$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 ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_sum T_b T_c)) (tptp.c_Sum__Type_OPlus T_b T_c 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_b) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) 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_b) 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_Pa_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_F $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) B_F)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) B_F))) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_F)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) B_x) B_F))))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_F_2)))))) (forall ((V_n $$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 $$unsorted) (V_m $$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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_b 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_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_n)))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_x))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (not (= V_m_2 V_na_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_m_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (= V_x_2 V_y_2)))))) (forall ((V_y $$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_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2))) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2))) (= V_x_2 V_y_2)))))) (forall ((V_y $$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_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_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_n)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_m)) (not (= V_m V_n)))) (forall ((V_t $$unsorted) (V_s $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_s) V_t)) (not (= V_s V_t)))) (forall ((V_y $$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)) (=> (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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))))) (forall ((V_a $$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_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_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.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 $$unsorted) (V_a $$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_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_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_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_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP 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(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_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_c V_f_2) (tptp.c_Finite__Set_Ofun__left__comm__idem T_d T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c T_c) T_d) V_f_2) V_g_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_c))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.class_Orderings_Oord T_c) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_f_2) V_g_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_g_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_g_2) V_f_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (not (tptp.hBOOL (tptp.hAPP 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(tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_x_2) V_y_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (not (= V_a V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) V_b)))))) (forall ((V_b $$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_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x) V_y)))))) (forall ((V_y $$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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_x_2) V_y_2))) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) V_b)) (=> (not (= V_a V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) V_b)))))) (forall ((V_a $$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_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_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.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.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_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_b) (= (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_c_2))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_b) (= (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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_c))) (=> (tptp.class_Groups_Oordered__ab__semigroup__add__imp__le T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (=> (= (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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (not (= V_A_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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 _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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_C_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_na_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_na_2)) (= V_m_2 V_na_2))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_m_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m_2))) (= V_na_2 V_m_2))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m))) (= V_m V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n)) (=> (not (= _let_1 V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_n))))))) (forall ((V_k $$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_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_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.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_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_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_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2)))))) (forall ((V_m $$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_m $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_j)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_j)))))) (forall ((V_k $$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_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_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_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_j)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i) V_k))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)) (not (= V_m_2 V_na_2))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_2)) (= V_m_2 V_na_2)))) (forall ((V_n $$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 $$unsorted)) (=> (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (= V_m V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)))) (forall ((V_n $$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)) (= (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 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless _let_1) V_A_2) V_B_2)))))))) (forall ((T_b $$unsorted)) (tptp.c_Finite__Set_Ofun__left__comm__idem T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Set_Oinsert T_b))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_b T_c)) (tptp.c_Sum__Type_OPlus T_b T_c V_A_2 V_B_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_b T_c)) (tptp.c_Sum__Type_OPlus T_b T_c V_A_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_b T_c)) (tptp.c_Sum__Type_OPlus T_b T_c V_A_2 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_sum T_b T_c)) (tptp.c_Sum__Type_OPlus T_b T_c V_A_2 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_c) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_d))))))))) (forall ((V_d $$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_na_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_na_2)) (exists ((B_k $$unsorted)) (= V_na_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) B_k)))))) (forall ((V_m $$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 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_m))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_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)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_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_na_2)) V_m_2)))) (forall ((V_na_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_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)))) (forall ((V_na_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_na_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_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_na_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_na_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2))) (= V_na_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)) (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_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_k_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i_2) V_k_2)) V_j_2))))) (forall ((V_c $$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_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_m_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_na_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_2 V_na_2) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))))) (forall ((V_na_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_na_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$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_b) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_A_2))) (not (tptp.c_Fun_Oinj__on T_c T_b V_f_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_b T_c V_A_2 V_f_2) V_x_2) (tptp.c_HOL_OThe T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_f_2)) V_x_2))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)))))) (forall ((V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (let ((_let_3 (tptp.c_Partial__Function_Oflat__lub T_b 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_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) _let_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Sum__Type_OPlus T_b T_c V_A_2 V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_sum T_b T_c) tptp.tc_HOL_Obool))) (and (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= V_B_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_sum T_b T_c)) (tptp.c_Sum__Type_OPlus T_b T_c V_A_2 V_B_2)))) (let ((_let_2 (and (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) 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_b) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2)))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_HOL_OThe T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2)) V_a_2)) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_HOL_OThe T_b (tptp.hAPP tptp.c_fequal V_x_2)) V_x_2)) (forall ((V_y_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (=> (tptp.c_Nitpick_Ofold__graph_H T_b T_c V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) V_y_2) (tptp.c_Nitpick_Ofold__graph_H T_b T_c V_f_2 V_z_2 V_A_2 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c 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_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_y_2))))))) (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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= V_n (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))) (forall ((V_n $$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 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (forall ((V_n $$unsorted)) (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)))) (= (tptp.c_Orderings_Obot__class_Obot tptp.tc_Nat_Onat) _let_12) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Ozero T_b) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Ozero__neq__one T_a) (not (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((T_a $$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 ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (tptp.c_Nitpick_Ofold__graph_H T_b T_c V_a1_2 V_a2_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_a2_2)) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (tptp.c_Nitpick_Ofold__graph_H T_b T_c 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 $$unsorted) (T_a $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) 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 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (= V_a_2 V_b_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_b) (= (= V_a_2 V_b_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (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_m $$unsorted)) (not (= (tptp.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (forall ((V_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.hAPP tptp.c_Nat_OSuc V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (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.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_m)))) (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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m) V_n)) (not (= V_n (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_n $$unsorted)) (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_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (not (= V_na_2 _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_na_2))))) (forall ((V_n $$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.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_na_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_na_2) _let_1) (and (= V_m_2 _let_1) (= V_na_2 _let_1))))) (forall ((V_m $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (forall ((V_n $$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_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_na_2) _let_1)) (= V_na_2 _let_1)))) (forall ((V_n $$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 $$unsorted) (V_m $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c))))))))) (forall ((V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_x_2)) (= V_x_2 V_z_2))) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c 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_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c 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_x_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (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 ((T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((T_a $$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 ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Nat_Onat_Onat__case T_b V_f1_2 V_f2_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_f1_2)) (forall ((T_a $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) _let_1))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_b) (=> (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_b) 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 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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (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_b) V_a_2) V_b_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_na_2 $$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_na_2)) (exists ((B_m $$unsorted)) (= V_na_2 (tptp.hAPP tptp.c_Nat_OSuc B_m))))) (forall ((V_na_2 $$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_na_2) (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= V_na_2 _let_1)))) (forall ((V_na_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_na_2))) (or (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (exists ((B_j $$unsorted)) (and (= V_m_2 (tptp.hAPP tptp.c_Nat_OSuc B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) V_na_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_na_2)) (or (and (= V_m_2 _let_2) (= V_na_2 _let_1)) (and (= V_m_2 _let_1) (= V_na_2 _let_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_na_2) _let_2) (or (and (= V_m_2 _let_2) (= V_na_2 _let_1)) (and (= V_m_2 _let_1) (= V_na_2 _let_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_na_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_m_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_na_2)))))) (forall ((T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_m_2 $$unsorted) (V_na_2 $$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_na_2) V_m_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_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_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_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_na_2 $$unsorted) (V_m_2 $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m_2) V_na_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_na_2)))) (= _let_19 _let_66) (forall ((V_y_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_y_2)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m_2))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.c_COMBI tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_na_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.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) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b)) _let_1)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_b))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_c)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_b) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_c)))))))) (forall ((V_b $$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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) _let_1) V_b)) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_a) V_b))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__idom 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_A_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) 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_b tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$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_b) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_n)) (= (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_n) (tptp.hAPP tptp.c_Nat_OSuc _let_1))) V_n)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_Pa_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_Pa_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((B_d $$unsorted)) (=> (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_b_2) B_d)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_d))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_Pa_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_Pa_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (exists ((B_d $$unsorted)) (and (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_b_2) B_d)) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_d))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))) (and (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))))) (forall ((V_n $$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_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_b V_f_2 V_a_2 V_A_2) V_x_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) V_x_2))))) (forall ((V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) 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_b) B_b) B_B)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_b) B_B))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) 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_b tptp.tc_HOL_Obool)))))))) (forall ((V_a5_2 $$unsorted) (V_a1_2 $$unsorted) (V_a2_2 $$unsorted) (V_a4_2 $$unsorted) (V_a3_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP V_a3_2 V_a1_2) V_a5_2))) (= (tptp.c_Nitpick_Ofold__graph_H T_b T_c V_a3_2 V_a4_2 V_a2_2 _let_1) (or (and (= V_a2_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= _let_1 V_a4_2)) (exists ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (and (= (tptp.hAPP (tptp.hAPP V_a3_2 V_a1_2) V_a5_2) (tptp.hAPP (tptp.hAPP V_a3_2 B_x) B_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_a2_2)) (tptp.c_Nitpick_Ofold__graph_H T_b T_c V_a3_2 V_a4_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_a2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) B_x) (tptp.c_Orderings_Obot__class_Obot _let_1))) B_y)))))))) (forall ((V_a $$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 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_b) (= (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_b_2) V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_na_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_na_2)) (exists ((B_k $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_k) V_na_2)) (forall ((B_i $$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_Pa_2 B_i))))) (tptp.hBOOL (tptp.hAPP V_Pa_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_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image tptp.tc_Nat_Onat T_b (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_COMBK tptp.tc_Nat_Onat T_b) (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_z_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_c V_f_2) V_g_2) V_z_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_z_2)) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_b V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_b V_f_2 V_A_2) V_x_2)) (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_c V_f_2) V_g_2) V_z_2) V_A_2))))) (forall ((V_d $$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) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_b_2)) (= V_a_2 V_b_2))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image tptp.tc_Nat_Onat T_b (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_COMBK tptp.tc_Nat_Onat T_b) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) _let_1) V_A_2))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.c_Int_Oring__1__class_OInts T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Oone__class_Oone T_b)) V_a_2)) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))))) (forall ((V_a3_2 $$unsorted) (V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a1_2) V_a2_2)) V_a3_2)) (exists ((B_a $$unsorted) (B_A $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (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_b T_b V_f_2 B_a B_A) V_a3_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_a) B_A)))))))) (forall ((V_x_2 $$unsorted) (V_X_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold1Set T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_X_2)) V_x_2)) (not (forall ((B_a $$unsorted) (B_A $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (=> (= (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_X_2) (tptp.hAPP (tptp.hAPP _let_1 B_a) B_A)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_b V_f_2 B_a B_A) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_a) B_A))))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Int_Oring__char__0 T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.c_Int_Oring__1__class_OInts T_b))) (not (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone T_b)) V_a_2)) V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_a_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oinsert T_b))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_4 (tptp.c_member T_b))) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_b _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_b T_b _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_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2))) (= (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) _let_1)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2) _let_1))))) (forall ((V_v $$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_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_b_2 $$unsorted) (V_a_2 $$unsorted) (V_fb_2 $$unsorted) (V_fa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.c_Nitpick_Opair__box_Opair__box__size T_b T_c V_fa_2 V_fb_2 (tptp.c_Nitpick_Opair__box_OPairBox T_b T_c V_a_2 V_b_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_fa_2 V_a_2)) (tptp.hAPP V_fb_2 V_b_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_w_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2)) (tptp.c_Int_Oring__1__class_OInts T_b))))) (forall ((V_ry $$unsorted) (V_rx $$unsorted) (V_ly $$unsorted) (V_lx $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ly)) (tptp.hAPP (tptp.hAPP _let_1 V_rx) V_ry)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_rx)) (tptp.hAPP (tptp.hAPP _let_1 V_ly) V_ry))))))) (forall ((V_ry $$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 _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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_lx))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ly)) V_rx) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ly) V_rx))))))) (forall ((V_z $$unsorted) (V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w)) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w))) V_z)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Int_Oring__char__0 T_b)) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_w)))))) (forall ((V_x_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (=> (tptp.class_Int_Onumber T_b) (= (= _let_1 V_x_2) (= V_x_2 _let_1))))) (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 T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_ry)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_ry)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_w $$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_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_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_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_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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_x $$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 $$unsorted)) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a) V_a))) (forall ((V_b_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Nitpick_Opair__box_OPairBox T_b T_c V_a_2 V_b_2) (tptp.c_Nitpick_Opair__box_OPairBox T_b T_c V_a_H_2 V_b_H_2)) (and (= V_a_2 V_a_H_2) (= V_b_2 V_b_H_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_b))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.class_Rings_Oring__1 T_b) (=> (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_b) V_a_2) V_b_2)) _let_1)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_x_2) V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (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_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_w $$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_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 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_v) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_a $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Omult__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) _let_1) _let_1)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Rings_Oring__no__zero__divisors T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) 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_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_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_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_c $$unsorted) (V_b $$unsorted) (V_e $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Osemiring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_e)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_e)) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_e)) V_c)))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_x))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (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_b))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_b) (= (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_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_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_z_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_w_2))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_b) (= (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_y_2)) (tptp.hAPP _let_2 V_z_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_z_2)) (tptp.hAPP _let_2 V_y_2))) (or (= V_w_2 V_x_2) (= V_y_2 V_z_2))))))))) (forall ((V_a $$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 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a) V_a))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_na_2 $$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_na_2) V_k_2)) (or (= V_m_2 V_na_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_na_2)) (or (= V_m_2 V_na_2) (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_na_2) _let_1) (or (= V_m_2 _let_1) (= V_na_2 _let_1))))) (forall ((V_m $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_na_2 $$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_na_2)) (= V_m_2 V_na_2)))) (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_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_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_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_na_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_na_2) _let_1) (and (= V_m_2 _let_1) (= V_na_2 _let_1))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_na_2)) (and (= V_m_2 _let_1) (= V_na_2 _let_1))))) (forall ((V_n $$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 ((T_b $$unsorted)) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_b) (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_b (tptp.c_Groups_Otimes__class_Otimes T_b)))) (forall ((V_w_2 $$unsorted) (V_v_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_v_2))) (let ((_let_3 (tptp.hAPP _let_1 V_w_2))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Orderings_Olinorder T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) _let_2) _let_3)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) _let_3) _let_2))))))))) (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_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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_b_2))) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__ring T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_a))))) (forall ((V_c $$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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)))))))) (forall ((V_a $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b)) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))) (forall ((V_d $$unsorted) (V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (V_r $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_r))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct T_a) (=> (not (= V_r (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (and (= V_a V_b) (not (= V_c V_d))) (not (= (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) (tptp.hAPP _let_1 V_d)))))))))) (forall ((V_n $$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 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_m) V_n)))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Rings_Oring T_b) (= (= (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_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Rings_Oring T_b) (= (= (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_b) V_a_2) V_b_2)) V_e_2)) V_c_2) V_d_2)))))) (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) (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 $$unsorted) (T_a $$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_na_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_na_2) _let_1) (and (= V_m_2 _let_1) (= V_na_2 _let_1))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j)))))))) (forall ((V_k $$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_na_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_2 V_na_2) V_k_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))) (forall ((V_na_2 $$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_na_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_na_2))))))) (forall ((V_na_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_na_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_m_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_na_2)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc V_k_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) (tptp.hAPP _let_1 V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_na_2 $$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_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2)))))) (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 ((T_b $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.c_Int_Oring__1__class_OInts T_b))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_b))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.class_Rings_Oring__1 T_b) (=> (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_b) V_a_2) V_b_2)) _let_1)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_b))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.class_Rings_Oring__1 T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) V_a_2) V_b_2)) _let_1)))))))) (forall ((V_b $$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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.c_Int_Oring__1__class_OInts T_b))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_y_2))) _let_1)) (and (= V_x_2 _let_1) (= V_y_2 _let_1))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_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 $$unsorted) (V_x $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Olinordered__ring__strict T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (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_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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Rings_Oordered__ring T_b) (= (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_b) V_a_2) V_b_2)) V_e_2)) V_c_2)) V_d_2)))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_e_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Rings_Oordered__ring T_b) (= (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_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Rings_Oordered__ring T_b) (= (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_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Rings_Oordered__ring T_b) (= (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_b) V_a_2) V_b_2)) V_e_2)) V_c_2)) V_d_2)))))))) (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_na_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_na_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_m_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_na_2)))))) (forall ((V_na_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_m_2) V_k_2)) (tptp.hAPP (tptp.hAPP _let_2 V_na_2) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))) (forall ((V_na_2 $$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_na_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_na_2))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_b _let_1 V_b_2 V_A_2) V_y_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_b _let_1 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_z_2) V_y_2)))))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$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.c_Finite__Set_Ofold__image T_c T_b _let_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) _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_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_z_2 $$unsorted) (V_g_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_c))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b _let_1) V_g_2) V_z_2))) (=> (tptp.class_Groups_Oab__semigroup__mult T_c) (=> (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_a_2) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) 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_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_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Int_Oring__char__0 T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.c_Int_Oring__1__class_OInts T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_a_2) _let_1) (= V_a_2 _let_1)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_j_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i_2) V_j_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j_2) V_i_2)) V_u_2)) V_na_2))))))))) (forall ((V_na_2 $$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_na_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_na_2)))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_j_2) V_i_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_i_2) V_j_2)) V_u_2)) V_m_2)) V_na_2)))))))) (forall ((V_n $$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_na_2 $$unsorted) (V_m_2 $$unsorted) (V_u_2 $$unsorted) (V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j_2) V_i_2)) (= (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_u_2)) V_m_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j_2) V_u_2)) V_na_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_i_2) V_j_2)) V_u_2)) V_m_2) V_na_2)))))) (forall ((V_na_2 $$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_na_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_na_2))))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_u $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_i) V_u)) V_m)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_u)) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_m) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_j) V_i)) V_u)) V_n)))))))) (forall ((V_na_2 $$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_na_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_na_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_m_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_m_2) V_na_2) _let_1) (and (= V_m_2 _let_1) (= V_na_2 _let_1)))))) (forall ((V_k $$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_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_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)) (forall ((V_z $$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_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_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_v $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_v)) (tptp.hAPP _let_2 V_w)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_v) V_w)))))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z) V_w) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))) (forall ((V_w $$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_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_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_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_z2 $$unsorted) (V_z1 $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z1) V_z2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z1)) (tptp.hAPP _let_1 V_z2)))))) (forall ((V_k $$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_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_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z))))))) (forall ((V_z3 $$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_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_i) V_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_i)) (tptp.hAPP _let_1 V_j))))))) (forall ((V_w $$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_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 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_w) V_w))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k_2) V_l_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (= V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_w_2 $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_w_H) V_w)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z_H) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_w_H) V_z_H)) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_z)))))))) (forall ((V_k $$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_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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (not (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (tptp.hBOOL (tptp.hAPP _let_53 _let_45)) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_z_2)) (= V_w_2 V_z_2))))) (forall ((V_z_2 $$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_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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_z)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_z)))))) (forall ((V_z_2 $$unsorted) (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 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_w_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) V_z_2)))) (forall ((V_z_2 $$unsorted) (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)) (= (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_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_na_2)) (or (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_m_2 V_na_2))))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_u $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_i) V_u)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_j) V_u)) V_k)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_i) V_j)) V_u)) V_k))))) (forall ((V_na_2 $$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_na_2)) (= V_m_2 V_na_2))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m_2)) (tptp.hAPP _let_2 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))) (forall ((V_q $$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 $$unsorted) (V_q $$unsorted) (V_r_H $$unsorted) (V_q_H $$unsorted) (V_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_q_H)) V_r_H)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_q)) V_r))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_r_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r_H) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_q_H) V_q))))))))))) (forall ((V_r_H $$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_H $$unsorted) (V_q_H $$unsorted) (V_b_H $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b_H) V_q_H)) V_r_H))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r_H) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_b_H)) (tptp.hBOOL (tptp.hAPP _let_2 V_q_H))))))))) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_k) V_k)) (not (= _let_44 _let_45)) (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))))))) (tptp.hBOOL (tptp.hAPP _let_69 _let_45)) (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_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_Nitpick_Opair__box T_a T_b)) (tptp.c_Nitpick_Opair__box_OPairBox T_a T_b V_a V_b)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.c_Nitpick_Opair__box_Opair__box__case T_c T_d T_b V_f1_2 (tptp.c_Nitpick_Opair__box_OPairBox T_c T_d 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_f1_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.c_Nitpick_Opair__box_Opair__box__rec T_c T_d T_b V_f1_2 (tptp.c_Nitpick_Opair__box_OPairBox T_c T_d V_a_2 V_b_2)) (tptp.hAPP (tptp.hAPP V_f1_2 V_a_2) V_b_2))) (tptp.hBOOL (tptp.hAPP _let_69 _let_44)) (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_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (=> (tptp.class_Nat_Osemiring__char__0 T_b) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_na_2)) (= V_m_2 V_na_2))))) (forall ((V_A_2 $$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_44 _let_76) (forall ((V_Pa_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_Pa_2 B_x)))) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_x)))))) (forall ((V_Pa_2 $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_x)))))) (forall ((V_z $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_z)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x_2) V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_x_2) V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_x) V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_x) V_y))))) (= _let_45 _let_75) (= _let_76 _let_44) (forall ((V_na_2 $$unsorted)) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_na_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (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)) (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_na_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_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_2))))) (forall ((V_na_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_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_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_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_z $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n))) V_z))))) (forall ((V_z_2 $$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_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 ((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))))) (forall ((T_a $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (=> (tptp.class_Rings_Olinordered__semidom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_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)))))) (= _let_75 _let_45) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (=> (tptp.class_Rings_Olinordered__semidom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP _let_1 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_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_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))) (forall ((T_a $$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_na_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b) V_na_2)) (tptp.c_Int_Oring__1__class_OInts T_b))))) (forall ((V_na_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_na_2)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hAPP _let_14 _let_66) _let_45) (forall ((V_z_2 $$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 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) (tptp.hAPP _let_1 V_m))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)))))) (forall ((V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 V_m)))))) (forall ((V_m $$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_na_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_na_2)))) (forall ((V_k $$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_na_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_b) (= (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.c_Nat_Osemiring__1__class_Oof__nat T_b) V_na_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_na_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_x_2) V_y_2)))) (and (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_y_2) V_x_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))))))) (forall ((V_na_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b) V_na_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b T_b) (tptp.c_Groups_Oplus__class_Oplus T_b)) (tptp.c_Groups_Oone__class_Oone T_b)) V_na_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_x $$unsorted) (V_y $$unsorted)) (let ((_let_1 (tptp.c_Nat__Transfer_Otsub V_x V_y))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_y) V_x)))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_x) V_y))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))))) (forall ((V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v)))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))))) (=> (tptp.class_Int_Onumber__ring T_a) (and (=> _let_2 (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_v)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_b _let_1))) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (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 _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_2 V_A_2)))))))))) (forall ((V_k $$unsorted) (V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v_H)) V_k)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_4 (= _let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_v_H))) V_k))))))))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_v_H)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_v) V_v_H))))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_c))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_c)))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b (tptp.c_Groups_Otimes__class_Otimes T_c)) V_f_2) _let_1) V_S_2) _let_1)))))) (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)))) (forall ((V_n $$unsorted)) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n))))) (forall ((V_A_2 $$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_na_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_na_2)) (= V_m_2 V_na_2)))) (= _let_58 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_14) _let_22)) (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)))))) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_60))) (= tptp.c_Int_OPls _let_44) (tptp.hBOOL (tptp.hAPP _let_64 tptp.c_Int_OPls)) (not (tptp.hBOOL (tptp.hAPP _let_62 tptp.c_Int_OPls))) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k) tptp.c_Int_OPls) V_k)) (forall ((V_k $$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_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_Ominus__class_Ominus tptp.tc_Int_Oint) V_k) tptp.c_Int_OPls) V_k)) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_44))) (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_45))) (forall ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Ozero__class_Ozero T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OPls)))) (forall ((T_a $$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 ((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))) (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))) (tptp.hBOOL (tptp.hAPP _let_69 _let_60)) (not (tptp.hBOOL (tptp.hAPP _let_62 _let_44))) (= _let_74 _let_12) (= _let_12 _let_74) (= _let_44 _let_60) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_a_2)) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_g_2 $$unsorted)) (=> (= V_g_2 (tptp.c_Finite__Set_Ofold1 T_b V_f_2)) (= (tptp.hAPP V_g_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_a_2))) (forall ((V_x_2 $$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 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg V_Z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_Z_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Nat__Transfer_Otsub (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_x) V_y))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b V_f_2) V_A_2))))) (forall ((V_i_2 $$unsorted) (V_inc_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b V_inc_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_i_2) V_i_2))) (forall ((V_i_2 $$unsorted) (V_na_2 $$unsorted) (V_inc_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b V_inc_2 (tptp.hAPP tptp.c_Nat_OSuc V_na_2) V_i_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat__aux T_b V_inc_2 V_na_2 (tptp.hAPP V_inc_2 V_i_2))))) (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)))) (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_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))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b V_f_2) V_A_2) (tptp.c_HOL_OThe T_b (tptp.c_Finite__Set_Ofold1Set T_b V_f_2 V_A_2)))) (forall ((V_v $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))) (let ((_let_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_1)))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_x_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_x_2) tptp.c_Int_OPls))))) (forall ((V_y_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) tptp.c_Int_OPls) V_y_2))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_x_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_x_2) tptp.c_Int_OPls))))) (forall ((V_y_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OPls) V_y_2))))) (forall ((V_v_2 $$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)))) (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))))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_b _let_1))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_b) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_2 V_A_2))))))))) (forall ((V_v_H_2 $$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_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_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 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_v)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_v))))))))) (forall ((V_v $$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 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_v)) _let_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v) tptp.c_Int_OPls)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_v)))))))))) (forall ((V_v $$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_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) 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_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Groups_Otimes__class_Otimes T_b)) V_A_2)) V_A_2))))))) (forall ((V_w_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_b) (=> (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_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2)) (tptp.c_Nat_Osemiring__1__class_ONats T_b)))))) (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)))))))) (forall ((V_x $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_Opred V_x)) V_x)) (forall ((V_l_2 $$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)) (= (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)))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc V_k) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (forall ((T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.c_Nat_Osemiring__1__class_ONats T_b))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_ONats T_b))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.class_Rings_Osemiring__1 T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_b_2)) _let_1)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_ONats T_b))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.class_Rings_Osemiring__1 T_b) (=> (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_b) V_a_2) V_b_2)) _let_1)))))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.c_Nat_Osemiring__1__class_ONats T_b))))) (forall ((V_na_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b) V_na_2)) (tptp.c_Nat_Osemiring__1__class_ONats T_b))))) (forall ((V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.c_Int_Osucc V_k)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Groups_Oone__class_Oone T_a)) (tptp.hAPP _let_1 V_k)))))) (forall ((V_v_2 $$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_na_2 $$unsorted) (V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (= (tptp.c_Nat_Onat_Onat__case T_b V_a_2 V_f_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) tptp.tc_Int_Oint) (tptp.c_If T_b)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Nat_Onat_Onat__case T_b V_a_2 V_f_2 V_na_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Int_Oint) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_na_2)))))))) (forall ((V_v_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (let ((_let_3 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fconj) _let_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fimplies) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_na_2)) tptp.c_Int_Onat))))))))))) (forall ((V_na_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_na_2))))))) (forall ((V_v_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (let ((_let_3 (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fconj) _let_2)) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 tptp.c_fimplies) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_na_2)) tptp.c_Int_Onat))))))))))) (forall ((V_na_2 $$unsorted) (V_v_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint) tptp.c_fNot) tptp.c_Nat__Numeral_Oneg))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_na_2))))))) (forall ((V_k_2 $$unsorted) (V_Pa_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_d_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_k_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_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_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w_2))))) (and (=> _let_3 (= _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_w_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat)) (tptp.c_COMBI tptp.tc_Nat_Onat))))))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Groups_Otimes__class_Otimes T_b)))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_b) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (= (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_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b 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_b T_b V_h_2) V_N_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_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_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_n $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n)) V_n)) (forall ((V_k $$unsorted)) (not (= (tptp.c_Int_OBit1 V_k) tptp.c_Int_OPls))) (forall ((V_l $$unsorted)) (not (= tptp.c_Int_OPls (tptp.c_Int_OBit1 V_l)))) (= _let_70 (tptp.hAPP tptp.c_Int_Onat _let_57)) (= _let_73 _let_12) (= _let_12 _let_73) (forall ((V_z_H_2 $$unsorted) (V_z_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_z_H_2)) (= (= (tptp.hAPP tptp.c_Int_Onat V_z_2) (tptp.hAPP tptp.c_Int_Onat V_z_H_2)) (= V_z_2 V_z_H_2)))))) (forall ((V_Pa_2 $$unsorted)) (= (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (exists ((B_x $$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_Pa_2 (tptp.hAPP tptp.c_Int_Onat B_x))))))) (forall ((V_Pa_2 $$unsorted)) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP tptp.c_Int_Onat B_x))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (= (= (tptp.hAPP tptp.c_Int_Onat V_x_2) (tptp.hAPP tptp.c_Int_Onat V_y_2)) (= V_x_2 V_y_2)))))) (forall ((V_w_2 $$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)))))) (forall ((V_v $$unsorted)) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v)))) (forall ((V_w $$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_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))))) (= _let_19 _let_72) (forall ((V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.c_Int_OBit1 V_k) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_k)) V_k)))) (forall ((V_z $$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 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg V_z))) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)) V_z))) (forall ((V_w_2 $$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_A_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_A_2)) V_A_2)) (= (tptp.c_Int_Osucc tptp.c_Int_OPls) _let_29) (= _let_22 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) _let_58)) (forall ((V_i_2 $$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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_w_2)) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_w_2) V_z_2)))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Groups_Oone__class_Oone T_a)) _let_2)) _let_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_y_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x_2)) (tptp.hAPP tptp.c_Int_Onat V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))))))) (= _let_72 _let_66) (forall ((V_a $$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 $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls))) V_a))) (forall ((V_z $$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.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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_Onat V_z)) V_z))) (forall ((V_z_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) (tptp.hAPP tptp.c_Int_Onat V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2)) V_z_2)))) (forall ((T_a $$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 ((T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.c_Groups_Oone__class_Oone T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (forall ((V_k_2 $$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 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Int_OBit1 V_k_2)) tptp.c_Int_OPls)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) tptp.c_Int_OPls)))) (= _let_45 (tptp.hAPP _let_54 _let_29)) (= _let_71 _let_19) (= _let_19 _let_71) (forall ((V_z_2 $$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_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_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_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_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_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 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP tptp.c_Int_Onat V_i_2))) (and (forall ((B_n $$unsorted)) (=> (= V_i_2 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) B_n)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_n)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_z_H $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_v_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v_2))) (= (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2) _let_1) (and (= V_m_2 (tptp.hAPP tptp.c_Int_Onat _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_1)))))) (forall ((V_z_H $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP tptp.c_Int_Onat V_z_H))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x)) (tptp.hAPP tptp.c_Int_Onat V_y)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_x) V_y))))))) (forall ((V_z $$unsorted) (V_z_H $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z_H) V_z)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP tptp.c_Int_Onat V_z_H))))))) (= _let_70 (tptp.hAPP tptp.c_Nat_OSuc _let_67)) (= _let_71 _let_66) (tptp.hBOOL (tptp.hAPP _let_69 _let_57)) (forall ((V_n $$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))) (= _let_57 (tptp.hAPP _let_14 _let_70)) (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_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)))))))))) (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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_w_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_w_2)) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_m_2)))))) (forall ((V_y_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) V_y_2))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_x_2)) (tptp.c_Groups_Oone__class_Oone T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_x_2) (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (forall ((V_z $$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_x_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_x_2)) (tptp.c_Groups_Oone__class_Oone T_b))) (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_y_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Rings_Olinordered__idom T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) 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_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_v_2 $$unsorted) (V_na_2 $$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_na_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_na_2)))))))))) (forall ((V_na_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_na_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_na_2)))))))))) (forall ((V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_b))) (= (tptp.c_Nat_Onat_Onat__case T_b 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_b) (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_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_1) tptp.tc_Int_Oint) (tptp.c_If T_b)) tptp.c_Nat__Numeral_Oneg)) V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Int_Oint) V_f_2) tptp.c_Int_Onat)))))) (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))) (forall ((V_na_2 $$unsorted) (V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat)) V_na_2))) (let ((_let_3 (tptp.c_Nat_Onat_Onat__rec T_b V_a_2 V_f_2))) (let ((_let_4 (tptp.tc_fun T_b T_b))) (let ((_let_5 (tptp.c_COMBS tptp.tc_Int_Oint T_b T_b))) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_b _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_4) tptp.tc_Int_Oint) (tptp.c_If T_b)) tptp.c_Nat__Numeral_Oneg)) (tptp.hAPP _let_3 V_na_2))) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_4 tptp.tc_Int_Oint) V_f_2) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Int_Oint) _let_3) _let_2))))))))))) (forall ((V_v_2 $$unsorted) (V_f_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Onat_Onat__rec T_b V_a_2 V_f_2))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.c_COMBS tptp.tc_Int_Oint T_b T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) tptp.tc_Int_Oint) (tptp.c_If T_b)) tptp.c_Nat__Numeral_Oneg)) V_a_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_2 tptp.tc_Int_Oint) V_f_2) tptp.c_Int_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Int_Oint) _let_1) tptp.c_Int_Onat))))))))) (tptp.c_Nat__Transfer_Otransfer__morphism tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat _let_69) (forall ((V_v_2 $$unsorted) (V_v_H_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP _let_2 V_v_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_3) (tptp.hAPP _let_2 V_v_H_2)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_v_H_2) tptp.c_Int_OPls)))) (and (=> _let_5 (= _let_4 _let_3)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_v_H_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_1) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) tptp.c_Int_Onat))))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat V_x)))) (=> (tptp.class_Rings_Osemiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x)) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_x))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_2) _let_2))))))) (forall ((V_v_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$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_b) V_v_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) _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_b))) (=> (tptp.class_Int_Onumber__ring T_b) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_b T_b) (tptp.hAPP _let_3 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_6) (tptp.hAPP _let_6 _let_2))) (tptp.c_COMBI T_b))))) (=> (not _let_5) (= _let_4 (tptp.c_Groups_Oone__class_Oone T_b)))))))))))) (forall ((V_na_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_na_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_na_2)))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (tptp.c_Fun_Oinj__on T_b T_b (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_A_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_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_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_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ouminus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_c T_b)) V_A_2) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.hAPP V_A_2 V_x_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_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_Omin T_a) V_a) V_a) V_a))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ouminus T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_c T_b)) V_A_2) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_y $$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_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= V_a_2 (tptp.hAPP _let_1 V_b_2)) (= V_b_2 (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_c)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_1 V_b_2)) (= V_a_2 V_b_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Lattices_Oboolean__algebra T_b) (= (= (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_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_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_z)) V_z))) (forall ((V_k $$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_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))) (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 $$unsorted) (V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP _let_1 V_w))) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w))) V_z)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (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_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a_2)) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b_2)) V_a_2))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Lattices_Oboolean__algebra T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_x_2)) (tptp.hAPP _let_2 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2))))))) (forall ((V_b $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_x)))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) V_y_2)) V_z_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_y $$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 $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y)) V_y)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) V_y_2) V_x_2)))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) 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_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_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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_y) V_x)))) (forall ((V_x $$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_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_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_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b) V_d)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_c) V_d))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b))) (not (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (not (tptp.hBOOL (tptp.hAPP _let_1 V_b))))))))) (forall ((T_a $$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_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) (= _let_1 V_a_2))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2) V_a_2) (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) (tptp.hAPP _let_1 V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) (tptp.hAPP _let_1 V_a_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (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_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b_2)) (tptp.hAPP _let_2 V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b)) V_x)))))) (forall ((V_a $$unsorted) (V_x $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_a) V_b)) V_x)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_z_2))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) V_y_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_x_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_y_2))))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) V_y_2)) V_z_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_b)))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (let ((_let_2 (tptp.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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Oring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (= (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_b) V_b_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_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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Omin T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_n $$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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_m)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))) (forall ((V_w_2 $$unsorted) (V_z_2 $$unsorted) (T_b $$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_b) V_z_2))) (=> (tptp.class_Int_Onumber__ring T_b) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_b T_b) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.c_Groups_Otimes__class_Otimes T_b)) (tptp.c_COMBI T_b)))))))) (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))))) (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))))) (= (tptp.hAPP _let_39 _let_44) _let_44) (= (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_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 $$unsorted)) (= (tptp.c_Int_OBit0 V_k) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_k) V_k))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p)) V_q) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_p) V_q))))))) (forall ((V_n $$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_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_y) V_z)))))) (forall ((V_w_2 $$unsorted) (V_z_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_z_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint)) (tptp.c_COMBI tptp.tc_Int_Oint))))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_k)) V_l) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (= (tptp.hAPP _let_39 tptp.c_Int_OPls) tptp.c_Int_OPls) (forall ((V_w $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z) V_w)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z)) (tptp.hAPP _let_1 V_w)))))) (forall ((V_w $$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_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_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_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)))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) 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_b) _let_1))))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_b (tptp.c_Orderings_Oord__class_Omin T_b)))) (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_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 $$unsorted) (T_a $$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_w_2 $$unsorted) (V_v_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_v_2)))) (=> (tptp.class_Int_Onumber__ring T_b) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_b T_b) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.c_Groups_Otimes__class_Otimes T_b)) (tptp.c_COMBI T_b)))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n))) _let_1)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_a_2) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$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)))) (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)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Osemiring__1 T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) _let_1) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_1)))) (forall ((V_n $$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))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_a_2))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a_2))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b)) V_a_2)))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b)) V_a_2)))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_a_2))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (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_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_b) (= (= V_a_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_b_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_a_2) V_b_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ogroup__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_a)) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_b $$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_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_v $$unsorted) (V_u $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (let ((_let_3 (tptp.hAPP _let_1 V_u))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) _let_3) _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) _let_3) _let_2)))) (=> (and (tptp.class_Int_Onumber T_b) (tptp.class_Orderings_Oord T_b)) (and (=> _let_5 (= _let_4 _let_3)) (=> (not _let_5) (= _let_4 _let_2)))))))))) (forall ((V_y $$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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Rings_Oring__1__no__zero__divisors T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_x_2) V_x_2) _let_1) (or (= V_x_2 _let_1) (= V_x_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) _let_1))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Rings_Ocomm__ring__1 T_a) (= (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone T_a))) V_x))))) (forall ((V_q $$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_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Int_OBit0 V_w_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_w_2) _let_1)))))) (forall ((V_n $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x) V_n)))))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Onat_Onat__rec T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Onat_Onat__rec T_b 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 ((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_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_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_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_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_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 $$unsorted) (V_k $$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 $$unsorted) (V_k $$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_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_x))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))) (forall ((V_z $$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 $$unsorted) (V_na_2 $$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_na_2)) (tptp.hAPP _let_2 V_m_2)) (and (= V_na_2 _let_1) (= V_m_2 _let_1)))))) (forall ((V_l $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) tptp.c_Int_OPls))) (= (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP _let_1 V_l))))) (forall ((V_l $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 V_l)) (tptp.c_Int_OBit1 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))))) (forall ((V_m $$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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_w) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))) (forall ((V_w $$unsorted) (V_z $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_z) V_w))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_b))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.class_Rings_Oring__1 T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) _let_1))))))) (forall ((V_na_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_OInts T_b))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.class_Rings_Oring__1 T_b) (=> (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_b) V_a_2) V_na_2)) _let_1))))))) (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_OBit0 V_k)) (tptp.c_Int_OBit1 (tptp.c_Int_Opred V_k)))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Opred (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 V_k))) (forall ((V_k $$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)))))) (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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_3 V_x) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y) _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_a)) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y))))))))) (forall ((V_y $$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_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_3 (tptp.c_Power_Opower__class_Opower T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2)) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))))) (forall ((V_w $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w))) (=> (and (tptp.class_Groups_Omonoid__mult T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) _let_1) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) _let_1) _let_1))))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit0 (tptp.c_Int_Osucc V_k)))) (forall ((V_k $$unsorted)) (= (tptp.c_Int_Osucc (tptp.c_Int_OBit0 V_k)) (tptp.c_Int_OBit1 V_k))) (forall ((V_n $$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_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) (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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_v))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v)) V_n))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_1)))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n))) (=> (not _let_4) (= _let_3 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) _let_1) V_n)))))))))) (forall ((V_y $$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 $$unsorted) (V_x $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (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_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1))) (tptp.c_Groups_Ozero__class_Ozero T_a)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (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_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (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 $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) _let_1))) (let ((_let_4 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_5 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_5 V_x) V_y)) _let_3) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x) _let_3)) (tptp.hAPP (tptp.hAPP _let_4 V_y) _let_3))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) _let_1)) V_x)) V_y)))))))))) (forall ((V_n $$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 $$unsorted) (V_a $$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) (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_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)))) (forall ((V_w $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w))))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) V_l) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) V_l))) V_l)))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP _let_2 V_na_2)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP _let_2 V_m_2)))) (and (= V_na_2 _let_1) (= V_m_2 _let_1)))))) (forall ((V_n $$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 $$unsorted)) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_1 V_v)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))))) (forall ((V_n $$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_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_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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n)))) _let_1))))))) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k)) (tptp.c_Int_OBit1 V_l)) (tptp.c_Int_OBit0 (tptp.hAPP (tptp.hAPP _let_1 V_k) (tptp.c_Int_Osucc V_l)))))) (forall ((V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_2) _let_2)) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w))))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) V_m_2) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP _let_1 V_na_2)) V_m_2)))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.c_Nat_Onat_Onat__case tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_na_2)) V_m_2)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Groups_Omonoid__mult T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_a)) V_a))))) (forall ((V_w $$unsorted) (V_c $$unsorted) (V_v $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_v)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_c) (tptp.hAPP _let_1 V_w))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_v) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_w)))) V_c)))))) (forall ((V_n $$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))))))))) (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_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_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_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 ((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 ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) _let_1) _let_1) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))) (= _let_67 _let_32) (= _let_32 _let_67) (tptp.hBOOL (tptp.hAPP _let_69 _let_55)) (forall ((V_n $$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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_n)))) (forall ((V_z $$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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_z) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_z) V_z))) (= _let_55 (tptp.hAPP _let_14 _let_32)) (= _let_32 _let_68) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) _let_19) _let_19) _let_32) (forall ((V_z_H $$unsorted) (V_z $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_z) V_z_H)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_z))) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_z_H))))))) (forall ((V_n $$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_68 _let_67) (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_z_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_z_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2))))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w_2))))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Int_Onumber__ring T_b) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet T_b T_b) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_b) _let_5) (tptp.hAPP _let_5 V_z_2))) (tptp.c_COMBI T_b))))) (=> (not _let_4) (= _let_3 (tptp.c_Groups_Oone__class_Oone T_b))))))))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) V_z_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_w_2))))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w_2))))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_w_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint) _let_5) (tptp.hAPP _let_5 V_z_2))) (tptp.c_COMBI tptp.tc_Int_Oint))))) (=> (not _let_4) (= _let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))))))) (forall ((V_v_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Int_Oint))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_v_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_Opred V_v_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Nat_Onat _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_Nat_Onat _let_3) tptp.tc_Int_Oint) (tptp.c_If tptp.tc_Nat_Onat)) tptp.c_Nat__Numeral_Oneg)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.hAPP _let_2 tptp.c_Nat_OSuc) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_na_2)) tptp.c_Int_Onat))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omin T_b)))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP _let_1 V_B_2)) (tptp.hAPP _let_1 V_A_2)))))))))) (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_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_x_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_A_2))) (forall ((B_x $$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 T_b) V_x_2) B_x))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_A_2)) V_x_2)) (exists ((B_x $$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_b) B_x) V_x_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_A_2))) (forall ((B_x $$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_b) V_x_2) B_x))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_A_2)) V_x_2)) (exists ((B_x $$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__eq T_b) B_x) V_x_2))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_A_2)) V_a_2)))))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) 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_b tptp.tc_HOL_Obool)) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b 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_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (let ((_let_3 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) V_x) V_x))) (forall ((V_n $$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_n $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) _let_1) V_n) _let_1))) (forall ((V_m_2 $$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) (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_na_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_na_2))) (or (tptp.hBOOL (tptp.hAPP _let_2 V_x_2)) (= V_na_2 _let_1)))))) (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) (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) (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_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_na_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (and (tptp.class_Power_Opower T_b) (tptp.class_Rings_Omult__zero T_b) (tptp.class_Rings_Ono__zero__divisors T_b) (tptp.class_Rings_Ozero__neq__one T_b)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_a_2) V_na_2) _let_1) (and (= V_a_2 _let_1) (not (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))) (forall ((V_n $$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_na_2 $$unsorted) (V_m_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_a_2))) (=> (tptp.class_Rings_Olinordered__semidom T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Oone__class_Oone T_b)) V_a_2)) (= (= (tptp.hAPP _let_1 V_m_2) (tptp.hAPP _let_1 V_na_2)) (= V_m_2 V_na_2)))))) (forall ((V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Power_Opower T_a) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP _let_1 V_n)))))) (forall ((V_n $$unsorted) (V_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 $$unsorted) (V_i $$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_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_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_b_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Rings_Olinordered__semidom T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_b)) V_b_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_x_2) V_y_2)))))))) (forall ((V_n $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_n) V_N)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Oone__class_Oone T_a)) V_a)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) (tptp.hAPP _let_1 V_N))))))))) (forall ((V_a $$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_y_2 $$unsorted) (V_x_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_b_2))) (=> (tptp.class_Rings_Olinordered__semidom T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Oone__class_Oone T_b)) V_b_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (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 $$unsorted) (V_m $$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_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (=> (tptp.class_Rings_Olinordered__semidom T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a) V_n)))))))) (forall ((V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Big__Operators_Osemilattice__big T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b V_f_2) V_A_2))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (tptp.class_Int_Onumber__ring T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.c_Int_Oiszero T_b (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))))) (forall ((V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (tptp.class_Int_Onumber__ring T_b) (= (= (tptp.c_Groups_Oone__class_Oone T_b) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Int_Oiszero T_b (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.c_Int_OBit1 tptp.c_Int_OPls)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_y_2)))))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_3 (tptp.hAPP _let_2 V_w))) (let ((_let_4 (tptp.hAPP _let_2 V_v))) (let ((_let_5 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_6 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_2 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (let ((_let_7 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_2 (tptp.c_Int_OBit1 V_v)) (tptp.hAPP _let_2 (tptp.c_Int_OBit0 V_w))))) (let ((_let_8 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) _let_3)))) (and (=> _let_8 (= _let_7 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_6 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_4 _let_3))) _let_1))) (=> (not _let_8) (= _let_7 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP _let_6 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_5 _let_4) _let_1) _let_3))) _let_1))))))))))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Orderings_Oord_Omax T_b (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (tptp.c_Orderings_Oord__class_Omin T_b)))) (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_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_A_2)) V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (= (tptp.hAPP _let_1 V_A_2) (tptp.hAPP _let_1 V_B_2)) (= V_A_2 V_B_2)))) (forall ((V_b $$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)) (=> (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_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)) (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_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_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 $$unsorted) (V_c $$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 $$unsorted) (V_c $$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 $$unsorted) (V_a $$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_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_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.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.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_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.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_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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_b))))) (forall ((V_a_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (= (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b) (tptp.c_Divides_Odiv__class_Omod T_a V_a_H V_b)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a) V_b) (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP _let_1 V_a_H) V_b)))))) (forall ((V_a $$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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_1 V_b) _let_1))) (forall ((V_c $$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_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_b) V_c) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c)) V_c)))) (forall ((V_m $$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 $$unsorted) (V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_y) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) V_m)))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))))) (forall ((V_m $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_x V_m)) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x) V_m)))) (forall ((V_y $$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 ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a) V_b) (tptp.c_Groups_Ozero__class_Ozero T_a)))) (forall ((V_b $$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_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_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_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_z_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.c_Int_Oiszero T_b V_z_2) (= V_z_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))) (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_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_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_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_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_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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_k) V_l) _let_1)) (not (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l) _let_1))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_a) V_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_4 (= _let_1 _let_3))) (and (=> _let_4 (= _let_2 _let_3)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_b) _let_1))))))))) (forall ((T_a $$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 ((V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (and (tptp.class_Int_Onumber__ring T_b) (tptp.class_Int_Oring__char__0 T_b)) (= (tptp.c_Int_Oiszero T_b (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w_2))) (tptp.c_Int_Oiszero T_b (tptp.hAPP _let_1 V_w_2)))))) (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_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 $$unsorted) (V_na_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_na_2 V_k_2))) (and (=> (= V_k_2 _let_1) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_na_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_k_2)) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_j) V_k_2)) (= V_na_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_j))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_k_2) _let_1)) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_k_2) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) B_j) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_na_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_j)))))))))) (forall ((V_w $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_v)) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 V_w))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_v) (tptp.hAPP _let_1 V_w)))))) (forall ((T_a $$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_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (tptp.class_Int_Onumber__ring T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Int_Oiszero T_b (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_y_2)))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (tptp.class_Int_Onumber__ring T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.c_Int_Oiszero T_b (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_x_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) tptp.c_Int_OPls)))))))) (forall ((V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (=> (tptp.class_Int_Onumber__ring T_b) (= (= (tptp.c_Groups_Ozero__class_Ozero T_b) (tptp.hAPP _let_1 V_y_2)) (tptp.c_Int_Oiszero T_b (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) tptp.c_Int_OPls) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_y_2)))))))) (forall ((V_k $$unsorted) (V_l $$unsorted)) (let ((_let_1 (tptp.c_SMT_Oz3mod V_k V_l))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_l)))) (and (=> _let_2 (= _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l))) (=> (not _let_2) (= _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_l)))))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$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_c) V_A_2)))) (=> (tptp.c_Big__Operators_Ocomm__monoid__big T_b T_c 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_b T_c V_f_2) V_g_2) V_z_2) V_A_2))) (=> (not _let_2) (= _let_1 V_z_2))))))) (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 ((T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.c_Orderings_Oord__class_Omin T_b) (tptp.c_Orderings_Oord_Omin T_b (tptp.c_Orderings_Oord__class_Oless__eq T_b))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Power_Opower T_b) (= (tptp.c_Power_Opower__class_Opower T_b) (tptp.c_Power_Opower_Opower T_b (tptp.c_Groups_Oone__class_Oone T_b) (tptp.c_Groups_Otimes__class_Otimes T_b))))) (forall ((V_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_1) V_b) _let_1))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m) V_n) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_n))) (forall ((V_n $$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 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_m))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)) V_k) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_n) V_k))))) (forall ((V_b $$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) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 V_a) (tptp.hAPP _let_1 V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_b))))) (forall ((V_a $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) _let_1) V_a) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_m $$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_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_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_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_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 $$unsorted)) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_a) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (forall ((V_d_2 $$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_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_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_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (and (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) _let_1) _let_1) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a _let_1) V_a)))) (forall ((V_m $$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_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_c)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c))) V_c)))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_x) (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_x V_y))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_a V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))) (forall ((V_b $$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 $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_b $$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_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_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_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_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_b $$unsorted) (V_b_H $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_3) V_b_H)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_H) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) (tptp.hAPP _let_1 V_b_H)))))))))) (forall ((V_b $$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_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) _let_1))))) (forall ((V_a_2 $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) V_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) _let_1))))))) (forall ((V_b $$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))) (=> (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 $$unsorted) (V_a_H $$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_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_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_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 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_c)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c)))))) (forall ((V_m $$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_k $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b))) V_k) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_k)))) (forall ((V_k $$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_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_a))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) V_c) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_b V_c))) V_c)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_m) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_m) V_n)) V_n)))) (forall ((V_a_2 $$unsorted) (V_times_2 $$unsorted) (V_one_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower_Opower T_b V_one_2 V_times_2) V_a_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_one_2)) (forall ((V_na_2 $$unsorted) (V_a_2 $$unsorted) (V_times_2 $$unsorted) (V_one_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower_Opower T_b V_one_2 V_times_2) V_a_2))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.hAPP (tptp.hAPP V_times_2 V_a_2) (tptp.hAPP _let_1 V_na_2))))) (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 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) V_b) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.c_Groups_Oone__class_Oone T_a)))))))) (forall ((V_a $$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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b)) V_b)) V_a))) (forall ((V_a $$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 $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b)) V_a))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b))) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_a) V_b)) V_b)) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b))) V_c) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c))))) (forall ((V_k_2 $$unsorted) (V_na_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_na_2 V_k_2))) (and (=> _let_1 (tptp.hBOOL (tptp.hAPP V_Pa_2 V_na_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_na_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_Pa_2 B_j)))))))))) (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_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.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_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_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Big__Operators_Ocomm__monoid__big T_b T_c V_f_2 V_z_2 V_F_2) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2))) (= (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) V_A_2) V_z_2)))) (forall ((V_k_2 $$unsorted) (V_na_2 $$unsorted) (V_Pa_2 $$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_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_na_2) V_k_2))) (and (=> (= V_k_2 _let_1) (tptp.hBOOL (tptp.hAPP V_Pa_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_na_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_Pa_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_na_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_Pa_2 B_i)))))))))) (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_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_b $$unsorted) (V_a $$unsorted) (V_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_c)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_m $$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_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_na_2 $$unsorted) (V_Pa_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_k_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_na_2) V_k_2)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_na_2 V_k_2))) (forall ((B_i $$unsorted) (B_j $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_j)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_j) V_k_2)) (= V_na_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_k_2) B_i)) B_j))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_i) B_j))))))) (forall ((V_na_2 $$unsorted) (V_Pa_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_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_na_2) V_k_2)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_na_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_na_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_Pa_2 B_i) B_j))))))) (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 $$unsorted) (V_w $$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_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_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) _let_1)) (= _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_v_H $$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_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 $$unsorted) (T_a $$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_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)))) (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_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_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))))) (forall ((V_c $$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_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))))) (tptp.hBOOL (tptp.hAPP _let_65 tptp.c_Int_OMin)) (not (tptp.hBOOL (tptp.hAPP _let_61 tptp.c_Int_OMin))) (not (= tptp.c_Int_OMin tptp.c_Int_OPls)) (not (= tptp.c_Int_OPls tptp.c_Int_OMin)) (= (tptp.c_Int_OBit1 tptp.c_Int_OMin) tptp.c_Int_OMin) (forall ((V_l_2 $$unsorted)) (= (= tptp.c_Int_OMin (tptp.c_Int_OBit1 V_l_2)) (= tptp.c_Int_OMin V_l_2))) (forall ((V_k_2 $$unsorted)) (= (= (tptp.c_Int_OBit1 V_k_2) tptp.c_Int_OMin) (= V_k_2 tptp.c_Int_OMin))) (forall ((V_l $$unsorted)) (not (= tptp.c_Int_OMin (tptp.c_Int_OBit0 V_l)))) (forall ((V_k $$unsorted)) (not (= (tptp.c_Int_OBit0 V_k) tptp.c_Int_OMin))) (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_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_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))))) (tptp.hBOOL (tptp.hAPP _let_61 _let_44)) (tptp.hBOOL (tptp.hAPP _let_65 tptp.c_Int_OPls)) (not (tptp.hBOOL (tptp.hAPP _let_64 tptp.c_Int_OMin))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2)) tptp.c_Int_OMin)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) tptp.c_Int_OMin))))) (forall ((V_k_2 $$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))))) (not (tptp.hBOOL (tptp.hAPP _let_62 tptp.c_Int_OMin))) (tptp.hBOOL (tptp.hAPP _let_61 tptp.c_Int_OPls)) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) tptp.c_Int_OMin))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Int_OBit1 V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2))))) (forall ((V_k_2 $$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_60 _let_59)) (= tptp.c_Int_OMin (tptp.hAPP _let_39 _let_45)) (forall ((V_k $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) tptp.c_Int_OMin) V_k) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_k))) (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_59)) (= (tptp.c_Int_Opred tptp.c_Int_OPls) tptp.c_Int_OMin) (forall ((V_k $$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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) tptp.c_Int_OMin) V_k) (tptp.c_Int_Opred V_k))) (= (tptp.c_Int_Opred tptp.c_Int_OMin) (tptp.c_Int_OBit0 tptp.c_Int_OMin)) (tptp.c_Nat__Transfer_Onat__set _let_58) (forall ((T_a $$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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_k) tptp.c_Int_OMin) (tptp.c_Int_Osucc V_k))) (forall ((V_S_2 $$unsorted)) (= (tptp.c_Nat__Transfer_Onat__set V_S_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)))))) (forall ((V_k $$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_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m) V_n)) V_n) V_m))) (forall ((V_m $$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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) V_m)))))) (forall ((V_z $$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 $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) V_z) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_z)))) (forall ((V_C_2 $$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 ((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)) (=> (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 ((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_k_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Int_OBit0 V_k_2)) tptp.c_Int_OMin)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_k_2) tptp.c_Int_OMin)))) (forall ((V_k_2 $$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_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_39 tptp.c_Int_OMin) _let_29) (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_na_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_na_2) _let_2) (or (and (= V_m_2 _let_2) (= V_na_2 _let_2)) (and (= V_m_2 _let_1) (= V_na_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 ((V_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_a))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_m) V_n))) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_n)))))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m) V_n)))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_5) V_n)) (and (=> _let_4 (= _let_2 _let_5)) (=> (not _let_4) (= _let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_n)))))))))))) (forall ((V_k_2 $$unsorted) (V_na_2 $$unsorted) (V_Pa_2 $$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_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_na_2) V_k_2))) (and (=> _let_2 (tptp.hBOOL (tptp.hAPP V_Pa_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_na_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_Pa_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_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) tptp.c_Int_OMin))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_1) V_b) _let_1)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x_2))))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) V_n))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m_2) V_na_2))) (or (and (= V_na_2 _let_1) (tptp.hBOOL (tptp.hAPP V_Pa_2 _let_1))) (exists ((B_q $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_na_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 B_q)) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc B_q)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_q))))))))) (forall ((V_m_2 $$unsorted) (V_q_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_na_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_na_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_na_2))))))) (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_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_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_m $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Nat_OSuc V_m))) _let_1) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_2 V_m) _let_1)))))) (forall ((V_m $$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_A_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2)) V_A_2))) (forall ((V_v_H $$unsorted) (V_v $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_v))) (let ((_let_3 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) (tptp.hAPP _let_3 V_v)) (tptp.hAPP _let_3 V_v_H)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP tptp.c_Nat__Numeral_Oneg _let_2)))) (and (=> _let_5 (= _let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not _let_5) (= _let_4 (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) _let_2) (tptp.hAPP _let_1 V_v_H)))))))))))) (forall ((V_b $$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))))) (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_n $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_n) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls))))))))) (forall ((V_v_H $$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_k $$unsorted)) (let ((_let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral) _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) _let_1)) (tptp.hAPP tptp.c_Code__Numeral_Oint__of (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Code__Numeral_Ocode__numeral) V_k) _let_2))))) (let ((_let_4 (tptp.hAPP tptp.c_Code__Numeral_Oint__of V_k))) (let ((_let_5 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_6 (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Code__Numeral_Ocode__numeral V_k _let_2) _let_5))) (let ((_let_7 (= V_k _let_5))) (and (=> _let_7 (= _let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (not _let_7) (and (=> _let_6 (= _let_4 _let_3)) (=> (not _let_6) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_3) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))))))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x_2) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Int_Oint) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Int_Oint V_l_2 V_u_2)) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_u_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_l_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))))) (= _let_20 (tptp.hAPP _let_18 _let_29)) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_b _let_3)))))))))))))) (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_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_z_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_b))) (=> (tptp.class_Int_Oring__char__0 T_b) (= (= (tptp.hAPP _let_1 V_w_2) (tptp.hAPP _let_1 V_z_2)) (= V_w_2 V_z_2))))) (= (tptp.hAPP (tptp.hAPP _let_15 _let_21) _let_20) _let_21) (forall ((V_k $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Int_Onumber__ring T_a) (= (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_k) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_k)))) (forall ((V_w $$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_n $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) V_n))) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int tptp.tc_Int_Oint) _let_1) _let_1))) (tptp.c_Nat__Transfer_Ois__nat _let_44) (forall ((V_y $$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_45) (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_Divides_Odiv__class_Omod 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.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_x) V_y))))) (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_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))))) (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_b $$unsorted)) (=> (tptp.class_Int_Oring__char__0 T_b) (= (= (tptp.c_Groups_Ozero__class_Ozero T_b) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_b) V_z_2)) (= V_z_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_z_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Int_Oring__char__0 T_b) (= (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_b) V_z_2) (tptp.c_Groups_Ozero__class_Ozero T_b)) (= V_z_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_z_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (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_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP _let_1 V_w_2)) (tptp.hAPP _let_1 V_z_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_w_2) V_z_2)))))) (forall ((V_z $$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_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_z $$unsorted) (V_w $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 V_z)))))) (forall ((T_a $$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 $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_a))) (=> (tptp.class_Rings_Oring__1 T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_w) V_z)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) (tptp.hAPP _let_1 V_w)) (tptp.hAPP _let_1 V_z)))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_n $$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_b $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_b) V_z_2)) (tptp.c_Int_Oring__1__class_OInts T_b))))) (forall ((V_n $$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_x_2 $$unsorted)) (= (tptp.c_Nat__Transfer_Ois__nat V_x_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_x_2)))) (forall ((V_k_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_k_2)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_k_2 V_l_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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_x_2 V_y_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b (tptp.hAPP _let_1 V_y_2) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2)) V_x_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_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= 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 T_b) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2)) V_A_2)))))) (= _let_21 (tptp.hAPP _let_18 tptp.c_Int_OPls)) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) _let_1) _let_1)))))) (forall ((T_a $$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_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_A_2))))) (tptp.c_Nat__Transfer_Ois__nat _let_57) (forall ((V_z_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_b) V_z_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))) (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 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (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.c_Int_Oring__1__class_Oof__int T_b) 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 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_b) V_z_2)) (tptp.c_Groups_Ozero__class_Ozero T_b))) (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_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (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.c_Int_Oring__1__class_Oof__int T_b) 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_N_2 $$unsorted) (V_M_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_M_2) V_N_2)) (=> (not (= V_M_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_M_2))))))))) (forall ((V_z $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Oring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_z)) (= (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) (tptp.hAPP tptp.c_Int_Onat V_z)) (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_a) V_z))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_B_2)) _let_1) _let_1)))))))) (tptp.c_Nat__Transfer_Ois__nat _let_55) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMin T_b _let_4)))))))))))))) (forall ((V_n $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_Code__Numeral_Onat__of__aux V_i V_n))) (let ((_let_2 (= V_i (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (and (=> _let_2 (= _let_1 V_n)) (=> (not _let_2) (= _let_1 (tptp.c_Code__Numeral_Onat__of__aux (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP tptp.c_Nat_OSuc V_n)))))))) (forall ((V_k_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_2 (tptp.c_COMBS T_b T_b T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.c_COMBI T_b)))) (let ((_let_4 (tptp.tc_fun T_b T_b))) (let ((_let_5 (tptp.c_COMBB T_b _let_4 T_b))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_7 (tptp.tc_fun T_b _let_4))) (let ((_let_8 (tptp.c_Int_Oring__1__class_Oof__int T_b))) (let ((_let_9 (tptp.tc_fun _let_4 T_b))) (let ((_let_10 (tptp.tc_fun tptp.tc_Int_Oint T_b))) (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_b) (and (=> _let_14 (= _let_12 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (not _let_14) (and (=> _let_13 (= _let_12 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (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_b) (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_b tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_9 tptp.tc_Int_Oint) (tptp.c_HOL_OLet T_b T_b)) _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_b)) (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_b T_b T_b) (tptp.hAPP (tptp.hAPP _let_5 _let_1) _let_3)) (tptp.c_Groups_Oone__class_Oone T_b))))) (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_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2)) V_A_2))))))) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__size _let_21) _let_12) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_5 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_4 V_A_2))) (let ((_let_6 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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)) (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_b _let_4 _let_3))))))))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_S_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_c) V_c_2))) (let ((_let_3 (tptp.tc_fun T_c T_c))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_3) T_b) (tptp.c_If T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b 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_b) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (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_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_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_b))) (=> (and (tptp.class_Groups_Ocomm__monoid__mult T_b) (tptp.class_Rings_Oring__1 T_b)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c tptp.tc_Int_Oint) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_b T_c) _let_1) V_f_2)) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (=> (and (tptp.class_Groups_Ocomm__monoid__mult T_b) (tptp.class_Rings_Osemiring__1 T_b)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c tptp.tc_Nat_Onat) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b T_c) _let_1) V_f_2)) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b tptp.tc_Nat_Onat) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_b) _let_1) V_f_2)) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (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.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_b) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_f_2)) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _let_2) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b) (tptp.hAPP (tptp.c_COMBK T_b T_c) _let_1)) V_A_2) _let_1)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Ocomm__monoid__mult T_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Nat_Onat T_b) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Int_Oint T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Int_Oint) V_f_2) tptp.c_Int_Onat)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_A_2))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_c)))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (and (tptp.class_Rings_Ocomm__semiring__1 T_c) (tptp.class_Rings_Ono__zero__divisors T_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_c)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c)))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b) V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Groups_Oone__class_Oone T_b)))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b) V_g_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Groups_Oone__class_Oone T_b)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2) V_A_2) (tptp.c_Groups_Oone__class_Oone T_c))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2) V_A_2) (tptp.c_Groups_Oone__class_Oone T_c))))) (forall ((V_f_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b tptp.tc_Nat_Onat) V_f_2) V_F_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_F_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Ocomm__monoid__mult T_b)) (=> (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_b) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Nat_Onat T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_b 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_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_c))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) _let_2) V_g_2)) V_h_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_h_2) V_A_2)))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b tptp.tc_Nat_Onat) V_f_2) V_S_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP V_f_2 B_x)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_c))) (let ((_let_2 (tptp.tc_fun T_c T_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_2) T_b) (tptp.c_If T_c)) (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_b) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_c))) (let ((_let_2 (tptp.tc_fun T_c T_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_2) T_b) (tptp.c_If T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_b_2)) _let_1)) V_S_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (and (=> _let_4 (= _let_3 (tptp.hAPP V_b_2 V_a_2))) (=> (not _let_4) (= _let_3 _let_1)))))))))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) (tptp.hAPP (tptp.c_COMBK T_c T_b) V_y_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_c) V_y_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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 _let_1 V_x_2) _let_2) _let_2))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b (tptp.c_Orderings_Oord__class_Omin T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (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 _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b _let_1) V_A_2)))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b (tptp.c_Groups_Otimes__class_Otimes T_c)) V_g_2) (tptp.c_Groups_Oone__class_Oone T_c)) V_A_2))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_c))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b (tptp.c_Groups_Otimes__class_Otimes T_c)) V_g_2) _let_1) V_A_2))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Nat_Onat V_l_2 V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_u_2) (tptp.hAPP tptp.c_Nat_OSuc V_l_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_B_2)) _let_2) _let_2))))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (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 _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (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_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) 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_b))) (let ((_let_6 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_5 (tptp.hAPP _let_3 V_A_2)))) (let ((_let_7 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b _let_5 _let_4))))))))))))))) (forall ((V_code__numeral $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__size (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__size V_code__numeral)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Big__Operators_Olattice_OInf__fin T_b (tptp.c_Orderings_Oord__class_Omin T_b) V_A_2)) V_A_2))))))) (forall ((V_b_2 $$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)))))))))))))))))))) (= 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_38 _let_38 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_33 tptp.tc_Int_Oint) tptp.c_fconj) _let_53))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint _let_38) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_51 _let_51 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_51 tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint _let_38 tptp.tc_Int_Oint) tptp.c_fequal)) tptp.c_Nitpick_Oint__gcd))) _let_45)))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_b 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_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)))) (let ((_let_6 (tptp.hAPP V_f_2 V_a_2))) (=> (tptp.class_Fields_Ofield T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= _let_6 (tptp.c_Groups_Ozero__class_Ozero T_c))) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_c) _let_2) _let_6))) (=> (not _let_5) (= _let_4 _let_2))))))))))))) (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)))))))))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) _let_1) _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) _let_1) V_a) _let_1)))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_c _let_3))) (let ((_let_5 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_b) (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_b T_c)))))) (= (= (tptp.hAPP _let_6 V_R_2) (tptp.hAPP _let_6 V_S_2)) (= V_R_2 V_S_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2) V_y_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_c _let_3))) (let ((_let_5 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_b) (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_b T_c)))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b _let_5)) (tptp.hAPP _let_6 V_R_2)) (tptp.hAPP _let_6 V_S_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_R_2) V_S_2)))))))))) (forall ((V_b $$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_r $$unsorted) (V_q $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r))) (=> (not (= V_a _let_1)) (= V_r _let_1))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odivmod__int__rel V_a V_b))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q_H) V_r_H))) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_q V_q_H))))))) (forall ((V_r_H $$unsorted) (V_q_H $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Divides_Odivmod__int__rel V_a V_b))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_q_H) V_r_H))) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= V_r V_r_H))))))) (forall ((V_code__numeral_H_2 $$unsorted) (V_code__numeral_2 $$unsorted)) (= (= (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_2) (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_H_2)) (= V_code__numeral_2 V_code__numeral_H_2))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_c $$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 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_a))) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a))) (forall ((V_b $$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 $$unsorted) (V_b $$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_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r))) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) V_q)))) (forall ((V_a $$unsorted) (V_b $$unsorted)) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_b) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2) V_b_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_b) (=> (not (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2) V_a_2) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2)))))) (forall ((V_a $$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 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Odivision__ring T_b) (=> (not (= V_b_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_a_2) V_b_2) (tptp.c_Groups_Oone__class_Oone T_b)) (= 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_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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.c_Groups_Oone__class_Oone T_a)))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_n)))))) (forall ((V_code__numeral_H_1 $$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_code__numeral_H $$unsorted)) (not (= (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_H)))) (forall ((V_br $$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_g_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_c))) (=> (tptp.class_Fields_Ofield__inverse__zero T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) _let_2) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_A_2)))))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_b)))) (=> (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b) V_q)) V_r)) (=> (and (=> _let_4 (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_r) V_b)))) (=> (not _let_4) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b) V_r)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r) _let_1))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q) V_r)))))))))) (forall ((V_w_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_3 (= _let_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) _let_2)) (and (=> (not _let_3) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) _let_2) V_b_2)) (=> _let_3 (= V_a_2 _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_3 (= V_c_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (and (=> (not _let_3) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) _let_2) V_c_2) V_b_2)) (=> _let_3 (= _let_2 _let_1))))))))) (forall ((V_w_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_3 (= V_c_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2) _let_2) (and (=> (not _let_3) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) _let_2) V_c_2))) (=> _let_3 (= _let_2 _let_1))))))))) (forall ((V_a_2 $$unsorted) (V_w_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_3 (= _let_2 _let_1))) (=> (and (tptp.class_Fields_Ofield__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) _let_2) V_a_2) (and (=> (not _let_3) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) _let_2))) (=> _let_3 (= V_a_2 _let_1))))))))) (forall ((V_a $$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_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 ((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_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_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_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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_b))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a V_b) (tptp.hAPP (tptp.hAPP _let_2 V_q) V_r))) (=> (not (= V_b _let_3)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_3) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__int__rel V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_q) V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_q V_c))) V_r))))))))))) (forall ((V_q $$unsorted) (V_p $$unsorted)) (= (tptp.c_Divides_Odivmod__int V_p V_q) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_p) V_q)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_p V_q)))) (forall ((V_a_2 $$unsorted) (V_w_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (let ((_let_4 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_4) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) _let_4))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP _let_3 _let_4)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) _let_4)) V_a_2)) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_6))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_6) V_b_2))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)))))))))))))))) (forall ((V_w_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_4 (tptp.hAPP _let_3 _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) _let_2)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) _let_1) V_c_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP _let_4 V_c_2)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) _let_1)) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_6))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_6) V_b_2))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_4 _let_1)))))))))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_4 (tptp.hAPP _let_3 _let_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) _let_2) V_c_2))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_c_2)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2))) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_6) V_b_2))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_6))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_4 _let_1)))))))))))))))) (forall ((V_w_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_4) _let_1)))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) _let_4))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) _let_4)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) _let_4))) (and (=> _let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_6) V_b_2))) (=> (not _let_7) (and (=> _let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_6))) (=> (not _let_5) (tptp.hBOOL (tptp.hAPP _let_3 _let_1)))))))))))))))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_a (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n)))))))) (forall ((V_x $$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)) (= (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_w_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) 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_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_c_2 $$unsorted) (V_b_2 $$unsorted) (V_w_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_4 (tptp.hAPP _let_3 _let_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) _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_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) _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_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_a_2 $$unsorted) (V_w_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) V_w_2))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) 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_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (=> (and (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (tptp.class_Int_Onumber__ring T_b)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_r_2) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_b) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))))) (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_q_2))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_3 (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_4 (tptp.tc_fun _let_3 _let_3))) (= (tptp.hAPP (tptp.c_Divides_Oadjust V_b_2) (tptp.hAPP (tptp.hAPP _let_2 V_q_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_r_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint _let_3 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_3 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun _let_3 _let_4) tptp.tc_Int_Oint) (tptp.c_If _let_3)) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_1) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r_2))))))))) (forall ((V_f_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_d) (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_b_2)) (tptp.hAPP (tptp.c_Set_Oimage _let_1 T_d (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) 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_w $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint))) (let ((_let_2 (tptp.hAPP _let_1 V_w))) (let ((_let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (let ((_let_4 (tptp.c_Divides_OnegDivAlg _let_3 _let_2))) (let ((_let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) _let_3) _let_2))) (let ((_let_6 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_6) _let_5)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_6) _let_2)) (and (=> _let_7 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP _let_1 tptp.c_Int_OMin)) _let_5))) (=> (not _let_7) (= _let_4 (tptp.hAPP (tptp.c_Divides_Oadjust _let_2) (tptp.c_Divides_OnegDivAlg _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP _let_1 (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) _let_2))))))))))))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_b_2 $$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_b T_c tptp.tc_HOL_Obool) V_f1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2))))) (forall ((T_d $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_z_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) 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_c T_d (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) V_a_2) V_b_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b (tptp.tc_fun T_c 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_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c (tptp.tc_fun T_d tptp.tc_HOL_Obool)) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) V_c_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_a_2) V_b_2) V_c_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_Pa_2) V_Q_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 V_x_2) V_y_2))))) (forall ((V_Q_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_x_2) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_Pa_2) V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 V_x_2) V_y_2))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) 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_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_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_c))) (= (= (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_c $$unsorted) (T_b $$unsorted) (V_Pa_2 $$unsorted)) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (forall ((B_a $$unsorted) (B_b $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) B_a) B_b)))))) (forall ((V_c_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_q_2 $$unsorted) (V_p_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) 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_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_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) 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 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_f1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) 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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_b_2)))) (forall ((V_l $$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_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) (tptp.tc_fun T_c T_d) T_b) (tptp.hAPP (tptp.c_COMBB _let_1 T_d T_c) V_f_2)) (tptp.c_Product__Type_OPair T_b T_c))) V_f_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.c_HOL_OThe (tptp.tc_prod T_b T_c) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 T_b) 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_b T_c) V_x_2) V_y_2))))) (forall ((V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.c_HOL_OThe _let_1 V_Pa_2) (tptp.c_HOL_OThe _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c) V_Pa_2)) (tptp.c_Product__Type_OPair T_b T_c))))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_b))) (tptp.c_Fun_Oinj__on (tptp.tc_prod T_b T_c) _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c _let_1) (tptp.hAPP (tptp.c_COMBC T_c T_b _let_1) (tptp.c_Product__Type_OPair T_c T_b))) V_A_2))) (forall ((V_p_2 $$unsorted) (V_g_2 $$unsorted) (T_f $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_f_2))) (let ((_let_2 (tptp.tc_prod T_c T_d))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_e T_f _let_2) V_g_2) V_p_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_e T_f T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_f _let_2) (tptp.tc_fun T_f T_b) T_e) (tptp.hAPP (tptp.c_COMBB _let_2 T_b T_f) _let_1)) V_g_2)) V_p_2))))) (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))))))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_FunDef_Oin__rel T_b T_c V_R_2) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2) V_y_2)) V_R_2)))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.c_Code__Numeral_Odiv__mod__code__numeral V_n V_m) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Code__Numeral_Ocode__numeral V_n V_m)))) (forall ((V_c $$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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_w $$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 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Fields_Ofield__inverse__zero T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a)) (tptp.hAPP _let_2 V_b)) (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_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_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_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_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_a_2) V_b_2)) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_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 $$unsorted) (V_b $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y))))))))) (forall ((V_y $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) _let_1)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.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_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_a_2) V_b_2)) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP _let_3 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP _let_3 V_b_2)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (let ((_let_3 (tptp.hAPP _let_2 _let_1))) (=> (tptp.class_Fields_Olinordered__field__inverse__zero T_b) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (= V_c_2 _let_1))) (=> (tptp.class_Fields_Ofield__inverse__zero T_b) (= (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) V_b_2) V_c_2)) (and (=> (not _let_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (= V_c_2 _let_1))) (=> (tptp.class_Fields_Ofield__inverse__zero T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_b))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Fields_Ofield T_b) (=> (not (= V_y_2 _let_3)) (=> (not (= V_z_2 _let_3)) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_y_2) (tptp.hAPP (tptp.hAPP _let_2 V_w_2) V_z_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2) (tptp.hAPP (tptp.hAPP _let_1 V_w_2) V_y_2)))))))))) (forall ((V_y $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) _let_1)))))))) (forall ((V_z $$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_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 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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_y) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_a) V_x) V_y)) _let_1)))))))) (forall ((V_y $$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_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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_b) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_b) V_a_2) V_c_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b)) 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_b) 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_b) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_a_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) 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_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) 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_b) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_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_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_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 (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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_y _let_4)) (=> (not (= V_z _let_4)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_3 V_w) V_z)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_w) V_y))) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ominus__class_Ominus T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Ofield T_a) (=> (not (= V_z (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_z) V_x)) V_y)) V_z))))))) (forall ((V_y $$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_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) 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_b) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_b))) (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_b) 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_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) 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_b) 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_b) V_a_2) V_c_2)) V_b_2))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_b) V_a_2) V_c_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))) (=> (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_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_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_c_2) (tptp.c_Groups_Ozero__class_Ozero T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) 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_b) V_a_2) V_c_2)))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Nitpick_Orefl_H T_b V_r_2) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)) V_r_2))))) (forall ((V_w $$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_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_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_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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_k)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_l)) (= (tptp.c_Divides_OposDivAlg V_k V_l) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_k) V_l)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_k V_l))))))) (forall ((V_b $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (= (tptp.c_Divides_Odivmod__nat V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) _let_1) V_n))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat _let_1 V_n))))))) (forall ((V_code__numeral $$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_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_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_b 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_b tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Nat_Onat T_b) tptp.c_Int_Onat) V_f_2)) V_A_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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_q))) (= (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.hAPP _let_1 V_r)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_r))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odivmod__nat V_m V_n) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_m) V_n)) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_qr $$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)))))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Code__Numeral_Ocode__numeral) _let_21) _let_12) (forall ((V_m_2 $$unsorted) (V_na_2 $$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_na_2))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_5 (or (= V_na_2 _let_4) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_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_na_2) V_na_2))))))))))) (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_Pa_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 $$unsorted)) (let ((_let_1 (tptp.hAPP V_Pa_2 B_a))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OnegDivAlg__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) B_a) B_b))) (=> (=> (not (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_a) B_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 B_b) _let_2)))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) B_b)))) (tptp.hBOOL (tptp.hAPP _let_1 B_b)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_a0_2) V_a1_2))))) (forall ((V_R2_2 $$unsorted) (V_R1_2 $$unsorted) (T_b $$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 T_b _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_b V_R2_2)) (tptp.c_Wellfounded_Oaccp T_b V_R1_2)))))) (forall ((V_c $$unsorted) (V_r $$unsorted) (V_q $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_b))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_a V_b) (tptp.hAPP (tptp.hAPP _let_2 V_q) V_r))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_b)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_a (tptp.hAPP _let_1 V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) V_q) V_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_q V_c))) V_r))))))))))) (forall ((V_qr_H $$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_na_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Divides_Odivmod__nat V_m_2 V_na_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_na_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_a_2 $$unsorted) (V_b_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oaccp T_b 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_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_b V_r_2) V_a_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 B_x) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_b V_r_2) B_x)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_b V_r_2) V_a_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 B_x) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_b V_r_2) B_x)))))) (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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ar) V_br))) (let ((_let_3 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_a V_c) (tptp.hAPP (tptp.hAPP _let_3 V_aq) V_ar))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel V_b V_c) (tptp.hAPP (tptp.hAPP _let_3 V_bq) V_br))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.c_Divides_Odivmod__nat__rel (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b) V_c) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 V_aq) V_bq)) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Nat_Onat) _let_2) V_c))) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat _let_2 V_c))))))))))) (forall ((V_a $$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_Pa_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 $$unsorted)) (let ((_let_1 (tptp.hAPP V_Pa_2 B_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Divides_OposDivAlg__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) B_a) B_b))) (=> (=> (not (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) B_a) B_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) B_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) (tptp.c_Int_OBit0 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) B_b)))) (tptp.hBOOL (tptp.hAPP _let_1 B_b)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_a0_2) V_a1_2))))) (forall ((V_nat $$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_w $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Int_Oint) V_w))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint _let_3 _let_1) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OnegDivAlg (tptp.hAPP _let_2 _let_3) (tptp.hAPP _let_2 _let_1)))))))))) (forall ((V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_Wellfounded_Omeasure T_b) V_f_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))))) (forall ((V_Pa_2 $$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_Pa_2 (tptp.hAPP _let_1 B_a)) (tptp.hAPP _let_1 B_b)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_a) B_b)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_a0_2) V_a1_2))))) (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_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_b) V_p_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_b T_b) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b T_b) T_c) (tptp.c_COMBI T_b))) V_p_2))) (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)))) (= (tptp.c_Nat_Onat_Onat__size _let_12) _let_12) (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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_a) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_b) _let_2)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Int_Oint V_a V_b) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OposDivAlg (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b)))))))))) (forall ((V_w $$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 ((V_R_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_R_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Wellfounded_Omlex__prod T_b V_f_2 V_R_2))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) (tptp.c_Wellfounded_Omlex__prod T_b V_f_2 V_R_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_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)))))))))) (forall ((V_nat $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_nat)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_nat)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_n $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat) V_n) V_n)) (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_a $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_a T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x) V_y)) V_a) (= V_x V_a))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_t_2 $$unsorted) (V_s_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_Osnd T_b T_c))) (let ((_let_2 (tptp.c_Product__Type_Ofst T_b T_c))) (= (= V_s_2 V_t_2) (and (= (tptp.hAPP _let_2 V_s_2) (tptp.hAPP _let_2 V_t_2)) (= (tptp.hAPP _let_1 V_s_2) (tptp.hAPP _let_1 V_t_2))))))) (forall ((V_q $$unsorted) (V_p $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_Osnd T_a T_b))) (let ((_let_2 (tptp.c_Product__Type_Ofst T_a T_b))) (=> (= (tptp.hAPP _let_2 V_p) (tptp.hAPP _let_2 V_q)) (=> (= (tptp.hAPP _let_1 V_p) (tptp.hAPP _let_1 V_q)) (= V_p V_q)))))) (forall ((V_p_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) V_p_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_b) (tptp.c_COMBK T_b T_c)) V_p_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_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_f_2) V_p_2) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_d) V_p_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) V_p_2)))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odivmod__int V_a V_b)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (= (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat V_m V_n) (tptp.hAPP (tptp.c_Product__Type_Osnd tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Divides_Odivmod__nat V_m V_n)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c T_d))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_2 T_c T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_e _let_1 _let_2) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_e _let_2) V_g_2) (tptp.c_Product__Type_Ofst T_b T_c)))) (tptp.c_Product__Type_Osnd T_b T_c)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_e _let_1 T_b) V_f_2) V_g_2)))))) (forall ((V_z_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_Pa_2) V_z_2) (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_d) V_z_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) V_z_2)))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize tptp.tc_Nat_Onat) _let_12) _let_12) (forall ((V_p_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_c_2) V_p_2) (tptp.hAPP (tptp.hAPP V_c_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_d) V_p_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) V_p_2)))) (forall ((V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.c_HOL_OThe _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) V_Pa_2)) (tptp.c_HOL_OThe _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool) _let_1) V_Pa_2) (tptp.c_Product__Type_Ofst T_b T_c))) (tptp.c_Product__Type_Osnd T_b T_c)))))) (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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) V_a)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b) _let_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint) V_a) V_b) (tptp.hAPP (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Divides_OnegateSnd (tptp.c_Divides_OnegDivAlg (tptp.hAPP _let_1 V_a) (tptp.hAPP _let_1 V_b))))))))))) (forall ((V_p_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.c_Product__Type_Oprod_Oprod__size T_b T_c 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_b T_c) V_p_2))) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_c) V_p_2)))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((T_b $$unsorted) (T_c $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) V_x_2) V_y_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_y_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_b) _let_1)) (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_b) _let_1)))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_q_2 $$unsorted) (V_Q_2 $$unsorted) (V_p_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_p_2) V_q_2))) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_p_2)) (=> (tptp.hBOOL (tptp.hAPP V_Q_2 V_q_2)) (and (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) _let_1))) (tptp.hBOOL (tptp.hAPP V_Q_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_c) _let_1)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.c_Product__Type_Oprod_Oprod__rec T_c T_d T_b V_f1_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP V_f1_2 V_a_2) V_b_2))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (V_b_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_H_2) V_b_H_2))) (tptp.c_Wellfounded_Olex__prod T_b T_c V_r_2 V_s_2))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_a_H_2)) V_r_2)) (and (= V_a_2 V_a_H_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_c) V_b_2) V_b_H_2)) V_s_2)))))))) (forall ((V_n $$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_na_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_na_2)))) (let ((_let_6 (= V_na_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_na_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_na_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_na_2)))) (=> (not _let_5) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_na_2) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)))))))))))))))) (forall ((T_b $$unsorted) (V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_y_H_2 $$unsorted) (T_c $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_c) V_y_H_2) V_y_2)) (tptp.hAPP V_R_2 V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP _let_1 V_y_H_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.c_Recdef_Osame__fst T_b T_c V_Pa_2 V_R_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b))) (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_b) V_B_2)))))) (forall ((V_na_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_na_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_na_2))))))))) (forall ((V_v_2 $$unsorted) (V_na_2 $$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_na_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_na_2)) tptp.c_Int_Onat)))))))))) (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_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_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_z_2))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) 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 $$unsorted) (V_b $$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_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_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_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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$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_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 $$unsorted) (V_b $$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_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat) V_n) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (forall ((V_n $$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_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_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_y_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_z_2))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) 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 $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y))))) (forall ((V_x $$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_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) V_y_2) V_y_2)))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) V_y_2)) V_z_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b))))))) (forall ((V_y $$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_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y) V_x)))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_a) V_b)) V_x))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_y) V_z)) V_x))))))) (forall ((V_d $$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_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_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 $$unsorted) (V_z $$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_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_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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_a) V_x) V_y)) V_x))) (forall ((V_y $$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 ((T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_b (tptp.c_Orderings_Oord__class_Omax T_b)))) (forall ((V_v $$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 $$unsorted) (V_y $$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_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_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_na_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_na_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_na_2)) V_m_2))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_na_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_na_2)) V_m_2))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.c_Orderings_Oord__class_Omax T_b) (tptp.c_Orderings_Oord_Omax T_b (tptp.c_Orderings_Oord__class_Oless__eq T_b))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Orderings_Oord_Omin T_b (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (tptp.c_Orderings_Oord__class_Omax T_b)))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_a_2) (tptp.c_Big__Operators_Olattice_OInf__fin T_b (tptp.c_Orderings_Oord__class_Omin T_b) V_A_2)) V_a_2))))) (forall ((V_t_2 $$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 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) V_XS_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_x_2) V_XS_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) tptp.c_FunDef_Opair__less)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_YS_2)) tptp.c_FunDef_Omin__strict)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_y_2) V_YS_2))) tptp.c_FunDef_Omin__strict)))))))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_s_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_t_2))) tptp.c_FunDef_Opair__leq)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_b _let_3)))))))))))))) (forall ((V_Y_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_Y_2)) (=> (not (= V_Y_2 _let_3)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_3) V_Y_2)) tptp.c_FunDef_Omax__strict)))))))) (forall ((V_X_2 $$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_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2)) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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 (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) _let_1) _let_1)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Orderings_Oord__class_Omax T_b)) V_A_2))))) (forall ((V_N_2 $$unsorted) (V_M_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_M_2) V_N_2)) (=> (not (= V_M_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_N_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_M_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_N_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_B_2)) _let_1) _let_1)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) (tptp.c_Big__Operators_Olinorder__class_OMax T_b _let_4)))))))))))))) (forall ((V_S_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (=> (not (= V_S_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_S_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) V_S_2)))))))) (forall ((V_S_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Olinordered__ab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (=> (not (= V_S_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_S_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) V_S_2)))))))) (forall ((V_YS_2 $$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 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_y_2) V_YS_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) tptp.c_FunDef_Opair__leq)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_XS_2) V_YS_2)) tptp.c_FunDef_Omax__weak)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_x_2) V_XS_2)) V_YS_2)) tptp.c_FunDef_Omax__weak)))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2)) V_A_2))))))) (forall ((V_X_2 $$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 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_X_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) tptp.c_FunDef_Omin__weak)))) (tptp.c_FunDef_Oreduction__pair _let_27 (tptp.hAPP (tptp.hAPP _let_50 tptp.c_FunDef_Omax__strict) tptp.c_FunDef_Omax__weak)) (tptp.c_FunDef_Oreduction__pair _let_27 (tptp.hAPP (tptp.hAPP _let_50 tptp.c_FunDef_Omin__strict) tptp.c_FunDef_Omin__weak)) (= tptp.c_FunDef_Omin__strict (tptp.c_Wellfounded_Omin__ext _let_24 tptp.c_FunDef_Opair__less)) (forall ((V_y_2 $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_3 _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_3 _let_3) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP _let_2 V_u_2) V_v_2))) tptp.c_Int_Ointrel)) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_v_2) (tptp.hAPP (tptp.hAPP _let_1 V_u_2) V_y_2))))))) (= tptp.c_FunDef_Opair__less (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than tptp.c_Wellfounded_Oless__than)) (forall ((V_R_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat T_b))) (= (tptp.c_Wellfounded_Omlex__prod T_b V_f_2 V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_1 T_b) (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat T_b tptp.c_Wellfounded_Oless__than V_R_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun T_b _let_1) T_b) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat T_b)) V_f_2)) (tptp.c_COMBI T_b)))))) (= tptp.c_FunDef_Omin__weak (tptp.hAPP (tptp.hAPP _let_49 (tptp.c_Wellfounded_Omin__ext _let_24 tptp.c_FunDef_Opair__leq)) _let_48)) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= 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_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Big__Operators_Olattice_OInf__fin T_b (tptp.c_Orderings_Oord__class_Omin T_b) V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b (tptp.c_Orderings_Oord__class_Omax T_b) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_5 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_4 V_A_2))) (let ((_let_6 (= _let_3 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_4 _let_3))))))))))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_FunDef_Oreduction__pair T_b V_Pa_2) (tptp.c_FunDef_Oreduction__pair T_c (tptp.hAPP (tptp.hAPP (tptp.c_FunDef_Orp__inv__image T_b T_c) V_Pa_2) V_f_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)) (=> (not (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))) (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2))))) (forall ((T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (=> (not (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2))) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (=> (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (or (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((T_b $$unsorted)) (=> (and (tptp.class_Lattices_Osemilattice__sup T_b) (tptp.class_Orderings_Olinorder T_b)) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) (tptp.c_Orderings_Oord__class_Omax T_b)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_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 _let_2 (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) 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_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c T_b)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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))) (= (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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_B_2) V_A_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2))))))) (forall ((V_a $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_x) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_a) V_a))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_x) V_x))) (forall ((V_b $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_y $$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_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.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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_c)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_c $$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 $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c T_b)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_A_2) V_A_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b 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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b 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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b 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_Pa_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (or (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_B_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))))) (forall ((V_Pa_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (and (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_B_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))))) (forall ((V_B_2 $$unsorted) (T_b $$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_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)))) (forall ((V_A_2 $$unsorted) (T_b $$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_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y))))) (forall ((V_y $$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_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y))))) (forall ((V_x $$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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2) V_y_2) V_y_2)))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Lattices_Osemilattice__sup T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2) V_y_2)) V_z_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_z_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_z_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_a)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b))))))) (forall ((V_y $$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_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y) V_x)))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_a) V_b)) V_x))))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_a))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_y) V_z)) V_x))))))) (forall ((V_d $$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_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_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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.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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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 _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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$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 _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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_B_2) V_B_2))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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.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_G_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_F_2) V_G_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_F_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_G_2)))))) (forall ((V_G_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_F_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_G_2)) (tptp.hBOOL (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_F_2) V_G_2))))))) (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_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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot T_b))) (=> (tptp.class_Lattices_Obounded__lattice__bot T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2) V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_b) (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b)))) (forall ((V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$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.c_member T_b))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBC T_b _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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (not (= V_A_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= V_B_2 _let_3)) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_B_2))))))))))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$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_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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_B_2) V_A_2)) V_B_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) V_C_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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 _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) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP _let_2 V_A_2)) V_B_2))))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oinv__image T_b T_c))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_c T_c) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c T_b))) (let ((_let_4 (tptp.tc_prod _let_2 _let_2))) (let ((_let_5 (tptp.tc_fun _let_2 _let_4))) (let ((_let_6 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun _let_3 _let_5))) (let ((_let_8 (tptp.tc_fun _let_3 _let_2))) (let ((_let_9 (tptp.tc_fun _let_3 _let_4))) (let ((_let_10 (tptp.tc_fun _let_8 _let_9))) (let ((_let_11 (tptp.tc_fun _let_6 _let_9))) (let ((_let_12 (tptp.tc_fun _let_6 _let_8))) (= (tptp.c_FunDef_Orp__inv__image T_b T_c) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_6 _let_6 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_6 _let_12 _let_11) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 (tptp.tc_fun _let_12 _let_11) _let_6) (tptp.c_COMBB _let_8 _let_9 _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 _let_10 _let_6) (tptp.c_COMBS _let_3 _let_2 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 _let_7 _let_6) (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_3) (tptp.c_Product__Type_OPair _let_2 _let_2))) _let_1)))) _let_1)))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_c))) (=> (tptp.c_Finite__Set_Ofolding__image__simple__idem T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_A_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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 _let_1 V_x_2) _let_2) _let_2))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b (tptp.c_Orderings_Oord__class_Omax T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b _let_1) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Orderings_Obot__class_Obot _let_3))) (let ((_let_5 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_b) (=> (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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b (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_b) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (not (= V_A_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= V_B_2 _let_3)) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_B_2))))))))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image _let_1 T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_Set_Oinsert T_c)) V_f_2)) _let_2)) _let_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_A_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) V_B_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_B_2)) _let_2) _let_2))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_a_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_b (tptp.c_Orderings_Oord__class_Omax T_b) V_A_2)) V_a_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_A_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) 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_b))) (let ((_let_6 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_5 (tptp.hAPP _let_3 V_A_2)))) (let ((_let_7 (= _let_4 _let_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b _let_5 _let_4))))))))))))))) (= tptp.c_FunDef_Omax__weak (tptp.hAPP (tptp.hAPP _let_49 (tptp.c_Wellfounded_Omax__ext _let_24 tptp.c_FunDef_Opair__leq)) _let_48)) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (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_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_c T_b) V_r_2) V_f_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_c) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) V_r_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) 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_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Big__Operators_Olattice_OSup__fin T_b (tptp.c_Orderings_Oord__class_Omax T_b) V_A_2)) V_A_2))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_u_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Lattices_Olattice T_b) (=> (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)) (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_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b _let_3)))))))))))))) (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)))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup tptp.tc_Nat_Onat) (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat)) (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))))) (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_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_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_k_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_k_2)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_k_2) V_l_2)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_k_2 V_l_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_k_2) V_l_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_2))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_l_2 V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_u_2) V_l_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_4 (tptp.c_member _let_1))) (let ((_let_5 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_fun T_c _let_5))) (let ((_let_7 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_2 _let_7) T_b) (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_6 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_c) _let_4)) _let_3))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b _let_7)) (tptp.hAPP _let_8 V_R_2)) (tptp.hAPP _let_8 V_S_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_R_2) V_S_2))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2))) (=> (tptp.class_Lattices_Olattice T_b) (=> (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 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2) _let_1) _let_1)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b)) V_A_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_B_2)) _let_1) _let_1)))))))) (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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_B_2)))))))))))) (= tptp.c_FunDef_Omax__strict (tptp.c_Wellfounded_Omax__ext _let_24 tptp.c_FunDef_Opair__less)) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)))))) (forall ((T_b $$unsorted)) (= (tptp.c_Wellfounded_Omeasure T_b) (tptp.hAPP (tptp.c_Relation_Oinv__image tptp.tc_Nat_Onat T_b) tptp.c_Wellfounded_Oless__than))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) 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_b (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b _let_4)))))))))))))) (forall ((V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_R_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Omax__ext T_b V_R_2))) (let ((_let_2 (tptp.tc_fun T_b 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)))))))))) (forall ((V_xa_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_prod T_b T_b))) (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_b _let_4))) (= (tptp.c_Wellfounded_Omax__extp T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_3 _let_1) T_b) (tptp.c_COMBC T_b _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_2) _let_5 T_b) (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 T_b) (tptp.c_member _let_2))) (tptp.c_Product__Type_OPair T_b T_b)))) 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_b V_R_2)))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Oirrefl T_b V_r_2) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)) V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Set_Oinsert T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) (tptp.hAPP (tptp.hAPP _let_3 V_a_2) (tptp.hAPP (tptp.hAPP _let_3 V_b_2) (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP _let_1 V_r_2))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2)) V_A_2))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (= (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_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_N_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_N_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b (tptp.tc_fun T_c 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 ((T_c $$unsorted) (T_b $$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_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_y_2 $$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_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_y_2 $$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_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)))) (forall ((T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OField T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b 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_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_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 V_s_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_b)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Relation_OField T_b) V_r_2))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (= (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_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_N_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_N_2))))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_N_2)))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (= (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_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_N_2)))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (= (tptp.hAPP V_h_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_h_2 B_x)) (tptp.hAPP V_h_2 B_y))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_N_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_N_2)))))))) (forall ((V_x $$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 $$unsorted)) (= V_x (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Nat_OSuc B_n))))))) (forall ((V_k_2 $$unsorted) (V_Pa_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_d_2))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_k_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_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_f_2 $$unsorted) (V_S_2 $$unsorted) (V_T_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_c)))) (= (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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_c)))) (= (tptp.hAPP _let_1 V_T_2) (tptp.hAPP _let_1 V_S_2)))))))) (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_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OPow T_b) 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_b) V_A_2))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_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_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_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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_b) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage 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 T_c 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_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (not (= (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool))))) (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_A_2 $$unsorted) (T_b $$unsorted)) (not (exists ((B_f $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) B_f) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2)))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_c T_b V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.c_Set_OPow T_b) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) _let_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.hAPP (tptp.c_Set_OPow T_b) V_B_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_4 (tptp.tc_fun T_b 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 ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_b) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (=> (= (tptp.hAPP _let_1 V_A_2) V_B_2) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_fun T_c tptp.tc_HOL_Obool) (tptp.tc_fun T_b tptp.tc_HOL_Obool) _let_1) (tptp.hAPP (tptp.c_Set_OPow T_c) V_A_2)) (tptp.hAPP (tptp.c_Set_OPow T_b) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_OPow T_b))) (let ((_let_3 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)))))))) (forall ((V_j_2 $$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_na_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_na_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_na_2)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_m_2) (=> (tptp.c_Nat__Transfer_Ois__nat V_na_2) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_m_2 V_na_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_na_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_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_na_2)))))) (forall ((V_A_2 $$unsorted) (V_A_H_2 $$unsorted) (V_r_2 $$unsorted) (V_r_H_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (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_r_H_2) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b 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_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Relation_OImage T_b T_c V_r_H_2) V_A_H_2)) (tptp.hAPP (tptp.c_Relation_OImage T_b T_c V_r_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool)) V_R_2) V_S_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Relation_OImage T_c T_b V_R_2) V_A_2)) (tptp.hAPP (tptp.c_Relation_OImage T_c T_b V_S_2) V_A_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (=> (tptp.c_Nat__Transfer_Ois__nat V_x_2) (tptp.c_Nat__Transfer_Onat__set (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_x_2 V_y_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_a_2) V_b_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_a_2) V_b_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_b_2) V_a_2)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_c_2 V_d_2))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_d_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (=> (= V_a_2 V_b_2) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_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))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_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)))) (and (= V_a_2 V_b_2) (= V_b_2 V_c_2))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_a_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)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_x_2 V_y_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b (tptp.hAPP _let_1 V_y_2) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (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_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_c tptp.tc_HOL_Obool) _let_2 _let_1) (tptp.hAPP (tptp.c_Set_OPow T_c) V_A_2))) (tptp.hAPP (tptp.c_Set_OPow T_b) V_B_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_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_A_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_c T_b V_r_2) V_A_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (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_x) V_b_2)) V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_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_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_b T_c V_r_2) V_A_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_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_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_c_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_c_2 V_d_2))) (and (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP _let_2 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_d_2)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_b_2) V_d_2))))) (tptp.hBOOL (tptp.hAPP _let_2 V_d_2))))))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2))))))) (forall ((V_i_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_i_2 V_j_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_j_2) V_i_2)))) (and (=> _let_2 (= _let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Int_Oint) V_i_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_j_2)))))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_c T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))) (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) V_a_2) V_b_2)) V_r_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))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.class_Rings_Oring__1 T_b) (= (tptp.hAPP (tptp.c_Int_Oring__1__class_Oof__int T_b) (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_b) (tptp.hAPP _let_1 V_i_2)) (tptp.hAPP _let_1 V_j_2))))))) (= _let_45 (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_43 (tptp.hAPP (tptp.hAPP _let_42 (tptp.hAPP (tptp.hAPP _let_41 _let_19) _let_12)) _let_40)))) (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_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_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_44 (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_43 (tptp.hAPP (tptp.hAPP _let_42 (tptp.hAPP (tptp.hAPP _let_41 _let_12) _let_12)) _let_40)))) (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 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) V_x_2) V_y_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_x_2) V_y_2)))) (forall ((V_v_2 $$unsorted) (V_u_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)))) (let ((_let_4 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.c_Set_Oinsert _let_2))) (let ((_let_6 (tptp.c_Relation_OImage _let_2 _let_2 tptp.c_Int_Ointrel))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)) _let_3)))) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_u_2) V_v_2)) _let_3))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_v_2)) (tptp.hAPP (tptp.hAPP _let_1 V_u_2) V_y_2))))))))))) (forall ((V_Pa_2 $$unsorted) (V_na_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_na_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_m)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_na_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_Pa_2 $$unsorted) (V_na_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_na_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_m)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_na_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_A_2 $$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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_B_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_D_2)))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_OPowp T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) 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_b) V_A_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))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) (tptp.c_Predicate_OPowp T_b V_A_2)) (tptp.c_Predicate_OPowp T_b V_B_2)))))) (forall ((V_A_2 $$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_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_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_B_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_x) V_D_2)) V_t_2)))))))) (forall ((V_t_2 $$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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_xb) V_A_2)) (not (= B_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) B_xb) B_xa))))))) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_x)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) B_x) V_D_2))))))))) (forall ((V_z_2 $$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))))))))))) (forall ((V_Pa_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_Pa_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_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 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (exists ((B_x $$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_Pa_2 B_x)))))))) (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_x_2 $$unsorted) (V_y_2 $$unsorted) (V_c_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_2) V_c_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_x_2) V_y_2)))) (let ((_let_4 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_c_2) V_y_2)))) (and (=> _let_6 (= _let_3 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_c_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_c_2)))) (=> (not _let_6) (and (=> _let_5 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) _let_1))) (=> (not _let_5) (= _let_3 _let_1)))))))))))) (forall ((V_na_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_na_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_na_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Relation_OImage T_c T_b V_r_2) V_A_2))) (not (forall ((B_x $$unsorted)) (=> (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_x) V_b_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_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)))) (forall ((V_k_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_k_2)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b V_a_2) V_b_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_c_2) V_d_2)) (and (= V_a_2 V_c_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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_c_2) V_d_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_d_2)) (= V_a_2 V_c_2))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b 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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatMost T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_Pa_2 $$unsorted) (V_na_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_na_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_m)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_na_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_Pa_2 $$unsorted) (V_na_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_na_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_m)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_x) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_na_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_l_2) (tptp.hAPP tptp.c_Nat_OSuc V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_l_2 V_u_2))) (forall ((V_m_2 $$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_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_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_b_2) V_a_2)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2) V_b_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oorder T_b) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2) V_b_2)))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_b_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b 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_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_m_2) V_u_2)) (tptp.hAPP _let_1 V_u_2)))))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_i_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_i_2) V_na_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_m_2)) (tptp.hAPP _let_1 V_na_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_na_2) V_m_2)))))) (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_na_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_na_2))) (forall ((V_j_2 $$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_u_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_u_2)) (tptp.hAPP tptp.c_Nat_OSuc V_u_2))) (forall ((V_k_2 $$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_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_i_2) V_k_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatMost T_b))) (=> (tptp.class_Orderings_Oorder T_b) (= (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_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)))))) (forall ((V_m_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_m_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_m_2 V_u_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2))))))) (forall ((V_k_2 $$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_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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_x_2 V_y_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b (tptp.hAPP _let_1 V_y_2)) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_x_2) V_y_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b (tptp.hAPP _let_1 V_y_2) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_na_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_na_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_na_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP _let_1 V_na_2)))) (=> (not _let_3) (= _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))))))) (= (tptp.hAPP _let_36 _let_12) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) _let_12) _let_22)) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatMost T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_2)) (tptp.hAPP _let_1 V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_m_2) V_u_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) 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_b V_l_2) V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_u_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_m_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_m_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_m_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_u_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_u_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_l_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_u_2))))) (forall ((V_u_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_u_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_u_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_u_2))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_c))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b _let_1) V_g_2) (tptp.c_Groups_Oone__class_Oone T_c)))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (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 ((T_b $$unsorted) (V_B_2 $$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_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_x_2))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (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) 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 ((V_k_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_k_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint V_l_2) V_u_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (not (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (not (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_G_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (or (tptp.hBOOL (tptp.hAPP _let_1 V_F_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_G_2))) (tptp.hBOOL (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_F_2) V_G_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_b))) (let ((_let_2 (tptp.tc_fun T_b 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)) (=> (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) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_B_2) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c T_b)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) (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_C_2 $$unsorted) (T_b $$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_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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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_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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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.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_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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_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 ((T_b $$unsorted)) (=> (and (tptp.class_Lattices_Osemilattice__inf T_b) (tptp.class_Orderings_Olinorder T_b)) (= (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) (tptp.c_Orderings_Oord__class_Omin T_b)))) (forall ((V_a $$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 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_x)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_b)) V_x)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_x) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a) V_a) V_a))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_x) V_x))) (forall ((V_b $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x))))) (forall ((V_y $$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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_y $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x))) (let ((_let_2 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_c)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_z)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_z)))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_b)) V_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (tptp.class_Lattices_Osemilattice__inf T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) V_z) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c T_b)) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_A_2) V_A_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b 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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b 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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b 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_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (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) V_x_2)) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (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) V_x_2)) (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y $$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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) V_x))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)) (tptp.hAPP _let_1 V_z))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) V_x) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x))))))) (forall ((V_x $$unsorted) (V_z $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a))) (=> (tptp.class_Lattices_Odistrib__lattice T_a) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_y) V_z)) V_x) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 V_C_2))))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2))) (tptp.hAPP (tptp.hAPP _let_3 V_C_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_C_2))) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) _let_2) V_B_2) _let_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) _let_2) _let_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_B_2)) (not (= B_x B_xa))))))))) (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 $$unsorted) (T_a $$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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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 _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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) V_B_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$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 _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) (V_C_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_D_2 $$unsorted) (V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_D_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 V_C_2) V_D_2))))))))) (forall ((V_b $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2))) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) 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_b $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) V_y_2) V_x_2)))) (forall ((V_y $$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 $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) V_x)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_c_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_l_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)))) (forall ((V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.c_member T_b))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) _let_3))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_4 V_R_2)) (tptp.hAPP _let_4 V_S_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_x_2) (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2))))))))) (forall ((V_u_2 $$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)) (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_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_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_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_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_x) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_y))))) (forall ((V_y $$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 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) (tptp.hAPP _let_1 V_x)) (tptp.hAPP _let_1 V_y)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.hAPP _let_1 V_B_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_B_2)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP _let_3 _let_2))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_B_2)) V_C_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_C_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP _let_3 _let_2))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_C_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_C_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_B_2)) V_C_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (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_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c 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 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_B_2) V_A_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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_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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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))) (= (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 $$unsorted) (T_b $$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))) (= (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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)) V_A_2))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_m_2) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_c T_b V_R_2))) (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_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c 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 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_na_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_na_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_na_2))) (= (tptp.hAPP _let_37 _let_12) _let_22) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat))) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_k_2 $$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))))) (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.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_u_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) _let_1) V_u_2)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint _let_1) V_u_2) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat V_u_2))))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_i_2) V_k_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_a_2) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2)) V_a_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (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_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_c 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) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_A_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_B_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_b_2) V_d_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_c_2) V_d_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_b_2) V_d_2))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (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_m_2)) (tptp.hAPP _let_1 V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_m_2) V_na_2)))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b 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_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_b_2) V_d_2))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_m_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_m_2) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_a_2 V_b_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_b_2) V_d_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_b_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_b_2) V_d_2))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_b_2) V_d_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_m_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_m_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_m_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_l_2)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (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_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_c))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c 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_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_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) V_A_2)) V_a_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) V_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_u_2) V_l_2))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint V_l_2) V_u_2))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= 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_Orderings_Oord__class_Oless__eq T_b) V_x_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) V_A_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) B_x))))))))) (forall ((V_u_2 $$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_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_c))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (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.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite 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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))) (let ((_let_3 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) _let_2)) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 _let_2)))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Int_Oint (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_l_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))) V_u_2) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Int_Oint V_l_2 V_u_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b))) (=> (tptp.c_Finite__Set_Ofolding__one 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)) (=> (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) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding__one T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) _let_2) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_c))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))))) (forall ((V_k_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_k_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) _let_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_k_2)) _let_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_u_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_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_u_2)) (tptp.hAPP _let_1 V_u_2)))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (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_c) (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 $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_2))) (let ((_let_4 (tptp.c_Set_Oimage T_b T_c V_f_2))) (= (tptp.c_Fun_Oinj__on T_b T_c 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_b T_c V_f_2 V_A_2) (tptp.c_Fun_Oinj__on T_b T_c 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_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b _let_2) V_g_2) (tptp.c_Groups_Oone__class_Oone T_c)))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2))) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Finite__Set_Ofold1 T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Orderings_Obot__class_Obot _let_3))) (let ((_let_5 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Olattice T_b) (=> (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_b (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_b) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) _let_2) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_2 (tptp.tc_fun T_b 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_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b (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_b _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olinorder__class_OMax T_b _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b (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_b) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2)) (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (not (= V_A_2 _let_3)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= V_B_2 _let_3)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) _let_3) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_B_2)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_2 (tptp.tc_fun T_b 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_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b _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_b _let_1 _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_B_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b _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_b _let_1 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 V_B_2)))))))))))))) (forall ((V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc) (tptp.hAPP _let_1 V_na_2)))))) (forall ((V_na_2 $$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_na_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_na_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_na_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) V_d_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_b))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_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.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_b 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_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Fields_Ofield T_c) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_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))) (not (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c))))) (= (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_c) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (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_f_2 $$unsorted) (V_T_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_c))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b _let_1) V_f_2) (tptp.c_Groups_Oone__class_Oone T_c)))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_S_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_c)))) (= (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_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))))))))))) (= (tptp.c_Lattices_Osemilattice__inf__class_Oinf tptp.tc_Nat_Onat) (tptp.c_Orderings_Oord__class_Omin tptp.tc_Nat_Onat)) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oab__group__add T_b) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_c) _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_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.c_COMBK T_b T_c) _let_1)) V_A_2) _let_1)))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _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_B_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.tc_fun T_d T_b))) (let ((_let_4 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_d T_b))) (let ((_let_5 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (=> (tptp.class_Rings_Osemiring__0 T_b) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_6 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_4 V_g_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_5 T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_5 T_b) T_c) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_3 _let_3) T_c) (tptp.c_COMBB T_b T_b T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_2 T_c) _let_1) V_f_2))) V_g_2))) V_B_2)) V_A_2)))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (=> (tptp.class_Rings_Osemiring__0 T_b) (= (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_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _let_1) V_f_2)) V_r_2)) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_r_2))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (=> (tptp.class_Rings_Osemiring__0 T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_c) _let_1) V_f_2)) V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_d T_b))) (let ((_let_4 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_4 T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d T_b) (tptp.tc_fun _let_4 T_b) T_c) _let_3) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_2 T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_b) (tptp.tc_fun _let_2 T_b) T_d) _let_1) (tptp.hAPP (tptp.c_COMBC T_c T_d T_b) V_f_2))) V_A_2)) V_B_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c tptp.tc_Nat_Onat) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b T_c) _let_1) V_f_2)) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Oring__1__class_Oof__int T_b))) (=> (tptp.class_Rings_Oring__1 T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c tptp.tc_Int_Oint) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_b T_c) _let_1) V_f_2)) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_Groups_Oab__group__add T_b) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _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_r_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Oinverse__class_Odivide T_b))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (=> (tptp.class_Fields_Ofield T_b) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2)) V_r_2) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _let_1) V_f_2)) V_r_2)) V_A_2)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (forall ((V_g_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) V_g_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_c))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2) V_A_2) (tptp.c_Groups_Ozero__class_Ozero T_c))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_4 (tptp.c_member _let_1))) (let ((_let_5 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_fun T_c _let_5))) (let ((_let_7 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_2 _let_7) T_b) (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_6 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_c) _let_4)) _let_3))))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b _let_7)) (tptp.hAPP _let_8 V_R_2)) (tptp.hAPP _let_8 V_S_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_R_2) V_S_2))))))))))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_c))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) _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_na_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_na_2))) (tptp.hAPP V_f_2 V_na_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Nat_Onat) V_f_2) tptp.c_Nat_OSuc)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) V_na_2)))))) (forall ((V_na_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$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_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (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_na_2) V_k_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b 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_na_2))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (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_na_2))) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b 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_na_2)))))) (forall ((V_na_2 $$unsorted) (V_k_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP _let_2 V_f_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2) (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_k_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Nat_Onat) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) V_k_2))) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_na_2))))))) (forall ((V_f_2 $$unsorted) (V_p_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$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_b) V_f_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m_2) V_na_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_na_2) V_p_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_na_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_na_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_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2))) (let ((_let_2 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat))) (=> (tptp.class_Groups_Oab__group__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_m_2) V_na_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_na_2) V_p_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_p_2))) (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_na_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_na_2) V_p_2)))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_c))) (let ((_let_2 (tptp.tc_fun T_c T_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_2) T_b) (tptp.c_If T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b 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_b) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_c))) (let ((_let_2 (tptp.tc_fun T_c T_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_2) T_b) (tptp.c_If T_c)) (tptp.hAPP tptp.c_fequal V_a_2))) V_b_2)) _let_1)) V_S_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_S_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (and (=> _let_4 (= _let_3 (tptp.hAPP V_b_2 V_a_2))) (=> (not _let_4) (= _let_3 _let_1)))))))))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.c_COMBK T_b T_c) V_y_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_A_2))) V_y_2)))) (forall ((V_na_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_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_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_na_2))) (tptp.hAPP V_f_2 _let_1)))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_F_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_F_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP _let_1 V_F_2)))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (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 _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_c))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_3) V_A_2) V_B_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Oab__group__add T_c) (=> (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 _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_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b (tptp.c_Groups_Oplus__class_Oplus T_c)) V_g_2) (tptp.c_Groups_Ozero__class_Ozero T_c)) V_A_2))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_c))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_c T_b (tptp.c_Groups_Oplus__class_Oplus T_c)) V_g_2) _let_1) V_A_2))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (= (tptp.hAPP V_f_2 _let_1) (tptp.c_Groups_Ozero__class_Ozero T_b)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc _let_1)) V_k_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat _let_1) V_k_2)))))))) (forall ((V_f_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2) V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_f_2 V_m_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2)) V_na_2)))))))) (forall ((V_f_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)) (= (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_f_2 V_m_2)) (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc V_m_2) V_na_2)))))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c T_c))) (let ((_let_3 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_2) T_b) (tptp.c_If T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_B_2))) V_f_2)) (tptp.c_Groups_Ozero__class_Ozero T_c))) V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (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_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Oab__group__add T_c) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2))))))))))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat V_m_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2))))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) V_m_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (and (=> _let_4 (= _let_3 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_na_2))) (tptp.hAPP V_f_2 V_na_2))))))))))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 _let_1)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_1) V_m_2)))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (and (=> _let_4 (= _let_3 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_na_2))) (tptp.hAPP V_f_2 _let_1))))))))))) (forall ((V_Pa_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_Pa_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)) (= (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.hAPP V_Pa_2 V_m_2)) (tptp.hAPP _let_1 (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_m_2 V_na_2)))))))) (forall ((V_p_2 $$unsorted) (V_f_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_na_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_b) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (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_b) (tptp.hAPP _let_4 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_na_2))) (tptp.hAPP _let_4 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat _let_3 _let_2))))))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (tptp.hAPP V_g_2 V_x_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (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_a_2) V_A_2)) (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (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_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Rings_Oring T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP V_f_2 V_a_2))))))))) (forall ((V_f_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_b 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_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)))) (=> (tptp.class_Groups_Oab__group__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_c) _let_2) (tptp.hAPP V_f_2 V_a_2)))) (=> (not _let_5) (= _let_4 _let_2))))))))))) (forall ((V_na_2 $$unsorted) (V_d_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_4 (tptp.hAPP _let_3 V_a_2))) (let ((_let_5 (tptp.c_Groups_Oone__class_Oone T_b))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_b) (= (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_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b tptp.tc_Nat_Onat) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) tptp.tc_Nat_Onat) _let_2) _let_1)) V_d_2))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_na_2))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_na_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_na_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) V_d_2)))))))))))) (forall ((V_na_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_b) V_x_2))) (=> (tptp.class_Fields_Ofield T_b) (=> (not (= V_x_2 _let_1)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) _let_3) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Oinverse__class_Odivide T_b) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_na_2)) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1))))))))) (forall ((V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_b))) (let ((_let_2 (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_na_2))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_b) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 _let_1) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) _let_2) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) V_na_2))) (tptp.hAPP (tptp.hAPP _let_5 _let_3) (tptp.hAPP (tptp.hAPP _let_4 _let_3) _let_1)))))))))) (forall ((V_f_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_b))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) tptp.tc_Nat_Onat) _let_3) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Nat_Onat) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_2) _let_1)))) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_m_2 V_na_2)))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)))) (=> (tptp.class_Groups_Oab__group__add T_b) (and (=> _let_5 (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP V_f_2 V_m_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP _let_2 V_na_2) _let_1))))) (=> (not _let_5) (= _let_4 (tptp.c_Groups_Ozero__class_Ozero T_b))))))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (=> (tptp.class_Groups_Oordered__comm__monoid__add T_c) (=> (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_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (forall ((B_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_b) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP V_f_2 B_b))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_g_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (= (tptp.hAPP V_g_2 B_x) (tptp.c_Groups_Oone__class_Oone T_c)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))))) (forall ((V_f_2 $$unsorted) (V_T_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_S_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone T_c)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (tptp.hAPP _let_1 V_S_2)) (tptp.hAPP _let_1 V_T_2)))))))))) (forall ((V_K_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (and (tptp.class_Groups_Oordered__ab__semigroup__add T_c) (tptp.class_Rings_Osemiring__1 T_c)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_i) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP V_f_2 B_i)) V_K_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_c) (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_c) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))) V_K_2)))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_A_2 V_x_2) V_y_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_x_2) V_y_2)))) (forall ((V_f_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_f_2) V_F_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_F_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBK tptp.tc_Nat_Onat T_b) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_A_2))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$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_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 tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_b) _let_1) V_f_2)) V_A_2)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_f_2) V_A_2) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_A_2)) (=> (not (= B_x B_xa)) (= (tptp.hAPP V_f_2 B_xa) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (= (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.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_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_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_f_2) V_A_2) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Int_Oint T_b) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint)) V_f_2)) V_A_2)))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Ocomm__monoid__mult T_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b 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) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_f_2))) (let ((_let_2 (tptp.hAPP _let_1 V_A_2))) (let ((_let_3 (tptp.tc_fun T_b 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_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_3)))))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) V_f_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite 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 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_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Ocomm__monoid__mult T_b)) (=> (tptp.c_Nat__Transfer_Onat__set V_A_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_b) V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint T_b tptp.tc_Nat_Onat) V_f_2) (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Int_Oint tptp.tc_Nat_Onat tptp.c_Int_Onat) V_A_2)))))) (forall ((V_na_2 $$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_na_2))) (tptp.hAPP (tptp.hAPP _let_1 V_na_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_na_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_d_2)))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_b 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_b tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Nat_Onat T_b) tptp.c_Int_Onat) V_f_2)) V_A_2))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_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) (tptp.c_Groups_Ozero__class_Ozero T_c)))) (= (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_Groups_Oplus__class_Oplus T_c) (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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_S_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c)))) (= (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_S_2) V_T_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (tptp.hAPP _let_1 V_S_2)) (tptp.hAPP _let_1 V_T_2)))))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_c) (tptp.class_Groups_Oordered__ab__semigroup__add T_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) _let_2)))) (let ((_let_4 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2))) (let ((_let_5 (= _let_3 _let_2))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (and (=> _let_5 (= _let_4 V_x_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b _let_3)))))))))))))) (forall ((V_i_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (V_s_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_s_2)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_i) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP V_f_2 B_i))))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2) V_s_2) V_B_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP V_f_2 V_i_2)) V_B_2)))))))) (forall ((V_i_2 $$unsorted) (V_f_2 $$unsorted) (V_s_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_c))) (=> (tptp.class_Groups_Oordered__ab__group__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_s_2)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_i) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP V_f_2 B_i))))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2) V_s_2) _let_1) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) V_s_2)) (= (tptp.hAPP V_f_2 V_i_2) _let_1)))))))) (forall ((V_na_2 $$unsorted) (V_Pa_2 $$unsorted) (V_Q_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_na_2))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (forall ((B_x $$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_Pa_2 B_x)))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_3 V_Pa_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_Pa_2)) V_Q_2)) _let_1))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_x_2))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_a_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)) V_a_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2))) (=> (tptp.class_Lattices_Olattice T_b) (=> (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 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) _let_1) _let_1)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) V_A_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_B_2)) _let_1) _let_1)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (not (= V_A_2 _let_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (=> (not (= V_B_2 _let_2)) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_B_2)))))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (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))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP _let_3 _let_2)))) (let ((_let_5 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP _let_3 V_A_2)))) (let ((_let_6 (= _let_4 _let_2))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (=> _let_6 (= _let_5 V_x_2)) (=> (not _let_6) (= _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b _let_4)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Olattice T_b) (=> (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_b (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_b) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_B_2))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2))) (let ((_let_4 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_4 V_B_2)) (=> (not (= _let_3 (tptp.c_Orderings_Obot__class_Obot _let_2))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2))) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b _let_3)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_B_2)))))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) B_x) B_y)) (tptp.hAPP (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP _let_1 B_y) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)) V_A_2))))))) (forall ((V_N_2 $$unsorted) (V_h_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b))) (= (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_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP V_h_2 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_N_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_N_2)))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (V_T_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c)))) (= (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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_T_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_T_2) V_S_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c)))) (= (tptp.hAPP _let_1 V_S_2) (tptp.hAPP _let_1 V_T_2)))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_c) (tptp.class_Groups_Oordered__cancel__ab__semigroup__add T_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$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 T_c) (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_c) (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_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_na_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_na_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (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_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod tptp.tc_Int_Oint T_b))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Ocomm__monoid__mult T_b)) (=> (= V_A_2 V_B_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (=> (forall ((B_x $$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_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeast T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_k_2) V_i_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeast T_b))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_y_2) V_x_2)))))) (forall ((V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_k_2)))) (forall ((V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_k_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeast T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_l_2) V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_a_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_c_2 V_d_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_a_2) V_c_2) V_d_2)))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_a_2 V_b_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_c_2)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_a_2) V_c_2) V_b_2)))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_SetInterval_Oord__class_OatLeast tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_2 (tptp.hAPP tptp.c_Nat_OSuc V_k_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_2 V_k_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_k_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_l_2))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_l_2))))) (forall ((V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_l_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeast T_b) V_l_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (let ((_let_2 (tptp.c_Groups_Osgn__class_Osgn T_a V_x))) (let ((_let_3 (tptp.c_Groups_Ozero__class_Ozero T_a))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_3) V_x)))) (let ((_let_5 (= V_x _let_3))) (=> (tptp.class_Groups_Osgn__if T_a) (and (=> _let_5 (= _let_2 _let_3)) (=> (not _let_5) (and (=> _let_4 (= _let_2 _let_1)) (=> (not _let_4) (= _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1))))))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OgreaterThan T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (= (tptp.hAPP _let_1 V_x_2) (tptp.hAPP _let_1 V_y_2)) (= V_x_2 V_y_2))))) (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_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) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (= (tptp.c_Groups_Osgn__class_Osgn T_b V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Groups_Osgn__if T_a) (= (tptp.c_Groups_Osgn__class_Osgn T_a _let_1) _let_1)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.c_Groups_Osgn__class_Osgn T_a (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Groups_Osgn__class_Osgn T_a V_a)) (tptp.c_Groups_Osgn__class_Osgn T_a V_b)))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Groups_Osgn__class_Osgn T_b V_a_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Groups_Osgn__class_Osgn T_b V_a_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_a_2)))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_k_2) V_i_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OgreaterThan T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_x_2)) (tptp.hAPP _let_1 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_y_2) V_x_2)))))) (forall ((V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_k_2)))) (forall ((V_k_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_k_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost T_b) V_k_2)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (= (tptp.c_Groups_Osgn__class_Osgn T_b V_a_2) (tptp.c_Groups_Oone__class_Oone T_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Groups_Ozero__class_Ozero T_b)) V_a_2))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (= (tptp.c_Groups_Osgn__class_Osgn T_a V_a) (tptp.c_Groups_Oone__class_Oone T_a))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OatLeastAtMost T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OgreaterThan T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_l_2) V_u_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_2)) (tptp.hAPP _let_1 V_u_2)) (tptp.hAPP _let_1 V_l_2)))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add T_b) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b _let_1) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan T_b) V_x_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) (tptp.hAPP _let_1 V_x_2)))))) (forall ((V_u_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Oorder T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost T_b V_l_2 V_u_2)) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OgreaterThan T_b) V_u_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_k_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP tptp.c_Nat_OSuc V_k_2))) (let ((_let_3 (tptp.c_SetInterval_Oord__class_OgreaterThan tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) (tptp.hAPP _let_3 V_k_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) _let_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_i $$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_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_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (= (tptp.c_Groups_Osgn__class_Osgn T_b V_a_2) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) (tptp.c_Groups_Oone__class_Oone T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_b)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_b))) (let ((_let_4 (tptp.c_Relation_OImage T_b T_b V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_b))) (let ((_let_4 (tptp.c_Relation_OImage T_b T_b V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_b_2) _let_2))) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_a_2) _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)) V_r_2)))))))))) (forall ((V_code__numeral_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__case T_b V_f1_2 V_f2_2 (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_2)) (tptp.hAPP V_f2_2 V_code__numeral_2))) (forall ((V_z $$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_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_b))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (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_b T_b T_c) _let_1) V_f_2)) V_A_2))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (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__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_c) (tptp.c_Groups_Oabs__class_Oabs T_b)) V_f_2)) V_A_2))))) (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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_c) _let_1) V_f_2)) V_A_2))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_b))) (let ((_let_2 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b))) (=> (tptp.class_Fields_Olinordered__field T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_c) _let_1) V_f_2)) V_A_2)))))) (forall ((V_b $$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 ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1) _let_1)))) (forall ((V_a $$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) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (= (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_b) V_a_2) _let_1) (= V_a_2 _let_1))))) (forall ((T_a $$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 ((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))))) (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_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Power_Opower__class_Opower T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_n)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) V_n)))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osemiring__1__class_Oof__nat T_a) V_n))) (=> (tptp.class_Rings_Olinordered__idom T_a) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1) _let_1)))) (forall ((V_m $$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_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 $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a))))) (forall ((V_a $$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_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_b) V_a_2)) _let_1)) (= V_a_2 _let_1))))) (forall ((V_a $$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 $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) V_a)) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a) V_a)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) _let_1) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_b) V_a_2))) (not (= V_a_2 _let_1)))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) (tptp.c_Groups_Ozero__class_Ozero T_a)))))) (forall ((V_b $$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_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 _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 $$unsorted) (V_a $$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 (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_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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_b) V_a_2)) V_b_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_a_2)) V_b_2))))))) (forall ((V_b $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_b) 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_b) V_a_2)) V_b_2))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (let ((_let_2 (tptp.c_Rings_Oinverse__class_Odivide T_a))) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (not (= V_b (tptp.c_Groups_Ozero__class_Ozero T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_a) V_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b)))))))) (forall ((V_n $$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_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 ((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_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_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_a) V_a) V_b))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) (tptp.hAPP _let_1 V_a)) (tptp.hAPP _let_1 V_b))))))) (forall ((V_d $$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_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)) (=> (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 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)) (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_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Olinordered__idom T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) V_a)) V_n))))) (forall ((V_x $$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 ((T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_z_2 $$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_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_i))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_i) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_i))) (=> (not _let_2) (= _let_1 V_i)))))) (forall ((V_z $$unsorted) (V_w $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint))) (= (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint) V_w) V_z))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_w))) (tptp.hAPP tptp.c_Int_Onat (tptp.hAPP _let_1 V_z)))))) (forall ((V_n $$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_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_a_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) V_x))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) _let_1))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) _let_1) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) _let_1))) (=> (not _let_3) (= _let_2 _let_1)))))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Int_Onumber__ring T_a) (tptp.class_Rings_Olinordered__idom T_a)) (= (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of T_a) tptp.c_Int_OMin)) V_n)) (tptp.c_Groups_Oone__class_Oone T_a)))) (forall ((V_na_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) V_x_2)) V_na_2))) (or (not (= V_x_2 _let_1)) (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_b))) (let ((_let_3 (tptp.c_Relation_OImage T_b T_b V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_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_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_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)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert T_b))) (let ((_let_4 (tptp.c_Relation_OImage T_b T_b V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2)) (and (= (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) _let_2)) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) _let_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)))))))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_b))) (let ((_let_3 (tptp.c_Relation_OImage T_b T_b 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_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) V_A_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_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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_b))) (let ((_let_3 (tptp.c_Relation_OImage T_b T_b V_r_2))) (let ((_let_4 (tptp.c_member T_b))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 V_y_2) V_A_2)) (= (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_b))) (let ((_let_4 (tptp.c_Relation_OImage T_b T_b V_r_2))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_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_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_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__case T_b V_f1_2 V_f2_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)) V_f1_2)) (forall ((V_x $$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_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_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_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_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_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_b V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_d) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2) (tptp.hAPP V_f_2 V_y_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_b V_f_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_d) V_x_2))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Osnd T_b T_d) V_x_2))) (= (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_b V_f_2) V_x_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_b V_g_2) V_x_2)) (= (tptp.hAPP V_f_2 _let_1) (tptp.hAPP V_g_2 _let_1))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_b) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_b T_c V_f_2) V_x_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) V_x_2)))) (= tptp.c_Divides_OnegateSnd (tptp.c_Product__Type_Oapsnd tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint _let_39)) (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_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))))))) (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_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)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Equiv__Relations_Oquotient T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_b _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_b) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_2)))) V_r_2) V_A_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_r_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oinsert T_b))) (let ((_let_3 (tptp.c_Equiv__Relations_Oquotient T_b))) (let ((_let_4 (tptp.c_member T_b))) (=> (tptp.c_Equiv__Relations_Oequiv T_b 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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2))))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) V_A_2) V_r_2)) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) (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_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (let ((_let_3 (tptp.c_member _let_1))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_X_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_Y_2) _let_2)) (or (= V_X_2 V_Y_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_X_2) V_Y_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_b))) (let ((_let_4 (tptp.hAPP (tptp.hAPP _let_3 V_a_2) _let_2))) (let ((_let_5 (tptp.c_Equiv__Relations_Oquotient T_b))) (let ((_let_6 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_7 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_7 _let_6) T_b) _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) _let_3) _let_2))) V_r_2) V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) _let_4)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_6) (tptp.hAPP (tptp.hAPP _let_5 V_A_2) V_r_2)) (tptp.hAPP (tptp.hAPP _let_5 _let_4) V_r_2))))))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (let ((_let_3 (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Equiv__Relations_Oequiv T_b 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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (let ((_let_3 (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_X_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_Y_2) _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_X_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_Y_2)) (= (= V_X_2 V_Y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2)))))))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) _let_2) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r_2) _let_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))) (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))))))) (forall ((V_Pa_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nitpick_Onat__gcd__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat) B_x) B_y))) (=> (=> (not (= B_y (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_y) (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat B_x B_y)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_x) B_y))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_a0_2) V_a1_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (V_X_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_X_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (not (forall ((B_x $$unsorted)) (=> (= V_X_2 (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_Set_Oinsert _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (tptp.c_Relation_OImage T_b T_b V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))) (forall ((T_c $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_B_2 B_x)))))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_c T_b V_r_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c 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_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Relation_OImage T_c T_b V_r_2))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_2 V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_3) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 _let_1) (tptp.c_Set_Oinsert T_c)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_B_2 $$unsorted) (V_b_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_c) 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))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_OPow T_b))) (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_c _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) _let_2) V_B_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) V_B_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_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_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun _let_1 _let_1) T_c) (tptp.c_Set_Oinsert T_b)) V_f_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_r2_2 $$unsorted) (V_A2_2 $$unsorted) (T_c $$unsorted) (V_r1_2 $$unsorted) (V_A1_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A1_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Oequiv T_c V_A2_2 V_r2_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_b T_c _let_1 V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a1_2) V_A1_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_a2_2) V_A2_2)) (= (tptp.hAPP (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_r1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a1_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_1 T_b) (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_c T_c V_r2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a2_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))) V_f_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_a1_2) V_a2_2))))))))) (forall ((V_Pa_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) (tptp.hAPP V_B_2 B_x))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_xa)))))))) (forall ((V_k_2 $$unsorted) (V_M_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun T_b 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_na_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_na_2) V_k_2))) V_M_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c T_b) V_A_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d T_b) V_B_2))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d T_b) T_b T_c) _let_2) V_f_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_b) T_b T_d) _let_1) (tptp.hAPP (tptp.c_COMBC T_c T_d T_b) V_f_2)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_2))) (let ((_let_4 (tptp.tc_fun T_d _let_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_COMBC T_c _let_4 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_4 _let_2) T_c) _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_c _let_1) V_A_2) V_B_2)) V_C_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_c _let_1))) (let ((_let_4 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_4) V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_1) T_d) _let_2) V_B_2)) V_C_2)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_c _let_1))) (let ((_let_4 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_4) V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_1) T_d) _let_2) V_B_2)) V_C_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 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_b T_c T_d V_r1_2 V_r2_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) V_y1_2) V_z1_2)) V_r1_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_c) 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_M_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_na_2)) V_M_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) V_na_2)) V_M_2)) (tptp.hAPP V_M_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_u_2 $$unsorted) (V_M_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (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_OSUPR T_c T_b) V_A_2) V_M_2)) V_u_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_M_2 B_x)) V_u_2))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (tptp.class_Orderings_Olinorder T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c T_b) V_A_2) V_f_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2) (tptp.hAPP V_f_2 B_x)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP V_B_2 B_x))))))) (forall ((V_M_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK T_c T_b) V_M_2)) V_M_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (forall ((V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (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) V_c_2)))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b 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_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_c) _let_2)) _let_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (= (tptp.c_Orderings_Obot__class_Obot _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_I_2) V_A_2)) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_I_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_A_2 B_x)) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_d T_c V_f_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_1 T_d) V_B_2) V_f_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c 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_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (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_d _let_1 T_c) V_B_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_c T_d V_f_2) V_A_2)) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_I_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_M_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c 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 $$unsorted) (V_J_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_3 (tptp.tc_fun _let_1 _let_1))) (let ((_let_4 (tptp.tc_fun T_d _let_1))) (let ((_let_5 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_1) V_J_2))) (let ((_let_6 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_I_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_6 V_A_2)) (tptp.hAPP _let_5 V_B_2)) (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_1 T_c) _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_4) T_c) (tptp.c_COMBB _let_1 _let_1 T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) _let_2) V_A_2))) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (T_b $$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_OSUPR T_c _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_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_2) V_A_2)) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_B_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_I_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) _let_2) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$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_OSUPR T_c _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_c) _let_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_C_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$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_c _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_c) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c (tptp.tc_fun T_d T_b)) V_A_2) V_f_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c T_b) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_d T_b) V_f_2) V_x_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_C_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (T_b $$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_OSUPR T_c _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_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_2) V_A_2)) V_B_2))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_c))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_f_2)) (tptp.hAPP (tptp.hAPP _let_1 V_B_2) V_f_2))))))) (forall ((V_M_2 $$unsorted) (V_A_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP V_M_2 V_i_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_c) V_A_2) V_M_2)))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_u_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_u_2) 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 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP V_B_2 V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (V_I_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_I_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_3) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1))) V_A_2))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _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_b 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_b) _let_1) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_3) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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__sup__class_Osup _let_1))) (let ((_let_4 (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)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b 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_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 V_A_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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_b _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_b 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_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_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.hAPP V_B_2 V_a_2)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_B_2)))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_r2_2 $$unsorted) (V_A2_2 $$unsorted) (T_c $$unsorted) (V_r1_2 $$unsorted) (V_A1_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A1_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Oequiv T_c V_A2_2 V_r2_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_b T_c _let_1 V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_a_2) V_A2_2)) (tptp.c_Equiv__Relations_Ocongruent T_b _let_1 V_r1_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_1 T_b) (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_c T_c V_r2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))) V_f_2)))))))) (forall ((V_z_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 _let_2))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (let ((_let_5 (tptp.tc_fun tptp.tc_Nat_Onat _let_2))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_z_2) (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_2) (tptp.hAPP tptp.c_Int_ORep__Integ V_z_2)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_5 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.c_Relation_OImage _let_1 _let_1 tptp.c_Int_Ointrel))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) tptp.tc_Nat_Onat) (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Nat_Onat _let_1) _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 tptp.tc_Nat_Onat) (tptp.c_Set_Oinsert _let_1))) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_1) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat))))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.c_Equiv__Relations_Ocongruent T_b _let_1 V_r_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) V_f_2) (tptp.hAPP V_f_2 V_a_2))))))) (forall ((T_d $$unsorted) (T_c $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 V_a_2) V_b_2) V_c_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c (tptp.tc_fun T_d tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Int_ORep__Integ V_x_2) (tptp.hAPP tptp.c_Int_ORep__Integ V_y_2)) (= V_x_2 V_y_2))) (forall ((V_x_2 $$unsorted)) (= (tptp.hAPP tptp.c_Int_OAbs__Integ (tptp.hAPP tptp.c_Int_ORep__Integ V_x_2)) V_x_2)) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Ocongruent T_b T_c 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) 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_z_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun _let_1 _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat _let_2))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_1))) (= (tptp.hAPP tptp.c_Int_Onat V_z_2) (tptp.c_Set_Othe__elem tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) (tptp.hAPP tptp.c_Int_ORep__Integ V_z_2)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_1 _let_4) tptp.tc_Nat_Onat) (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_3 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_2 tptp.tc_Nat_Onat) (tptp.c_Set_Oinsert tptp.tc_Nat_Onat))) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat)))) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_r2_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (V_r1_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_b T_c T_d V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.c_Equiv__Relations_Ocongruent T_c T_d V_r2_2 (tptp.hAPP V_f_2 V_a_2)))))) (forall ((V_Y_2 $$unsorted) (V_X_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (let ((_let_2 (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_3))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.c_Equiv__Relations_Ocongruent T_b _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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_d $$unsorted) (V_r2_2 $$unsorted) (V_A2_2 $$unsorted) (T_c $$unsorted) (V_r1_2 $$unsorted) (V_A1_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A1_2 V_r1_2) (=> (tptp.c_Equiv__Relations_Oequiv T_c V_A2_2 V_r2_2) (=> (tptp.c_Equiv__Relations_Ocongruent2 T_b T_c _let_1 V_r1_2 V_r2_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_X1_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A1_2) V_r1_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_X2_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_c) V_A2_2) V_r2_2))) (=> (forall ((B_x1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x1) V_A1_2)) (forall ((B_x2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x2) V_A2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_d tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP V_f_2 B_x1) B_x2)) V_B_2)))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_X1_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_1 T_b) (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_X2_2)) V_f_2))) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_X_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (tptp.c_Equiv__Relations_Ocongruent T_b _let_1 V_r_2 V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_X_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP V_f_2 B_x)) V_B_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_X_2) V_f_2)) V_B_2)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c (tptp.tc_fun T_d tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2) V_c_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Relation_OId__on T_b V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_Set_Oinsert _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b _let_1) (tptp.c_Product__Type_OPair T_b T_b)) (tptp.c_COMBI T_b)))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_b T_c V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_e T_c T_d V_g_2) V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_d T_e) V_x_2))) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_d T_e) V_x_2))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_b V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_e T_b T_d V_g_2) V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_e T_d) V_x_2))) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_e T_d) V_x_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_b T_c V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_d T_c) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) (tptp.hAPP V_f_2 V_x_2)) V_y_2))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_Ofst T_d T_c) V_x_2))) (= (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_b T_c V_f_2) V_x_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_b T_c V_g_2) V_x_2)) (= (tptp.hAPP V_f_2 _let_1) (tptp.hAPP V_g_2 _let_1))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_b T_c V_f_2) V_x_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_d T_c) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_b) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_c T_b V_f_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_d T_b) V_x_2))) (forall ((V_p_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_b V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_e T_b T_d V_g_2) V_p_2)) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_e T_b T_c V_g_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_e V_f_2) V_p_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b _let_1))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Relation_OId__on T_b V_A_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b 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_b) (tptp.c_COMBC T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 T_b) tptp.c_fconj)) tptp.c_fequal))) V_A_2))))))) (forall ((T_b $$unsorted)) (= (tptp.c_Relation_OId__on T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_b T_b (tptp.c_Relation_OId__on T_b 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)) V_A_2) V_B_2))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (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_x_2) V_y_2)) (tptp.c_Relation_OId__on T_b V_A_2))) (and (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_b $$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_b) V_a_2) V_A_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_a_2) V_b_2)) (tptp.c_Relation_OId__on T_b V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) V_c_2) (tptp.c_Relation_OId__on T_b V_A_2))) (not (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (not (= V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Oboolean__algebra T_a) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_a) V_x) V_y) (tptp.c_Orderings_Obot__class_Obot T_a)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a) V_x) V_y) (tptp.c_Orderings_Otop__class_Otop T_a)) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_a) V_x) V_y))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_b))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_b) (tptp.class_Groups_Ocomm__monoid__mult T_b)) (=> (= V_A_2 V_B_2) (=> (tptp.c_Nat__Transfer_Onat__set V_B_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x)))) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_A_2) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_B_2)))))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Otop T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.c_Orderings_Otop__class_Otop T_a))))) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) _let_17))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_OPow T_b) (tptp.c_Orderings_Otop__class_Otop _let_1)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))) (= (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_19) (forall ((T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Sum__Type_OPlus T_b T_c (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_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun (tptp.tc_sum T_b T_c) tptp.tc_HOL_Obool)))) (forall ((V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Otop T_b) (= (tptp.hAPP (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c T_b)) V_x_2) (tptp.c_Orderings_Otop__class_Otop T_b)))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) _let_2) V_B_2) _let_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) _let_2) _let_2)))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (not (= (tptp.c_Orderings_Otop__class_Otop _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((V_A_2 $$unsorted) (T_b $$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 _let_1) V_A_2) (tptp.c_Orderings_Otop__class_Otop _let_1))))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_c))) (=> (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_c) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_Option_Ooption T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Finite__Set_Ofinite T_b) (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_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_sum T_b T_c))) (= (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_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 T_c) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_d T_b T_c) V_f_2) V_g_2)) _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_d T_b V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_d V_g_2) _let_1))))) (tptp.c_Equiv__Relations_Oequiv _let_24 _let_28 tptp.c_Int_Ointrel) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Int_Oint) (tptp.c_Orderings_Otop__class_Otop _let_38)))) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_14 _let_17) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b 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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (= (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (= V_x_2 V_y_2)))) (forall ((V_x $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop T_b))) (=> (tptp.class_Lattices_Obounded__lattice__top T_b) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))) (= (tptp.hAPP _let_35 _let_37) _let_17) (= (tptp.hAPP _let_35 _let_36) _let_17) (forall ((T_c $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (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_c T_b V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_b_2 $$unsorted)) (=> (= V_b_2 (tptp.hAPP V_f_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))))))) (= (tptp.hAPP _let_35 _let_34) _let_17) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.c_Orderings_Otop__class_Otop _let_1)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Finite__Set_Ocard T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) _let_1)) (=> (= (tptp.hAPP _let_2 V_A_2) (tptp.hAPP _let_2 _let_1)) (= V_A_2 _let_1)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b 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 ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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 ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (let ((_let_3 (tptp.c_member T_c))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (tptp.hAPP (tptp.c_COMBC T_c _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 $$unsorted) (T_d $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_g_2) _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_d) (tptp.hAPP (tptp.c_Set_Oimage T_c T_d (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_d T_c) V_f_2) V_g_2)) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_b T_c _let_1 V_f_2) (tptp.hAPP V_f_2 V_x_2)) V_x_2)))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (=> (not (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Nat_Osemiring__char__0 T_b) (tptp.c_Fun_Oinj__on tptp.tc_Nat_Onat T_b (tptp.c_Nat_Osemiring__1__class_Oof__nat T_b) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (= (tptp.hAPP _let_34 _let_12) _let_17) (forall ((T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_fun T_d T_c))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (tptp.c_Fun_Oinj__on T_b _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_2 T_b) (tptp.c_COMBK T_c T_d)) V_f_2) _let_1))))) (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_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_b_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_b_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c 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_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) _let_1)) (=> (tptp.c_Fun_Oinj__on T_b T_b V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_f_2) _let_1) _let_1))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) _let_1)) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_f_2) _let_1) _let_1) (tptp.c_Fun_Oinj__on T_b T_b V_f_2 _let_1))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oimage T_b T_c V_f_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_c 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_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_c 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 $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oimage T_c T_b V_f_2))) (let ((_let_3 (tptp.tc_fun T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b 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 ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((T_b $$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_b) _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_b) _let_1)))))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard tptp.tc_HOL_Obool) (tptp.c_Orderings_Otop__class_Otop _let_33)) _let_32) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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))))))))) (= (tptp.hAPP _let_23 _let_12) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.c_Nat_OSuc) _let_17)) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_prod T_c T_d))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Product__Type_OPair T_c T_d))) (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_d _let_6))) (let ((_let_8 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.tc_fun T_c _let_8))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_9) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_9 T_b) (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_3 _let_8) T_c) (tptp.c_COMBC T_d _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d _let_2) _let_7 T_c) (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 T_d) _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_b _let_3) _let_1) V_r_2)))))))))))))) (forall ((V_B_2 $$unsorted) (V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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 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 $$unsorted)) (let ((_let_1 (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_1 B_n)) V_A_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_n) V_k_2))) V_B_2))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_k_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (=> (forall ((B_n $$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_b 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (forall ((B_n $$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 _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_x_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_x_2))) (= tptp.c_Int_OInteg (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient _let_24) _let_28) tptp.c_Int_Ointrel)) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b 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.c_Equiv__Relations_Oequiv T_b (tptp.c_Orderings_Otop__class_Otop _let_5) V_A_2) (tptp.c_Equiv__Relations_Oequivp T_b (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_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_b T_b)))) V_A_2))))))))) (forall ((T_b $$unsorted)) (tptp.c_Equiv__Relations_Oequivp T_b tptp.c_fequal)) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.hAPP V_R_2 B_x))) (= (tptp.hBOOL (tptp.hAPP _let_1 B_y)) (= _let_1 (tptp.hAPP V_R_2 B_y))))))) (forall ((V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_b 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 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_b 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_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_R_2 V_x_2))) (=> (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_y_2) V_z_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_z_2))))))) (forall ((V_x_2 $$unsorted)) (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_27 tptp.c_Int_ORep__Integ tptp.c_Int_OAbs__Integ tptp.c_Int_OInteg) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_b T_c V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) (tptp.c_HOL_OThe T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_2 T_b) tptp.c_fequal) V_f_2)) V_a_2))) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (let ((_let_5 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_b) (tptp.c_COMBC T_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_b T_b)))) V_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Wellfounded_Oacc T_b V_r_2)))))))))) (forall ((T_c $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted)) (=> (= (tptp.hAPP V_f_2 V_a_2) V_b_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_f_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c 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_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (tptp.hAPP V_f_2 V_a_2)) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (tptp.hAPP V_f_2 V_a_2)) V_A_2)))) (forall ((T_c $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_a_2)) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_f_2) V_A_2))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_c T_b V_f_2))) (=> (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)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c 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_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) V_A_2) V_x_2)) (tptp.hBOOL (tptp.hAPP V_A_2 (tptp.hAPP V_f_2 V_x_2))))) (forall ((V_Y_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_b (tptp.c_COMBI T_b)) V_Y_2) V_Y_2)) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_c tptp.tc_HOL_Obool)) 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) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c 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_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_f_2) V_A_2))) V_A_2))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (=> (= (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_f_2) V_A_2)) V_A_2)))) (forall ((V_R2_2 $$unsorted) (V_R1_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) 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_b tptp.tc_HOL_Obool)) (tptp.c_Wellfounded_Oacc T_b V_R2_2)) (tptp.c_Wellfounded_Oacc T_b V_R1_2))))) (forall ((V_b_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))))) (= (tptp.hAPP V_f_2 V_a_2) V_b_2))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oinsert T_c) 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_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))))) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_F_2 $$unsorted) (V_h_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_h_2) V_F_2))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_h_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_c tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_F_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop _let_3)) (tptp.c_Orderings_Otop__class_Otop _let_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_3) (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_f_2) V_B_2)) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) V_B_2) (tptp.hAPP _let_1 V_A_2))))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))))))) (forall ((V_h_2 $$unsorted) (T_c $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_F_2)) (=> (tptp.c_Fun_Oinj__on T_c T_b V_h_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b V_h_2) V_F_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_A_2))) (forall ((T_c $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b (tptp.hAPP (tptp.c_COMBK T_b T_c) V_c_2)) V_A_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_c_2) V_A_2)))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_3) (= _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_2 V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_3) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 _let_1) (tptp.c_Set_Oinsert T_c)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.c_Wellfounded_Oacc T_b V_r_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) V_a_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.c_Wellfounded_Oacc T_b V_r_2))))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_b V_r_2))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_c tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c 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_b T_c V_f_2) V_B_2)) V_A_2)))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_d_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Ovimage T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_2) T_c) (tptp.c_If T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_B_2))) V_c_2)) V_d_2)) V_A_2))) (let ((_let_4 (tptp.c_member T_b))) (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_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_b T_c V_Rep_2 V_Abs_2 V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_Abs_2) V_A_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_b T_c V_Rep_2 V_Abs_2 V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_Rep_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_A_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_member T_c))) (=> (tptp.c_Typedef_Otype__definition T_b T_c V_Rep_2 V_Abs_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_A_2)) (= (= (tptp.hAPP V_Abs_2 V_x_2) (tptp.hAPP V_Abs_2 V_y_2)) (= V_x_2 V_y_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_Abs_2 $$unsorted) (V_Rep_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_b T_c 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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_b T_c 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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_b T_c V_Rep_2 V_Abs_2 V_A_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Typedef_Otype__definition T_b T_c V_Rep_2 V_Abs_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_y_2) V_A_2)) (= (tptp.hAPP V_Rep_2 (tptp.hAPP V_Abs_2 V_y_2)) V_y_2)))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (= (tptp.c_Wellfounded_Omax__extp T_b 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_b tptp.tc_HOL_Obool)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_a1_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_a2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_xa)))))))))) (forall ((V_R_2 $$unsorted) (V_Y_2 $$unsorted) (V_X_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_X_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_Y_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) 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_b V_R_2)))))))))) (forall ((T_a $$unsorted)) (= (tptp.c_Nitpick_Ozero__frac T_a) (tptp.c_Nitpick_OAbs__Frac T_a (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (forall ((T_a $$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))))) (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))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nitpick_Ofrac T_a) V_a) V_b) (tptp.c_Nitpick_OAbs__Frac T_a (tptp.c_Nitpick_Onorm__frac V_a V_b)))) (forall ((V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.c_Set_Oimage T_d _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_d T_c _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c _let_3) T_d) (tptp.c_Product__Type_OPair T_b T_c)) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.hAPP (tptp.c_Set_Oimage T_d T_b V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) (tptp.c_Set_Oimage T_d T_c V_g_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 T_b) (tptp.c_Set_Ovimage T_d T_b V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1))))) V_A_2)))))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_na_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (tptp.hAPP V_f_2 B_i)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_k_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) (tptp.hAPP V_f_2 V_na_2))) (exists ((B_i $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_i) V_na_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2)))))))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_na_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_na_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_na_2))) (exists ((B_i $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_i) V_na_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2)))))))) (forall ((V_B_2 $$unsorted) (V_b_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_b_2) (tptp.hAPP V_B_2 V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_I_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_b) (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_b T_c) 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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_b T_c))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) _let_1)))))) (forall ((V_C_2 $$unsorted) (V_J_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_b T_c))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b 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_b T_c) 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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_C_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_c) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_c T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_B_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d T_b) (tptp.tc_fun _let_1 T_b) T_c) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_d T_b)) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum (tptp.tc_prod T_c T_d) T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_d) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_c) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_B_2)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))) (or (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (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) (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBK _let_1 T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2)))))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBK _let_1 T_b) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (let ((_let_3 (tptp.c_Product__Type_OSigma T_b T_c))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun (tptp.tc_prod T_b T_c) 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_b tptp.tc_HOL_Obool)) V_A_2)) _let_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d T_b) (tptp.tc_fun _let_1 T_b) T_c) (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_d T_b)) 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_c T_d) T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_d) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_c) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_I_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2)) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_b T_c))) (= (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)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_c) 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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_b T_c))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_I_2) V_J_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_I_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_1 V_J_2) V_C_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_c T_b) _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_b _let_1) (tptp.hAPP (tptp.c_COMBC T_b T_c _let_1) (tptp.c_Product__Type_OPair T_b T_c)))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_c) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_B_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_A_2))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_B_2))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_b 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_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_I_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2)) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_b T_c))) (= (tptp.hAPP (tptp.hAPP _let_2 (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)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_c) 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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_b T_c))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_I_2) V_J_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_I_2) V_C_2)) (tptp.hAPP (tptp.hAPP _let_1 V_J_2) V_C_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) (tptp.c_Relation_OId__on T_b V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) V_A_2))))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) (tptp.c_Orderings_Otop__class_Otop _let_1))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_b_2) (tptp.hAPP V_B_2 V_a_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_b_2) (tptp.hAPP V_B_2 V_a_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2))) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_b_2) (tptp.hAPP V_B_2 V_a_2)))))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_A_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_C_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_c) V_C_2))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_c T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_C_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_B_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.c_Relation_OImage T_b T_c V_r_2) V_C_2)) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_B_2)))) (=> (not (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) V_B_2)))) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_B_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_c T_b) T_c (tptp.c_Product__Type_Ofst T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_c) V_B_2))))) (let ((_let_3 (= V_B_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (not _let_3) (= _let_2 V_A_2))))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_b T_c) T_c (tptp.c_Product__Type_Osnd T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_B_2))))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_3 (= _let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not _let_3) (= _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b))) (let ((_let_2 (tptp.c_Product__Type_OSigma T_b T_c))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)))) (let ((_let_4 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_b_2) V_B_2)))) (let ((_let_5 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP _let_3 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_5 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) _let_4)) (tptp.hAPP _let_3 (tptp.hAPP _let_1 V_B_2))))))))))) (forall ((V_X_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_2 T_b) V_A_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) V_X_2) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_X_2)))))))) (forall ((V_F_2 $$unsorted) (V_E_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_e $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBK _let_1 T_b))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Product__Type_OSigma T_b T_c))) (let ((_let_5 (tptp.tc_fun T_b _let_1))) (let ((_let_6 (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun _let_5 _let_6))) (let ((_let_8 (tptp.tc_fun T_e _let_6))) (let ((_let_9 (tptp.tc_fun T_e _let_5))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR (tptp.tc_prod T_d T_e) _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_d T_e) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_e tptp.tc_HOL_Obool) T_d) V_B_2))) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_e _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_9 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_9 _let_8) T_d) (tptp.c_COMBB _let_5 _let_6 T_e)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_7 T_d) _let_4) V_E_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_e) _let_2) V_F_2)))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_3) V_A_2) V_E_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_e _let_1) V_B_2) V_F_2)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 _let_2))) (let ((_let_4 (tptp.tc_fun T_c _let_3))) (let ((_let_5 (tptp.tc_fun T_c _let_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_5 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) (tptp.tc_fun _let_5 _let_2) T_b) (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_2)) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_b) (tptp.c_COMBC T_c _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) (tptp.c_Set_Oinsert _let_1))) (tptp.c_Product__Type_OPair T_b T_c)))) (tptp.c_Orderings_Obot__class_Obot _let_2))))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (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 (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_A_2)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (exists ((B_f $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_I_2) V_A_2))) (let ((_let_2 (tptp.tc_prod T_c T_b))) (and (tptp.c_Fun_Oinj__on T_b _let_2 B_f _let_1) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b _let_2 B_f) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_b) V_I_2) V_A_2)))))))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_d))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_d))) (let ((_let_3 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_3 T_d))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_d) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_d T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_3 T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_d _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_d) (tptp.tc_fun T_d _let_4) T_b) (tptp.c_Finite__Set_Ofold__image T_d T_c _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_d (tptp.tc_prod T_b T_c) _let_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) V_g_2)) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_B_2 B_a))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_d) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_d) (tptp.tc_fun _let_1 T_d) T_b) (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_d)) 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_d) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2)))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_d) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_d) (tptp.tc_fun _let_1 T_d) T_b) (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_d)) V_f_2)) V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum (tptp.tc_prod T_b T_c) T_d) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2)))))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_B_2 B_x))))) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) tptp.tc_Nat_Onat T_b) (tptp.c_Finite__Set_Ocard T_c)) V_B_2)) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) V_A_2) V_B_2))) (not (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_y) (tptp.hAPP V_B_2 B_x))) (not (= V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) B_x) B_y)))))))))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_d))) (let ((_let_2 (tptp.tc_prod T_c T_e))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_e T_d V_g_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_e tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Oimage _let_2 _let_1 (tptp.c_Product__Type_Omap__pair T_c T_b T_e T_d V_f_2 V_g_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) (forall ((V_B_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.c_Fun_Oinj__on T_d T_e V_g_2 V_B_2) (tptp.c_Fun_Oinj__on (tptp.tc_prod T_b T_d) (tptp.tc_prod T_c T_e) (tptp.c_Product__Type_Omap__pair T_b T_c T_d T_e V_f_2 V_g_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_d) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_d tptp.tc_HOL_Obool) T_b) V_B_2)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_d T_e))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) 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_d T_e) (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_b T_d T_c T_e V_f_2 V_g_2)) V_R_2))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Ofst T_b T_c) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_d T_b T_e T_c V_f_2 V_g_2) V_x_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_d T_e) V_x_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_e $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_d T_b T_e T_c V_f_2 V_g_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_d T_e) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) (tptp.hAPP V_f_2 V_a_2)) (tptp.hAPP V_g_2 V_b_2)))) (forall ((V_z_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_b T_b T_c T_c (tptp.c_COMBI T_b) (tptp.c_COMBI T_c)) V_z_2) V_z_2)) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_b) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_d T_c T_e T_b V_f_2 V_g_2) V_x_2)) (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_d T_e) V_x_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_e $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_d T_e))) (let ((_let_2 (tptp.tc_fun T_e _let_1))) (let ((_let_3 (tptp.tc_fun T_c _let_1))) (let ((_let_4 (tptp.tc_fun T_c T_e))) (= (tptp.c_Product__Type_Omap__pair T_b T_d T_c T_e V_f_2 V_g_2) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_b) (tptp.c_COMBB T_e _let_1 T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_d _let_2 T_b) (tptp.c_Product__Type_OPair T_d T_e)) V_f_2))) V_g_2)))))))) (forall ((V_B_H_2 $$unsorted) (V_B_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (T_e $$unsorted) (V_A_H_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2) V_A_H_2) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_e T_d V_g_2) V_B_2) V_B_H_2) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_c T_e) (tptp.tc_prod T_b T_d) (tptp.c_Product__Type_Omap__pair T_c T_b T_e T_d V_f_2 V_g_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_e) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_e tptp.tc_HOL_Obool) T_c) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_d) V_A_H_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_d tptp.tc_HOL_Obool) T_b) V_B_H_2)))))) (forall ((V_R_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_c_2) (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_d T_e) _let_1 (tptp.c_Product__Type_Omap__pair T_d T_b T_e T_c V_f_2 V_g_2)) V_R_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (= V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_y))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_d T_e)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_d T_e) B_x) B_y)) V_R_2))))))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Orefl__on T_b 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_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) V_A_2)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)) V_r_2))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b tptp.tc_Nat_Onat) V_f_2) V_S_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Orefl__on T_b V_A_2 (tptp.c_Relation_OId__on T_b V_A_2))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Orefl__on T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (forall ((V_s_2 $$unsorted) (V_B_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (=> (tptp.c_Relation_Orefl__on T_b V_B_2 V_s_2) (tptp.c_Relation_Orefl__on T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))) (forall ((V_s_2 $$unsorted) (V_B_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (=> (tptp.c_Relation_Orefl__on T_b V_B_2 V_s_2) (tptp.c_Relation_Orefl__on T_b (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 (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_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_a_2) V_a_2)) V_r_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_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_x_2) V_y_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) V_A_2)))) (=> (forall ((B_x $$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 (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)) V_r_2)))) (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_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_Pa_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 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP tptp.c_Int_ORep__Integ B_x)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_y_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_g_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_c))) (=> (tptp.c_Finite__Set_Ofolding__image__simple T_b T_c V_f_2 V_z_2 V_g_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_B_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c 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_c 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_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 ((T_b $$unsorted) (V_a_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2))))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (or (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_Pa_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_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) V_Pa_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_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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_b T_c)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_b) tptp.c_fconj) V_Pa_2))) V_Q_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (tptp.c_Set_OCollect T_c)) V_Q_2)))))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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_b T_c)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_b) tptp.c_fconj) V_Pa_2))) V_Q_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2)) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) (tptp.hAPP (tptp.c_Set_OCollect T_c) V_Q_2))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (= (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_Pa_2)) (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_Pa_2))))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) V_Pa_2)) V_Q_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_Pa_2)) (tptp.hAPP _let_1 V_Q_2))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2)) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2) V_Pa_2)) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (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_fdisj) V_Pa_2)) V_Q_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_Pa_2)) (tptp.hAPP _let_1 V_Q_2))))) (forall ((V_Pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.hAPP _let_1 V_Pa_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2)))) V_Pa_2))))) (forall ((T_b $$unsorted)) (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool T_b) tptp.c_fFalse)))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fdisj) V_Pa_2)) V_Q_2)))) (and (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_Pa_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_Q_2)))))))) (forall ((T_b $$unsorted)) (= (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool T_b) tptp.c_fTrue)))) (forall ((V_Pa_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) (tptp.hAPP (tptp.c_Set_OCollect T_c) V_Pa_2)) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c tptp.tc_HOL_Obool T_b) V_Pa_2) V_f_2)))) (forall ((V_Pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)))) (forall ((V_Pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)))) (forall ((V_Pa_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)) V_A_2)) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (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_fdisj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fdisj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (= (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_b) (tptp.hAPP (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_fdisj) (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_Pa_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (= (tptp.hBOOL (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) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2)))) (and (tptp.hBOOL (tptp.hAPP _let_1 V_A_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP _let_2 V_A_2))) (tptp.hAPP _let_2 V_B_2))))))) (forall ((T_b $$unsorted) (V_a_2 $$unsorted) (V_Pa_2 $$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_OCollect T_b) (tptp.hAPP (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 T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_Pa_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) _let_1))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((T_b $$unsorted) (V_a_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP tptp.c_fequal V_a_2))) V_Pa_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) _let_1))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP tptp.c_fequal V_a_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (= (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_b) (tptp.hAPP (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 _let_2 V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) (tptp.hAPP _let_2 V_B_2)))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (let ((_let_2 (tptp.c_Set_OCollect T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_Pa_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_b) tptp.c_fNot) V_Pa_2)))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_a_2))) V_B_2))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus _let_1) V_A_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) T_b) (tptp.c_member T_c)) V_f_2)) V_B_2))))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (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) V_Pa_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_Pa_2))) (tptp.hAPP _let_1 V_Q_2)))))) (forall ((V_i_2 $$unsorted) (V_Pa_2 $$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_Pa_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 $$unsorted) (T_c $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_COMBB _let_1 _let_1 T_c))) (let ((_let_4 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_d))) (let ((_let_5 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_d))) (let ((_let_6 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.c_COMBB _let_6 _let_6 T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_d) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_T_2)) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b _let_6 T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_d) (tptp.tc_fun _let_6 T_d) T_b) _let_5) V_f_2)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.c_Set_OCollect T_c)) (tptp.hAPP (tptp.hAPP _let_7 (tptp.hAPP (tptp.c_COMBS T_c tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_c) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_6 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_T_2)))) V_R_2)))) V_S_2) (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c _let_1 T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b T_d) (tptp.tc_fun _let_1 T_d) T_c) _let_4) (tptp.hAPP (tptp.c_COMBC T_b T_c T_d) V_f_2))) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Set_OCollect T_b)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_S_2)))) (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) V_R_2))))) V_T_2))))))))))))) (forall ((V_S_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_c))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_S_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b _let_3) T_c) (tptp.c_If T_b)) V_Pa_2)) V_f_2)) V_g_2)) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_Pa_2)))) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_g_2) (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_c) tptp.c_fNot) V_Pa_2)))))))))) (forall ((V_na_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat T_b V_f_2) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) V_na_2)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (let ((_let_2 (tptp.hAPP _let_1 V_B_2))) (let ((_let_3 (tptp.tc_fun T_c T_c))) (let ((_let_4 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_4 V_f_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (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 _let_1 V_A_2))) _let_2))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_3) T_b) (tptp.c_If T_c)) _let_2)) V_f_2)) (tptp.c_Groups_Ozero__class_Ozero T_c))) V_A_2))))))))) (forall ((V_f_2 $$unsorted) (T_d $$unsorted) (V_g_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBB _let_1 _let_1 T_d))) (let ((_let_3 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_g_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (= (tptp.hAPP _let_3 V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_d T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 T_c T_d) _let_3) (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_S_2)))) (tptp.hAPP (tptp.c_COMBC T_b T_d tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_d (tptp.tc_fun T_d tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_f_2)))))) (tptp.hAPP (tptp.c_Set_Oimage T_b T_d V_f_2) V_S_2))))))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Relation_OImage T_c T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _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_b) (tptp.c_member _let_1)) (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) V_a_2))) V_r_2)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_Pa_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2))) (let ((_let_2 (tptp.tc_fun T_b 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_b T_c))) (let ((_let_5 (tptp.tc_fun T_c T_c))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_c _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_c _let_5) T_b) (tptp.c_If T_c)) V_Pa_2)) V_f_2)) V_g_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (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_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 $$unsorted) (V_R_2 $$unsorted) (V_T_2 $$unsorted) (T_c $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ocard T_c))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_COMBB _let_2 _let_2 T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_T_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_T_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_S_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) V_R_2) B_x)))) V_k_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_Nat_Onat T_b) _let_1) (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Set_OCollect T_c)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.c_COMBS T_c tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_c) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_T_2)))) V_R_2)))) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP _let_1 V_T_2)))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b _let_6 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_6 (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_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_b) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (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_b) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_7 (tptp.tc_fun T_b T_b))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_b _let_1))) (let ((_let_10 (tptp.tc_fun T_b _let_7))) (let ((_let_11 (tptp.tc_fun T_b _let_3))) (let ((_let_12 (tptp.tc_fun T_b _let_11))) (let ((_let_13 (tptp.tc_fun T_b _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_b)))) (let ((_let_15 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_16 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_17 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b _let_15 V_A_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_15 V_B_2)) (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_15 (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_b) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_b) (tptp.c_COMBS T_b _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_b) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_b) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_b) (tptp.c_COMBB _let_7 _let_1 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_A_2)))) (tptp.hAPP _let_2 V_B_2))))))))))))))))))))))))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq _let_1)) V_A_2))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_COMBB _let_3 tptp.tc_HOL_Obool _let_2))) (let ((_let_5 (tptp.c_Groups_Ouminus__class_Ouminus _let_2))) (let ((_let_6 (tptp.tc_fun _let_2 _let_2))) (let ((_let_7 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_8 (tptp.tc_fun _let_2 _let_7))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_5 V_A_2)) (tptp.hAPP (tptp.c_Set_OCollect _let_2) (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_HOL_OEx _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_3 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_3 _let_3) _let_2) (tptp.c_COMBS _let_2 tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_8 _let_2) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_7 _let_2) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_6 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_6 _let_3) _let_2) (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool _let_2)) tptp.c_fequal)) _let_5)))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.c_member _let_2) V_A_2)) _let_1)))))))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq _let_1)) V_A_2))))) (forall ((V_Q_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) V_Pa_2))) V_Q_2))))) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_y)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_b tptp.tc_HOL_Obool) V_Q_2) B_y)))))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b T_c))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_b _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_1 _let_1) T_c) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 T_c) (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_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) (tptp.tc_fun _let_2 _let_1) T_c) (tptp.c_COMBB T_c tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) V_f_2)))) V_Pa_2))))))))))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_c))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod T_b T_c))) (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_b _let_2) _let_4 _let_1) (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_COMBB T_c _let_2 T_b) (tptp.hAPP (tptp.c_COMBC T_c T_c tptp.tc_HOL_Obool) tptp.c_fequal)))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))))) (forall ((V_Pa_2 $$unsorted)) (tptp.c_Nat__Transfer_Onat__set (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) V_Pa_2)))) (forall ((T_b $$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))) (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_b) (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_b)))))))))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_c))) (let ((_let_2 (tptp.tc_fun T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (let ((_let_4 (tptp.tc_fun T_b T_c))) (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_b _let_6))) (let ((_let_8 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_c T_b) V_r_2) V_f_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_5 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_5 _let_8) T_b) (tptp.c_COMBC T_b _let_5 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_7 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_6 T_b) (tptp.c_member _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_b) (tptp.c_COMBB T_c _let_1 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_2 T_b) (tptp.c_Product__Type_OPair T_c T_c)) V_f_2))) V_f_2)))) V_r_2))))))))))))) (forall ((V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Int_Oint))) (= (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)))) V_Pa_2)) (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_1) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_HOL_Obool tptp.tc_Nat_Onat) V_Pa_2) _let_1)))))) (forall ((V_Pa_2 $$unsorted)) (= (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) V_Pa_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_Pa_2) tptp.c_Int_Onat)))))) (forall ((V_k_2 $$unsorted) (V_N_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_k_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (=> (tptp.class_Groups_Olinordered__ab__semigroup__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_N_2)) (=> (not (= V_N_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_b) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_N_2))))))))))))))) (forall ((V_k_2 $$unsorted) (V_N_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) V_k_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (=> (tptp.class_Groups_Olinordered__ab__semigroup__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b V_N_2)) (tptp.c_Big__Operators_Olinorder__class_OMax 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_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_b) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_N_2))))))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (=> (tptp.class_Lattices_Odistrib__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.hAPP _let_2 (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_b) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (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_b) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (let ((_let_7 (tptp.tc_fun T_b T_b))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_b _let_1))) (let ((_let_10 (tptp.tc_fun T_b _let_7))) (let ((_let_11 (tptp.tc_fun T_b _let_3))) (let ((_let_12 (tptp.tc_fun T_b _let_11))) (let ((_let_13 (tptp.tc_fun T_b _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_b)))) (let ((_let_15 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_16 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Odistrib__lattice T_b) (=> (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_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_B_2)) (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_b) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_b) (tptp.c_COMBS T_b _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_b) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_b) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_b) (tptp.c_COMBB _let_7 _let_1 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_A_2)))) (tptp.hAPP _let_2 V_B_2)))))))))))))))))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (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_b) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b))) (let ((_let_7 (tptp.tc_fun T_b T_b))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_b _let_1))) (let ((_let_10 (tptp.tc_fun T_b _let_7))) (let ((_let_11 (tptp.tc_fun T_b _let_3))) (let ((_let_12 (tptp.tc_fun T_b _let_11))) (let ((_let_13 (tptp.tc_fun T_b _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_b)))) (let ((_let_15 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_16 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Lattices_Odistrib__lattice T_b) (=> (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_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_B_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_b) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_b) (tptp.c_COMBS T_b _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_b) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_b) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_b) (tptp.c_COMBB _let_7 _let_1 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (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 $$unsorted) (T_b $$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 T_b) V_x_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (=> (tptp.class_Lattices_Odistrib__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b V_A_2)) (tptp.c_Big__Operators_Olattice__class_OSup__fin 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_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_b) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (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_b) tptp.c_fconj))) (let ((_let_5 (tptp.tc_fun _let_1 _let_1))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_7 (tptp.tc_fun T_b T_b))) (let ((_let_8 (tptp.tc_fun _let_7 _let_1))) (let ((_let_9 (tptp.tc_fun T_b _let_1))) (let ((_let_10 (tptp.tc_fun T_b _let_7))) (let ((_let_11 (tptp.tc_fun T_b _let_3))) (let ((_let_12 (tptp.tc_fun T_b _let_11))) (let ((_let_13 (tptp.tc_fun T_b _let_5))) (let ((_let_14 (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_b)))) (let ((_let_15 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_16 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_17 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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_b _let_15 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_15 V_B_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_15 (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP _let_14 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_1 T_b) _let_14) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_9 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun _let_9 _let_9) T_b) (tptp.c_COMBS T_b _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 _let_13 T_b) (tptp.hAPP (tptp.c_COMBB _let_11 _let_5 T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_12 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_11 T_b) _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_10 _let_9) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_10 _let_9) T_b) (tptp.c_COMBB _let_7 _let_1 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_8 T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal))) _let_6))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_5 T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b)) (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2))) (let ((_let_3 (tptp.tc_fun T_b T_b))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (let ((_let_6 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b _let_6 V_A_2)) (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_6 (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_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 T_b) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) _let_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (= (tptp.c_Nitpick_Oprod T_b T_c V_A_2 V_B_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_c)) (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 (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_B_2)))))))) (forall ((V_d_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_HOL_Obool))) (= (tptp.c_Int_Oint__ge__less__than V_d_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) tptp.tc_Int_Oint) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 tptp.tc_Int_Oint) tptp.c_fconj) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_d_2)))) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint)))))))) (forall ((V_d_2 $$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))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Oreflp T_b V_r_2) (tptp.c_Relation_Orefl__on T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) V_r_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b 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_d))) (let ((_let_6 (tptp.tc_fun T_d T_d))) (let ((_let_7 (tptp.tc_fun T_c T_d))) (let ((_let_8 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_7 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_7 _let_7) T_b) (tptp.c_COMBB T_d T_d T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_d _let_6 T_b) _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_b _let_1))) (let ((_let_11 (tptp.tc_fun T_b _let_7))) (let ((_let_12 (tptp.tc_fun T_c _let_3))) (let ((_let_13 (tptp.tc_fun T_b _let_12))) (let ((_let_14 (tptp.tc_fun T_b _let_4))) (=> (tptp.class_Rings_Osemiring__0 T_d) (=> (tptp.c_Fun_Oinj__on (tptp.tc_prod T_b T_c) T_d (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) _let_8) (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))) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_d) V_f_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_d) V_g_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_d T_d) (tptp.c_Fun_Oid T_d)) (tptp.hAPP (tptp.c_Set_OCollect T_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_d) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_2 T_d) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_c))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_10 _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_14 (tptp.tc_fun _let_10 _let_10) T_d) (tptp.c_COMBS T_b _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 _let_14 T_d) (tptp.hAPP (tptp.c_COMBB _let_12 _let_4 T_b) (tptp.c_COMBS T_c tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_10 _let_13 T_d) (tptp.hAPP (tptp.c_COMBB _let_1 _let_12 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 T_c) tptp.c_fconj))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_11 _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_11 _let_10) T_d) (tptp.c_COMBB _let_7 _let_1 T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d tptp.tc_HOL_Obool) _let_9 T_d) (tptp.c_COMBB T_d tptp.tc_HOL_Obool T_c)) tptp.c_fequal))) _let_8))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_4 T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 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_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_B_2))))))))))))))))))))))))) (forall ((V_k_2 $$unsorted) (V_R_2 $$unsorted) (V_t_2 $$unsorted) (T_c $$unsorted) (V_s_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_COMBB _let_1 _let_1 T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_s_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_t_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_t_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (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 (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_s_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c 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_b tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_Nat_Onat T_b) (tptp.c_Finite__Set_Ocard T_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Set_OCollect T_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_COMBS T_c 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_c) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_t_2)))) V_R_2)))) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c tptp.tc_Nat_Onat) V_k_2) V_t_2)))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Fun_Oinj__on T_b T_b (tptp.c_Fun_Oid T_b) V_A_2)) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_b (tptp.c_Fun_Oid T_b)) V_A_2) V_A_2)) (forall ((T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Oapfst T_b T_b T_c (tptp.c_Fun_Oid T_b)) (tptp.c_Fun_Oid (tptp.tc_prod T_b T_c)))) (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_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Predicate_Oreflp T_b V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_x_2) V_x_2)))) (forall ((T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_Product__Type_Oapsnd T_c T_c T_b (tptp.c_Fun_Oid T_c)) (tptp.c_Fun_Oid (tptp.tc_prod T_b T_c)))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Omap__pair T_b T_b T_c T_c (tptp.c_Fun_Oid T_b) (tptp.c_Fun_Oid T_c)) (tptp.c_Fun_Oid (tptp.tc_prod T_b T_c)))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Omap__pair T_b T_b T_c T_c (tptp.c_COMBI T_b) (tptp.c_COMBI T_c)) (tptp.c_Fun_Oid (tptp.tc_prod T_b T_c)))) (forall ((V_Y_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_b (tptp.c_Fun_Oid T_b)) V_Y_2) V_Y_2)) (= (tptp.c_Int_Oring__1__class_Oof__int tptp.tc_Int_Oint) (tptp.c_Fun_Oid tptp.tc_Int_Oint)) (forall ((T_b $$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_b (tptp.c_Fun_Oid T_b)) _let_1) _let_1))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Oapfst T_b T_d T_c V_f_2) (tptp.c_Product__Type_Omap__pair T_b T_d T_c T_c V_f_2 (tptp.c_Fun_Oid T_c)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.c_Product__Type_Oapsnd T_c T_d T_b V_f_2) (tptp.c_Product__Type_Omap__pair T_b T_b T_c T_d (tptp.c_Fun_Oid T_b) V_f_2))) (forall ((T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c _let_1) (tptp.c_Product__Type_OPair T_b T_c)) (tptp.c_Fun_Oid _let_1)))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_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_Oimage T_b T_c V_f_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_c) (tptp.c_Fun_Oid T_c)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_B_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (exists ((B_n $$unsorted) (B_f $$unsorted)) (= V_A_2 (tptp.hAPP (tptp.c_Set_Oimage tptp.tc_Nat_Onat T_b 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_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_d 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_d _let_3))) (let ((_let_5 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_prod T_b T_d))) (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_d _let_8))) (let ((_let_10 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_11 (tptp.tc_fun T_d _let_10))) (let ((_let_12 (tptp.tc_fun _let_5 _let_5))) (let ((_let_13 (tptp.tc_fun T_c _let_5))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) V_r_2) V_s_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_13 (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) (tptp.hAPP (tptp.c_COMBB _let_5 tptp.tc_HOL_Obool T_c) (tptp.c_HOL_OEx T_d))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_13 _let_13) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_12 (tptp.tc_fun _let_13 _let_13) T_b) (tptp.c_COMBB _let_5 _let_5 T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_11 _let_12 T_b) (tptp.c_COMBS T_d tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_11 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_10 T_d) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_7 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_7 _let_5) T_b) (tptp.c_COMBC T_d _let_7 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d _let_6) _let_9 T_b) (tptp.hAPP (tptp.c_COMBB _let_6 _let_8 T_d) (tptp.c_member _let_6))) (tptp.c_Product__Type_OPair T_b T_d)))) V_r_2))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_c) (tptp.c_COMBC T_d _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d _let_1) _let_4 T_c) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_d) (tptp.c_member _let_1))) (tptp.hAPP (tptp.c_COMBC T_d T_c _let_1) (tptp.c_Product__Type_OPair T_d T_c))))) V_s_2)))))))))))))))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_b tptp.tc_HOL_Obool) V_Pa_2))) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_c)) (tptp.hAPP (tptp.c_COMBC T_c T_b tptp.tc_HOL_Obool) V_Pa_2))))) (forall ((V_s_2 $$unsorted) (V_c_2 $$unsorted) (T_d $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (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_a_2) V_b_2)) V_r_2)) (=> (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) V_b_2) V_c_2)) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_d) V_a_2) V_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_c T_d) V_r_2) V_s_2)))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_b_2) (tptp.hAPP (tptp.c_Relation_ORange T_b T_c) V_r_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (V_I_2 $$unsorted) (T_e $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_b T_d T_c))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_d T_c) tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun (tptp.tc_prod T_b T_d) tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_e _let_4) V_I_2) V_r_2)) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_e _let_2) V_I_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_e _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_2) T_e) _let_1) V_r_2)) V_s_2)))))))) (forall ((V_r_2 $$unsorted) (V_I_2 $$unsorted) (T_e $$unsorted) (V_s_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) V_s_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_d T_c) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_e _let_3) V_I_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_e _let_2) V_I_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_e) _let_1) V_r_2))))))) (forall ((V_s_2 $$unsorted) (V_s_H_2 $$unsorted) (T_d $$unsorted) (V_r_2 $$unsorted) (V_r_H_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_b T_c T_d))) (=> (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_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_c T_d) 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_b T_d) 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_R_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_d) tptp.tc_HOL_Obool))) V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)))) (forall ((V_R_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_d T_c) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)))) (forall ((V_T_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) 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_d T_c) 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_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_S_2)) (tptp.hAPP _let_1 V_T_2))))) (forall ((V_R_2 $$unsorted) (V_T_2 $$unsorted) (V_S_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_b T_d T_c))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_d) 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_b T_c) 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 $$unsorted) (T_e $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_e T_d) V_R_2) V_S_2)) V_T_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_e T_c) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_e T_d T_c) V_S_2) V_T_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_b) (tptp.c_Relation_OId__on T_b V_A_2)) V_A_2)) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_A_2))) (=> (= (tptp.hAPP _let_3 V_B_2) _let_2) (=> (= (tptp.hAPP _let_3 V_C_2) _let_2) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2) V_C_2)) _let_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Relation_Orel__comp T_b T_b T_b))) (=> (= (tptp.hAPP (tptp.hAPP _let_3 V_A_2) V_C_2) _let_2) (=> (= (tptp.hAPP (tptp.hAPP _let_3 V_B_2) V_C_2) _let_2) (= (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) V_C_2) _let_2))))))) (forall ((T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (= (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= V_r_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool))))) (forall ((V_C_2 $$unsorted) (V_s_2 $$unsorted) (T_d $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (=> (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_r_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.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_c T_d) tptp.tc_HOL_Obool)) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_c T_d) V_B_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_c) 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_d) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_c T_d) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_d) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_C_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_c T_b))) (= (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_A_2) V_B_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_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_c)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Relation_ORange T_b T_c) V_r_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_c T_b) T_b (tptp.c_Product__Type_Osnd T_c T_b)) V_R_2) (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) V_R_2))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) V_r_2))) (exists ((B_y $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) B_y) V_a_2)) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_c T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_b_2) (tptp.hAPP _let_1 V_r_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_c T_b))) (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_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_c 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_c T_b))) (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_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_c T_b) tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Omax__ext T_b V_R_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod _let_2 _let_2))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.c_Orderings_Obot__class_Obot _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_R_2) V_S_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp _let_2 _let_2 _let_2) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_4) (tptp.c_Wellfounded_Omax__ext T_b V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_5) _let_5)) (tptp.c_Orderings_Obot__class_Obot _let_4))))) _let_1))))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Omin__ext T_b V_R_2))) (let ((_let_2 (tptp.tc_fun T_b 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_b T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) 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_b 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_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$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))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ORangeP T_b T_c (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_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_x_2) (tptp.hAPP (tptp.c_Relation_ORange T_b T_c) V_r_2)))))))))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) V_r_2))) (not (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) B_x) V_b_2)) V_r_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_d))) (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_d _let_3))) (let ((_let_5 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_prod T_b 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 T_c tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Opred__comp T_b T_c T_d (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_7 _let_10) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 (tptp.tc_fun _let_7 _let_10) T_b) (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_b) (tptp.hAPP (tptp.c_COMBB _let_6 _let_8 T_c) (tptp.c_member _let_6))) (tptp.c_Product__Type_OPair T_b T_c)))) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_c) (tptp.c_COMBC T_d _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d _let_1) _let_4 T_c) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_d) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_c T_d)))) V_s_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_d) V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_c T_d) V_r_2) V_s_2))))))))))))))) (forall ((T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_c_2 $$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_b T_c T_d V_r_2 V_s_2) V_a_2) V_c_2))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ORangeP T_b T_c V_r_2) V_b_2)))) (forall ((V_a3_2 $$unsorted) (V_a1_2 $$unsorted) (V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Opred__comp T_b T_c T_d V_r_2 V_s_2) V_a1_2) V_a3_2)) (exists ((B_b $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a1_2) B_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_s_2 B_b) V_a3_2)))))) (forall ((V_a2_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ORangeP T_b T_c V_r_2) V_a2_2)) (exists ((B_a $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 B_a) V_a2_2))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_b V_R_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_R_2) V_S_2)) V_R_2)) (tptp.c_FunDef_Oreduction__pair T_b (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_R_2) V_S_2)))))) (forall ((V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) V_Pa_2))) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx T_c)) V_Pa_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (exists ((B_x $$unsorted)) (let ((_let_1 (tptp.c_member T_b))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 B_x) V_A_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) _let_1) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_f_2)) (tptp.hAPP V_f_2 B_x))))))))))))) (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than) (forall ((T_b $$unsorted)) (tptp.c_Wellfounded_Owf T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf T_c (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_b T_c) V_r_2) V_f_2)))) (forall ((V_rb_2 $$unsorted) (T_c $$unsorted) (V_ra_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_ra_2) (=> (tptp.c_Wellfounded_Owf T_c V_rb_2) (tptp.c_Wellfounded_Owf (tptp.tc_prod T_b T_c) (tptp.c_Wellfounded_Olex__prod T_b T_c V_ra_2 V_rb_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.c_Wellfounded_Omeasure T_b) V_f_2))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_r_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_Owf T_b V_R_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_R_2) V_R_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Omin__ext T_b V_r_2)))) (forall ((V_p_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b 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_b) tptp.tc_HOL_Obool)) V_p_2) V_r_2)) (tptp.c_Wellfounded_Owf T_b V_p_2)))) (forall ((T_b $$unsorted)) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Ofinite__psubset T_b))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_r_H_2)))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_H_2) V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Omax__ext T_b V_r_2)))) (tptp.c_Wellfounded_Owf _let_24 tptp.c_FunDef_Opair__less) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_b) (tptp.c_Relation_OId__on T_b V_A_2)) V_A_2)) (forall ((V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_R_2) (tptp.c_Wellfounded_Owf T_b (tptp.c_Wellfounded_Omlex__prod T_b V_f_2 V_R_2)))) (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat (tptp.hAPP _let_26 (tptp.hAPP _let_25 (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_a_2)) V_r_2))))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_x_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_a_2)) V_r_2)))))))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_x_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_a_2)) V_r_2)))))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (not (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_a_2) V_a_2)) V_r_2))))) (forall ((V_d_2 $$unsorted)) (tptp.c_Wellfounded_Owf tptp.tc_Int_Oint (tptp.c_Int_Oint__ge__less__than2 V_d_2))) (forall ((V_d_2 $$unsorted)) (tptp.c_Wellfounded_Owf tptp.tc_Int_Oint (tptp.c_Int_Oint__ge__less__than V_d_2))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (=> (tptp.c_Wellfounded_Owf T_b V_s_2) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_b) V_r_2)) (tptp.hAPP (tptp.c_Relation_ORange T_b T_b) V_s_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_Owf T_b V_r_2) (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.c_Wellfounded_Oacc T_b V_r_2)))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Wellfounded_Oacc T_b V_r_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_R_2) V_R_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (tptp.c_Wellfounded_Owf T_b V_R_2))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Orel__comp T_b T_b T_b))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (= (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2)) (tptp.c_Wellfounded_Owf T_b (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 ((T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= V_r_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 V_s_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_b T_c))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2))))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_c)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_r_2))))) (forall ((V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_b T_c) T_b (tptp.c_Product__Type_Ofst T_b T_c)) V_R_2) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_R_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_Owf T_b V_r_2) (not (exists ((B_f $$unsorted)) (forall ((B_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) (tptp.hAPP B_f (tptp.hAPP tptp.c_Nat_OSuc B_i))) (tptp.hAPP B_f B_i))) V_r_2))))))) (forall ((V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_b 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_b T_b 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_b V_R_2) (=> (tptp.c_Wellfounded_Owf T_b V_S_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_R_2) V_S_2)) V_R_2)) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_R_2) V_S_2))))))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_r_2))) (exists ((B_y $$unsorted)) (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_a_2) B_y)) V_r_2))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Orderings_Owellorder T_b) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless T_b)))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_b T_c))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.hAPP _let_1 V_r_2))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) V_Pa_2))) (= (tptp.c_FunDef_Oreduction__pair T_b V_Pa_2) (and (tptp.c_Wellfounded_Owf T_b _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_b T_b T_b) _let_2) (tptp.hAPP (tptp.c_Product__Type_Osnd _let_1 _let_1) V_Pa_2))) _let_2))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1))) (=> (tptp.c_FunDef_Oreduction__pair T_b V_Pa_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_R_2) (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) V_Pa_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_S_2) (tptp.hAPP (tptp.c_Product__Type_Osnd _let_1 _let_1) V_Pa_2))) (=> (tptp.c_Wellfounded_Owf T_b V_S_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_R_2) V_S_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_b T_c))) (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_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_c) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_b T_c))) (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_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_c) tptp.tc_HOL_Obool)) V_A_2) V_B_2))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OField T_b) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_b) V_r_2)) (tptp.hAPP (tptp.c_Relation_ORange T_b T_b) V_r_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Wellfounded_Owf T_c (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod T_b T_b) (tptp.tc_prod T_c T_c) (tptp.c_Product__Type_Omap__pair T_b T_c T_b T_c V_f_2 V_f_2)) V_r_2))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$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))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ODomainP T_b T_c (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_r_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_r_2)))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_List_Olist T_b) (tptp.c_List_Olenlex T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_r_2))) (not (forall ((B_y $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) B_y)) V_r_2))))))) (forall ((T_c $$unsorted) (T_b $$unsorted) (V_b_2 $$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_b T_c V_r_2) V_a_2)))) (forall ((V_a1_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_ODomainP T_b T_c V_r_2) V_a1_2)) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a1_2) B_x1))))) (= tptp.c_Wellfounded_Opred__nat (tptp.hAPP _let_26 (tptp.hAPP _let_25 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_16 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.c_fequal)) tptp.c_Nat_OSuc)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_List_Olist T_b) (tptp.c_List_Olex T_b V_r_2)))) (forall ((V_na_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_List_Olist T_b) (tptp.hAPP (tptp.c_List_Olexn T_b V_r_2) V_na_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Olex T_b 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_b V_r_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_List_Olexn T_b 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))))) (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat) (forall ((T_b $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (tptp.c_Wellfounded_Owf T_c (tptp.hAPP V_R_2 B_x)))) (tptp.c_Wellfounded_Owf (tptp.tc_prod T_b T_c) (tptp.c_Recdef_Osame__fst T_b T_c V_Pa_2 V_R_2)))) (forall ((T_b $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b T_b))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_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_b (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (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) V_Pa_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) V_g_2))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_OwfP T_b V_r_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) V_r_2))))) (forall ((T_b $$unsorted)) (tptp.c_Wellfounded_OwfP T_b (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool T_b) tptp.c_fFalse)))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_OwfP T_b V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_b V_r_2) V_x_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_OwfP T_b V_r_2) (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp T_b V_r_2) B_x1))))) (forall ((V_p_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_OwfP T_b V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_p_2) V_r_2)) (tptp.c_Wellfounded_OwfP T_b V_p_2)))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_R_2) (tptp.c_Wellfounded_OwfP T_b (tptp.c_FunDef_Oin__rel T_b T_b V_R_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b 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.c_Wellfounded_OwfP T_b (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_b _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_b) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_b T_b)))) V_r_2)) (tptp.c_Wellfounded_Owf T_b V_r_2)))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_OwfP T_b V_r_2) (tptp.c_Wellfounded_Oacyclic T_b (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (and (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) 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_b T_b)) B_x) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_1 _let_1) T_b) (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_b) tptp.c_fconj) _let_2))) _let_2)) B_x))))))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)) V_r_2))))))) (forall ((V_x_2 $$unsorted) (V_Pa_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_Pa_2 (tptp.hAPP tptp.c_Int_OAbs__Integ B_y))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted) (B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Oacyclic T_b V_r_2))) (forall ((V_r_2 $$unsorted) (V_s_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Oacyclic T_b V_s_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.c_Wellfounded_Oacyclic T_b V_r_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Predicate_OPowp T_b V_A_2) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_B_2)) (tptp.hBOOL (tptp.hAPP V_A_2 B_x)))))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Equiv__Relations_Ocongruent T_b T_c V_r_2 V_f_2) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) B_x) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_COMBB T_c tptp.tc_HOL_Obool T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_1 T_b) tptp.c_fequal) V_f_2))) V_f_2)) B_x))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_b)) V_r_2)) (=> (tptp.c_Wellfounded_Oacyclic T_b V_r_2) (tptp.c_Wellfounded_Owf T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_b)) V_r_2)) (= (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Oacyclic T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Nitpick_Owf_H T_b V_r_2) (and (tptp.c_Wellfounded_Oacyclic T_b V_r_2) (or (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_b)) V_r_2)) (tptp.c_Nitpick_Ounknown tptp.tc_HOL_Obool))))) (forall ((V_x_2 $$unsorted)) (not (forall ((B_y $$unsorted)) (=> (= V_x_2 (tptp.hAPP tptp.c_Int_OAbs__Integ B_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool)) B_y) tptp.c_Int_OInteg))))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_B_2 B_a))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2)))))) (forall ((T_b $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted)) (=> (forall ((B_i $$unsorted)) (tptp.c_Wellfounded_OwfP T_c (tptp.hAPP V_r_2 B_i))) (=> (forall ((B_i $$unsorted) (B_j $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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_c T_c _let_3)) (tptp.c_Predicate_ORangeP T_c T_c _let_2)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))) (tptp.c_Wellfounded_OwfP T_c (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) V_r_2))))) (forall ((V_p_2 $$unsorted) (V_c_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (V_z_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d (tptp.tc_fun T_b 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_c T_d) B_x) B_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_z_2) (tptp.hAPP (tptp.hAPP V_c_2 B_x) B_y))))))))) (forall ((V_c_2 $$unsorted) (V_z_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_p_2 $$unsorted)) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (=> (= V_p_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) B_a) B_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_d) V_z_2) (tptp.hAPP (tptp.hAPP V_c_2 B_a) B_b))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_d) V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c (tptp.tc_fun T_d tptp.tc_HOL_Obool)) V_c_2) V_p_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_d))) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b _let_1 V_f_2) (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.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 T_c T_b (tptp.c_Product__Type_Ofst T_c T_d)) V_f_2)) V_A_2)) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 T_d T_b (tptp.c_Product__Type_Osnd T_c T_d)) V_f_2)) V_B_2))))) (forall ((V_na_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (= (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.c_Nat_OSuc V_na_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP V_f_2 B_x))))))) (forall ((V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.c_Hilbert__Choice_OEps _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) V_Pa_2)) (tptp.c_Hilbert__Choice_OEps _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool) _let_1) V_Pa_2) (tptp.c_Product__Type_Ofst T_b T_c))) (tptp.c_Product__Type_Osnd T_b T_c)))))) (forall ((V_g_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_b (tptp.c_Fun_Oid T_c)) V_g_2) V_g_2)) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_b V_f_2) (tptp.c_Fun_Oid T_b)) V_f_2)) (forall ((V_v_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_a_2) V_b_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_b (tptp.c_Fun_Oid T_c)) V_c_2)) (= (tptp.hAPP V_a_2 (tptp.hAPP V_b_2 V_v_2)) (tptp.hAPP V_c_2 V_v_2)))) (forall ((T_b $$unsorted) (V_c_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (=> (= V_c_2 (tptp.c_Hilbert__Choice_OEps T_b V_Pa_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_c_2))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_g_2) V_f_2)) V_x_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_d V_f_2) (tptp.hAPP (tptp.c_Set_Ovimage T_d T_c V_g_2) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapfst T_d T_b T_c V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_e T_d T_c V_g_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Oapfst T_e T_b T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_b T_e V_f_2) V_g_2)) V_x_2))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_c V_f_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_1) _let_1) _let_1)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_d V_f_2) V_g_2) V_x_2) (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)))) (forall ((T_b $$unsorted) (V_Pa_2 $$unsorted)) (=> (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Hilbert__Choice_OEps T_b V_Pa_2))))) (forall ((T_b $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Hilbert__Choice_OEps T_b V_Pa_2))) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Hilbert__Choice_OEps T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_x_2)) V_x_2)) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Hilbert__Choice_OEps T_b (tptp.hAPP tptp.c_fequal V_x_2)) V_x_2)) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_d T_b V_g_2) V_h_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_c T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_e V_f_2) V_g_2)) V_h_2))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_d V_f_2) V_g_2) V_x_2) (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 V_x_2)))) (forall ((V_v_2 $$unsorted) (V_d_2 $$unsorted) (V_c_2 $$unsorted) (T_e $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_a_2) V_b_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_c T_b 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_d_2 $$unsorted) (V_c_2 $$unsorted) (T_e $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_a_2) V_b_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_c T_b V_c_2) V_d_2)) (forall ((B_v $$unsorted)) (= (tptp.hAPP V_a_2 (tptp.hAPP V_b_2 B_v)) (tptp.hAPP V_c_2 (tptp.hAPP V_d_2 B_v)))))) (forall ((T_b $$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_b B_P))))) (forall ((T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Hilbert__Choice_OEps T_b V_Pa_2))))) (forall ((T_f $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$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_c T_b T_d V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_b T_f V_f_H_2) V_g_H_2) V_x_H_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_d T_c T_b V_f_2))) (= (tptp.hAPP _let_1 (tptp.c_Fun_Ofun__upd T_b T_d V_g_2 V_x_2 V_y_2)) (tptp.c_Fun_Ofun__upd T_b T_c (tptp.hAPP _let_1 V_g_2) V_x_2 (tptp.hAPP V_f_2 V_y_2))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_d T_c T_b V_f_2) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_e T_d T_b V_g_2) V_x_2)) (tptp.hAPP (tptp.c_Product__Type_Oapsnd T_e T_c T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_e V_f_2) V_g_2)) V_x_2))) (forall ((V_v_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_a_2) V_b_2) V_c_2) (= (tptp.hAPP V_a_2 (tptp.hAPP V_b_2 V_v_2)) (tptp.hAPP V_c_2 V_v_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_c_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_d (tptp.hAPP (tptp.c_COMBK T_b T_c) V_c_2)) V_f_2) V_x_2) V_c_2)) (forall ((V_prod_2 $$unsorted) (V_i_2 $$unsorted) (V_h_2 $$unsorted) (T_g $$unsorted) (T_f $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_e $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_d T_b T_e T_c V_f_2 V_g_2) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_f T_d T_g T_e V_h_2 V_i_2) V_prod_2)) (tptp.hAPP (tptp.c_Product__Type_Omap__pair T_f T_b T_g T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_b T_f V_f_2) V_h_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_e T_c T_g V_g_2) V_i_2)) V_prod_2))) (forall ((V_g2_2 $$unsorted) (V_g1_2 $$unsorted) (T_g $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_f $$unsorted) (T_e $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Omap__pair T_b T_d T_c T_e (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_d T_b V_f1_2) V_f2_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_g T_e T_c V_g1_2) V_g2_2)) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_f T_g) (tptp.tc_prod T_d T_e) (tptp.tc_prod T_b T_c) (tptp.c_Product__Type_Omap__pair T_f T_d T_g T_e V_f1_2 V_g1_2)) (tptp.c_Product__Type_Omap__pair T_b T_f T_c T_g V_f2_2 V_g2_2)))) (forall ((V_i_2 $$unsorted) (V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_g $$unsorted) (T_f $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_f T_g) (tptp.tc_prod T_d T_e) (tptp.tc_prod T_b T_c) (tptp.c_Product__Type_Omap__pair T_f T_d T_g T_e V_f_2 V_g_2)) (tptp.c_Product__Type_Omap__pair T_b T_f T_c T_g V_h_2 V_i_2)) (tptp.c_Product__Type_Omap__pair T_b T_d T_c T_e (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_d T_b V_f_2) V_h_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_g T_e T_c V_g_2) V_i_2)))) (forall ((V_r_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_b T_c V_f_2) V_g_2)) V_r_2) (tptp.hAPP (tptp.c_Set_Oimage T_d T_b V_f_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_d V_g_2) V_r_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (V_f_H_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_f_H_2) V_f_2) V_A_2) (tptp.c_Fun_Oinj__on T_b T_d V_f_2 V_A_2))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_g_2) V_f_2) V_A_2) (tptp.c_Fun_Oinj__on T_d T_c V_g_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_d V_f_2) V_A_2)))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.c_Fun_Oinj__on T_c T_d V_g_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (tptp.c_Fun_Oinj__on T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_g_2) V_f_2) V_A_2)))) (forall ((V_f_H_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (= (tptp.c_Fun_Oinj__on T_c T_d V_f_H_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (tptp.c_Fun_Oinj__on T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_f_H_2) V_f_2) V_A_2)))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_d tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_d T_b V_g_2 _let_1) (tptp.c_Fun_Oinj__on T_d T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_d V_f_2) V_g_2) _let_1))))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_d tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) _let_2) _let_3) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_d V_g_2) _let_3) _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_d T_c V_g_2) V_f_2)) _let_2) _let_1))))))) (forall ((V_h_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_d) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_d) V_h_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_d) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_h_2) V_f_2)) V_B_2))))) (forall ((V_h_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_d) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_B_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_d) V_h_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_d) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b 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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_d) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (=> (= V_g_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_h_2) V_f_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_d) V_h_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_d) V_g_2) V_A_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.c_Hilbert__Choice_OEps (tptp.tc_prod T_b T_c) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_1 T_b) 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_b T_c) V_x_2) V_y_2))))) (forall ((V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.c_Hilbert__Choice_OEps _let_1 V_Pa_2) (tptp.c_Hilbert__Choice_OEps _let_1 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c) V_Pa_2)) (tptp.c_Product__Type_OPair T_b T_c))))))) (forall ((V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (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 _let_1 V_g_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_b V_g_2) V_h_2)) V_A_2))))))) (forall ((V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_b T_b V_h_2 V_A_2) (= (tptp.hAPP (tptp.hAPP _let_1 V_g_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_b V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_b V_g_2) V_h_2)) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_d T_b V_g_2) V_A_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (=> (tptp.c_Fun_Oinj__on T_d T_b V_g_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_x_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) _let_1))) (= (tptp.hAPP (tptp.c_Fun_Othe__inv__into T_d T_c V_A_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_d V_f_2) V_g_2)) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_d T_c (tptp.c_Fun_Othe__inv__into T_d T_b V_A_2 V_g_2)) (tptp.c_Fun_Othe__inv__into T_b T_c _let_1 V_f_2)) V_x_2))))))) (forall ((V_z_2 $$unsorted) (V_g_2 $$unsorted) (V_h_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_d))) (=> (tptp.class_Groups_Oab__semigroup__mult T_d) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_h_2 V_A_2) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_d T_c _let_1) V_g_2) V_z_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_d T_b _let_1) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_g_2) V_h_2)) V_z_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP V_F_2 V_A_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))) (tptp.hAPP V_f_2 V_x_2)))))))) (forall ((V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image T_b T_c V_f_2 V_z_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (=> (tptp.c_Fun_Oinj__on T_c T_c V_h_2 V_A_2) (= (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_c V_h_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_F_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_c V_g_2) V_h_2)) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_B_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_F_2 V_B_2)))))))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (T_g $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_e $$unsorted) (T_f $$unsorted) (V_F_2 $$unsorted) (V_fa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_fa_2 V_F_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_e T_d V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_g T_f T_d V_g_2) V_h_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_g T_e T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_f T_e T_g V_f_2) V_g_2)) V_h_2)))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$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_b T_c V_f_2 V_F_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_2) _let_1)))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_d $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_c T_c T_d (tptp.hAPP V_f_2 V_y_2)))) (let ((_let_2 (tptp.c_Fun_Ocomp T_c T_c T_d (tptp.hAPP V_f_2 V_x_2)))) (=> (tptp.c_Finite__Set_Ofolding T_b T_c 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_f_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_d _let_1 T_c V_g_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_c T_d V_f_2) V_A_2)) V_g_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_e $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_d T_e) T_d _let_1 (tptp.c_Product__Type_Ofst T_d T_e)) (tptp.c_Product__Type_Omap__pair T_b T_d T_c T_e V_f_2 V_g_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_d _let_1 V_f_2) (tptp.c_Product__Type_Ofst T_b T_c))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_e $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod T_e T_d) T_d _let_1 (tptp.c_Product__Type_Osnd T_e T_d)) (tptp.c_Product__Type_Omap__pair T_b T_e T_c T_d V_f_2 V_g_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d _let_1 V_g_2) (tptp.c_Product__Type_Osnd T_b T_c))))) (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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image T_b T_c V_f_2 V_z_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) 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 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_e $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_d T_b) (tptp.hAPP (tptp.c_Fun_Ocomp T_e (tptp.tc_fun T_d T_b) T_c V_f_2) V_g_2)) V_x_2) (tptp.hAPP (tptp.hAPP V_f_2 (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_d) V_x_2))) (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_d) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_2) _let_1))))))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_c T_c T_d (tptp.hAPP V_f_2 V_x_2)))) (let ((_let_2 (tptp.c_Fun_Ocomp T_c T_c T_d (tptp.hAPP V_F_2 V_A_2)))) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_g_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_g_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_B_2))) (let ((_let_2 (tptp.hAPP V_F_2 V_A_2))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_2) _let_1))))))))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_c T_c T_d (tptp.hAPP V_F_2 V_B_2)))) (let ((_let_2 (tptp.c_Fun_Ocomp T_c T_c T_d (tptp.hAPP V_F_2 V_A_2)))) (let ((_let_3 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_A_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_B_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_g_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_g_2)))))))))) (forall ((V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (= (tptp.hAPP V_F_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Fun_Oid T_c)))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image T_b T_c V_f_2 V_z_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) 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_b T_c V_f_2) V_g_2) V_z_2) V_A_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_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.c_Fun_Ocomp T_c T_c T_c (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_f_2 V_x_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$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 T_b T_c 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_c T_c T_c (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_c T_c T_c (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_c $$unsorted) (T_b $$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 T_b T_c V_f_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.c_Fun_Ocomp T_c T_c T_c (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 ((T_c $$unsorted) (V_h_2 $$unsorted) (V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_d) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_S_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 B_x))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_d))))))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_d) V_h_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_d) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_h_2) V_f_2)) V_S_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.c_Finite__Set_Ofolding__idem T_b T_c 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_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c (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_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__idem T_b T_c V_f_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.c_Fun_Ocomp T_c T_c T_c _let_1) (tptp.hAPP V_F_2 V_B_2)) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (=> (tptp.c_Finite__Set_Ofolding__idem T_b T_c V_f_2 V_F_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_1) _let_1) _let_1)))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (T_d $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Fun_Ocomp T_c T_c T_d (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_b T_c V_f_2 V_F_2) (= (tptp.hAPP _let_1 _let_2) _let_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_F_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofolding__idem T_b T_c V_f_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.c_Fun_Ocomp T_c T_c T_c _let_1) (tptp.hAPP V_f_2 V_x_2)) _let_1)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__idem T_b T_c V_f_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.c_Fun_Ocomp T_c T_c T_c (tptp.hAPP V_F_2 V_A_2)) (tptp.hAPP V_f_2 V_x_2)))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__mult T_d) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (= (tptp.hAPP V_g_2 B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_h_2) V_f_2) B_x)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_c T_d) V_h_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_d) V_g_2) V_A_2))))))) (forall ((V_y $$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 $$unsorted)) (= V_x (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_y) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_n) B_q))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_A $$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 _let_1) B_A) (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_R_2) B_A))) (= B_A (tptp.c_Orderings_Obot__class_Obot _let_1))))) (tptp.c_Wellfounded_Owf T_b V_R_2))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_b T_c))) (=> (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_C_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP V_B_2 B_x)) (tptp.hAPP V_D_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP _let_1 V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP _let_1 V_C_2) V_D_2))))))) (forall ((V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Wellfounded_Oacc T_b V_R_2)))) (not (forall ((B_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_z) V_x_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_z) (tptp.c_Wellfounded_Oacc T_b V_R_2)))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_y) V_x_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_y) (tptp.c_Wellfounded_Oacc T_b V_r_2))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Wellfounded_Oacc T_b V_r_2))))) (forall ((V_R_2 $$unsorted) (V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Finite__Set_Ofinite T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_a1_2) V_a2_2)) (tptp.c_Wellfounded_Omax__ext T_b V_R_2))) (and (tptp.hBOOL (tptp.hAPP _let_2 V_a1_2)) (tptp.hBOOL (tptp.hAPP _let_2 V_a2_2)) (not (= V_a2_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_a1_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_a2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_xa)) V_R_2))))))))))) (forall ((V_i_2 $$unsorted) (V_Pa_2 $$unsorted) (V_p_2 $$unsorted)) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_p_2)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_i)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Divides_Odiv__class_Omod tptp.tc_Nat_Onat (tptp.hAPP tptp.c_Nat_OSuc B_i) V_p_2)))))) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_i_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) V_p_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_z_2 $$unsorted) (V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold__image T_c T_b (tptp.c_Groups_Otimes__class_Otimes T_c)))) (=> (tptp.class_Groups_Oab__semigroup__mult T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_y) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) B_y) V_x_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b V_A_2) V_x_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_y) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) B_y)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b V_A_2) V_x_2)))))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (not (forall ((B_k $$unsorted)) (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 (tptp.hAPP tptp.c_Nat_OSuc B_k))) (tptp.hAPP V_f_2 B_k))) V_r_2)))))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_d) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (= V_B_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) (=> (forall ((B_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_a) V_A_2)) (= (tptp.hAPP V_g_2 B_a) (tptp.hAPP V_h_2 (tptp.hAPP V_f_2 B_a))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_c T_d) V_h_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_d) V_g_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_c T_d T_e T_b) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP V_g_2 (tptp.hAPP (tptp.c_Product__Type_Ofst T_d T_e) _let_1)) (tptp.hAPP (tptp.c_Product__Type_Osnd T_d T_e) _let_1))))) (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_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c))) (=> (and (tptp.class_Groups_Ocomm__monoid__add T_c) (tptp.class_Orderings_Oord T_b)) (=> (= V_a_2 V_c_2) (=> (= V_b_2 V_d_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_c_2) B_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) B_x) V_d_2)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))))) (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan T_b V_c_2) V_d_2))))))))) (forall ((V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_d T_b T_c) (tptp.hAPP (tptp.c_Product__Type_OPair T_d T_b) V_x_2)) V_f_2) (tptp.hAPP V_f_2 V_x_2))) (forall ((V_x_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_c T_d (tptp.tc_prod T_c T_d)) V_x_2) (tptp.c_Product__Type_OPair T_c T_d)) V_x_2)) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_g $$unsorted) (T_f $$unsorted) (T_c $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_g T_c))) (let ((_let_2 (tptp.tc_fun T_d (tptp.tc_fun T_e T_c)))) (let ((_let_3 (tptp.tc_prod T_d T_e))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_d T_e T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_f T_g _let_3) V_f_2) V_g_2)) V_h_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_f T_g T_c) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_f _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_g _let_3) (tptp.tc_fun _let_2 _let_1) T_f) (tptp.c_Product__Type_Oscomp T_g T_d T_e T_c)) V_g_2)) V_h_2))))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_c T_d T_e T_b) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_e T_b) V_g_2) (tptp.hAPP V_f_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_e $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_c T_d T_e T_b) V_f_2) V_g_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_e T_b) V_g_2) (tptp.hAPP V_f_2 V_x_2)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Random_Oiterate T_c T_b 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_c T_b) V_x_2))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp T_b T_c T_b (tptp.tc_prod T_c T_b)) (tptp.hAPP V_f_2 V_x_2)) (tptp.c_Random_Oiterate T_c T_b (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_na_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_na_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_na_2)))) (forall ((V_k_2 $$unsorted) (V_f_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint))) (=> (forall ((B_i $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) B_i)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_i) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (tptp.hAPP V_f_2 B_i)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_m_2) V_na_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_m_2)) V_k_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) (tptp.hAPP V_f_2 V_na_2))) (exists ((B_i $$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_na_2)) (= (tptp.hAPP V_f_2 B_i) V_k_2)))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b V_r_2) (tptp.c_Wellfounded_Owf T_b (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))) (= tptp.c_Wellfounded_Oless__than (tptp.c_Transitive__Closure_Otrancl tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat)) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_Oacyclic T_b V_r_2) (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)) (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Relation_OField T_b) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) _let_1) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) _let_1)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_b) 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_b T_b T_b) (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 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (= (tptp.c_Transitive__Closure_Otrancl T_b _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_b T_b))) (= (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)) (tptp.hAPP _let_1 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_b T_b))) (= (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)) (tptp.hAPP _let_1 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_b)))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b 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_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_b_2)) V_R_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c_2)) (tptp.c_Transitive__Closure_Otrancl T_b V_R_2))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (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_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (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_s_2 $$unsorted) (V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (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_b 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_b V_s_2)))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) _let_1) V_r_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) V_A_2)))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)) _let_1)))))) (forall ((V_i $$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_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.tc_prod T_b T_b))) (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_b) (tptp.c_member _let_3)))) (let ((_let_6 (tptp.c_COMBC T_b _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_b tptp.tc_HOL_Obool))) (= (tptp.c_Transitive__Closure_Otrancl T_b (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_b V_r_2)) (tptp.hAPP (tptp.c_Set_OCollect _let_3) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_8 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_8 _let_8) T_b) (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_7 T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b _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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_b 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))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b _let_7)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_7) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_2 _let_7) T_b) (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_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_5 T_b) _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_b))))))))))))) (forall ((V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_r_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (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_a_2) V_a_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (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_a_2) V_a_2)) (tptp.c_Relation_OId T_b)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (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_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_s_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Orderings_Oord__class_Oless__eq _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.c_Relation_OId T_b)) V_s_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) _let_1) V_s_2)) V_r_2)) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 _let_1) V_s_2)))))))) (forall ((V_X_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (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_b T_b V_r_2) V_X_2)) V_X_2)) (= (tptp.hAPP (tptp.c_Relation_OImage T_b T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)) V_X_2) V_X_2))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_s_2)))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) 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_b V_S_2) _let_1)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_s_2))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_r_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)) _let_1)))))) (forall ((V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_c T_c) V_R_2) (tptp.c_Relation_OId T_c)) V_R_2)) (forall ((V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_c) (tptp.c_Relation_OId T_b)) V_R_2) V_R_2)) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_OImage T_b T_b (tptp.c_Relation_OId T_b)) V_A_2) V_A_2)) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (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_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_b_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_c_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_c_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (= (tptp.c_Transitive__Closure_Ortrancl T_b _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b))) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_R_2) (tptp.c_Relation_OId T_b))) (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (= (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_S_2))) (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.hAPP _let_1 V_R_2) V_S_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) _let_1) (tptp.c_Relation_OId T_b)) _let_1))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) _let_1) _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Relation_Orel__comp T_b T_b T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 V_r_2) _let_1) (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_r_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) (tptp.c_Relation_OId T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) _let_1) V_r_2))))) (forall ((T_b $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (tptp.c_Relation_OId T_b))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)) (tptp.c_Relation_OId T_b)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Transitive__Closure_Otrancl T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b))) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (= (tptp.c_Transitive__Closure_Otrancl T_b _let_1) _let_1))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.c_Transitive__Closure_Otrancl T_b V_R_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (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_b V_R_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b V_S_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b (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 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_b) (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_b) (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Orefl__on T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Transitive__Closure_Otrancl T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)) V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Transitive__Closure_Otrancl T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_r_2) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_S_2))) (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.hAPP _let_2 V_R_2) V_S_2))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (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_a_2) V_b_2)) (tptp.c_Relation_OId T_b))) (= V_a_2 V_b_2))) (forall ((V_z_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_x_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_z_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 V_z_2)) _let_1))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_a_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (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_b 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_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_b_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c_2)) (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))))))))) (forall ((V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)))) (let ((_let_2 (= V_a_2 V_b_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (or _let_2 (and (not _let_2) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_R_2))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (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_b V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c_2)) (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))))))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (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_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_b V_R_2))) (or _let_2 (and (not _let_2) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_R_2))))))))) (forall ((T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_b) (tptp.c_Relation_OId T_b)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_b T_b) (tptp.c_Relation_OId T_b)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Orefl__on T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.c_Relation_OId T_b))) (= tptp.c_FunDef_Opair__leq (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod _let_24 _let_24) tptp.tc_HOL_Obool)) tptp.c_FunDef_Opair__less) (tptp.c_Relation_OId _let_24))) (forall ((V_y_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_b) V_R_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (= V_x_2 V_y_2)))) (forall ((V_b_2 $$unsorted) (V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_b V_r_2))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_b_2) V_a_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_b V_r_2))) (let ((_let_2 (tptp.c_member T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_b_2) V_a_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_a_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) _let_1))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.tc_prod T_b T_b))) (= (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) V_r_2)) (and (tptp.c_Wellfounded_Owf T_b V_r_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.tc_prod T_b T_b))) (= (tptp.c_Wellfounded_Oacyclic T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) V_r_2)) (and (tptp.c_Wellfounded_Oacyclic T_b V_r_2) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Orefl__on T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b)))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b _let_1))) (= (tptp.c_Relation_OId T_b) (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_b)) (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_b)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b _let_1) (tptp.c_Product__Type_OPair T_b T_b)) (tptp.c_COMBI T_b)))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Oirrefl T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b)))) (forall ((V_na_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_na_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_na_2)))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) V_A_2)))) (or (= V_a_2 V_b_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (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_b T_b) V_r_2)) (tptp.c_Transitive__Closure_Ortrancl T_b 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_b T_b) V_x_2) V_x_2)) (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted) (V_r_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_i $$unsorted)) (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 (tptp.hAPP tptp.c_Nat_OSuc B_i))) (tptp.hAPP V_f_2 B_i))) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_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 (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_b V_r_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Ototal__on T_b V_A_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b))) (tptp.c_Relation_Ototal__on T_b V_A_2 V_r_2))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OId T_b))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_b))) (= (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_b))) (= (tptp.c_Transitive__Closure_Otrancl T_b (tptp.hAPP _let_1 V_r_2)) (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_Oacyclic T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.c_Wellfounded_Oacyclic T_b V_r_2))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Orefl__on T_b V_A_2 (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_c T_b)) V_r_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OId__on T_b V_A_2))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.hAPP _let_1 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_c) V_r_2)) V_r_2)) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Ototal__on T_b V_A_2 (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.c_Relation_Ototal__on T_b V_A_2 V_r_2))) (forall ((V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oinv__image T_c T_b))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) (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_c T_c) V_R_2)) V_f_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_c T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_c T_b) 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_b T_c) 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_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_c T_b))) (= (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_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_c) 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_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_c T_d T_b) V_r_2) V_s_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) (tptp.hAPP (tptp.c_Relation_Oconverse T_d T_b) V_s_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_d) V_r_2)))) (forall ((V_r_2 $$unsorted) (V_S_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_c T_b))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_c) 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_S_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_2) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) V_b_2) V_a_2)) V_r_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (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_a_2) V_b_2)) V_r_2)) (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) V_b_2) V_a_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_c) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) V_b_2) V_a_2)) V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_b))) (=> (tptp.c_Wellfounded_Owf T_b (tptp.hAPP _let_1 V_r_2)) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) V_r_2) V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) V_r_2) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2)))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2)) (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Relation_ORange T_c T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_c) V_r_2)) (tptp.hAPP (tptp.c_Relation_ODomain T_b T_c) V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Ototal__on T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_r_2)) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_b (tptp.hAPP _let_1 V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_b T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Otrancl T_b (tptp.hAPP _let_1 V_r_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c 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_c T_b 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_b T_c (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) 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_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) V_r_2))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) V_r_2) V_r_2) (=> (= (tptp.hAPP (tptp.c_Relation_ODomain T_b T_b) V_r_2) V_A_2) (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_prod T_b T_b)) V_r_2)) (=> (tptp.c_Wellfounded_Oacyclic T_b V_r_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Ototal__on T_b V_A_2 V_r_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (forall ((B_xa $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_A_2)) (=> (not (= B_x B_xa)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_xa)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_xa) B_x)) V_r_2)))))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$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))) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2) (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) (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_r_2)))))))))) (forall ((V_R_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_member _let_1) V_p_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool)) V_R_2))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (not (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) B_x) B_x)) (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))))) (tptp.c_Wellfounded_Oacyclic T_b V_r_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_c T_b V_R_2))) (=> (tptp.c_Relation_Osingle__valued T_b T_c (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_R_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (=> (tptp.c_Relation_Osingle__valued T_b T_c V_s_2) (tptp.c_Relation_Osingle__valued T_b T_c V_r_2)))) (forall ((V_f_2 $$unsorted) (V_na_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_c))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 _let_1) V_na_2) (tptp.c_Fun_Ocomp T_c T_c T_b V_f_2)) (tptp.c_Fun_Ocomp T_c T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow T_c T_c) V_na_2) V_f_2))))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool) V_na_2) V_R_2)) (tptp.c_Wellfounded_Owf T_b V_R_2))) (forall ((V_f_2 $$unsorted) (V_m_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow T_b T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 V_na_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_na_2)) V_f_2)))) (forall ((V_x_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow T_b T_b) V_na_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_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_R_2) V_R_2)) (forall ((V_na_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osingle__valued T_b T_b V_R_2) (tptp.c_Relation_Osingle__valued T_b T_b (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool) V_na_2) V_R_2)))) (forall ((V_na_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool) V_na_2) V_R_2))) (let ((_let_2 (tptp.c_Relation_Orel__comp T_b T_b T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 V_R_2) _let_1) (tptp.hAPP (tptp.hAPP _let_2 _let_1) V_R_2))))) (forall ((V_s_2 $$unsorted) (T_d $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osingle__valued T_b T_c V_r_2) (=> (tptp.c_Relation_Osingle__valued T_c T_d V_s_2) (tptp.c_Relation_Osingle__valued T_b T_d (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_c T_d) V_r_2) V_s_2))))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Osingle__valued T_b T_b (tptp.c_Relation_OId T_b))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (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_na_2) V_R_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))))))) (forall ((V_R_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) V_p_2) (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (exists ((B_n $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (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_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Osingle__valued T_b T_b (tptp.c_Relation_OId__on T_b V_A_2))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_R_2)) V_R_2)))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_na_2)) V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_R_2)) (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_R_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_R_2) (tptp.c_Relation_OId T_b))) (forall ((V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_na_2 $$unsorted) (V_z_2 $$unsorted) (V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_b T_b))) (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_na_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_na_2)) V_R_2))))))))))) (forall ((V_z_2 $$unsorted) (V_R_2 $$unsorted) (V_na_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Product__Type_OPair T_b T_b))) (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_na_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_na_2)) V_R_2))))))))))) (forall ((V_f_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow T_b T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2)) V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b V_f_2) (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_f_2))))) (forall ((V_f_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Ocompow T_b T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_na_2)) V_f_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_b T_b (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_f_2)) (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_f_2))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_c)))) (=> (tptp.c_Relation_Osingle__valued T_b T_c V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z_2)) V_r_2)) (= V_y_2 V_z_2))))))) (forall ((V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Osingle__valued T_b T_c V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (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) B_x) B_y)) V_r_2)) (forall ((B_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) B_x) B_z)) V_r_2)) (= B_y B_z))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow T_b T_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_f_2) (tptp.c_Fun_Oid T_b))) (forall ((V_R_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) V_p_2) (tptp.c_Transitive__Closure_Otrancl T_b V_R_2))) (exists ((B_n $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) B_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_p_2) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) B_n) V_R_2)))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR tptp.tc_Nat_Onat _let_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool)) V_R_2)))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_b T_b)))) (let ((_let_4 (tptp.hAPP _let_2 V_x_2))) (=> (tptp.c_Relation_Osingle__valued T_b T_b 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 ((T_b $$unsorted)) (= (tptp.c_Nat_Ofunpow T_b) (tptp.c_Nat_Ocompow T_b T_b))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_na_2) V_R_2))) (=> (=> (= V_na_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_b T_b))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.c_member _let_1))) (=> (= V_na_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_R_2 $$unsorted) (V_na_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_na_2) V_R_2))) (=> (=> (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (not (= V_x_2 V_z_2))) (not (forall ((B_y $$unsorted) (B_m $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.tc_prod T_b T_b))) (let ((_let_3 (tptp.c_member _let_2))) (=> (= V_na_2 (tptp.hAPP tptp.c_Nat_OSuc B_m)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_2 tptp.tc_HOL_Obool) B_m) V_R_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 B_y) V_z_2)) V_R_2)))))))))))))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_Pa_2) V_Q_2)))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) B_x)) (tptp.c_Transitive__Closure_Ortrancl T_b V_Pa_2))) (forall ((B_y $$unsorted)) (=> (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) B_x) B_y)) V_Q_2)) (= B_x B_y))))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b V_Pa_2)))))))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_Pa_2) V_Q_2)))) (=> (forall ((B_x $$unsorted)) (=> (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) B_x) V_b_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_Pa_2))) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_y) B_x)) V_Q_2)) (= B_y B_x))))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl T_b V_Pa_2)))))))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_na_2)) V_R_2))) (exists ((B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.c_member _let_1))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) B_y)) V_R_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 B_y) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_na_2) V_R_2))))))))))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_na_2)) V_R_2))) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.tc_prod T_b T_b))) (let ((_let_3 (tptp.c_member _let_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_y)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_2 tptp.tc_HOL_Obool) V_na_2) V_R_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 B_y) V_z_2)) V_R_2)))))))))))) (forall ((V_R_2 $$unsorted) (V_na_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_na_2)) V_R_2))) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_3 (tptp.c_member _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) B_y)) V_R_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_2 B_y) V_z_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 tptp.tc_HOL_Obool) V_na_2) V_R_2))))))))))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Otrancl T_b V_R_2))) (exists ((B_z $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (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_b V_R_2))))))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (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_x_2) V_y_2)) (tptp.c_Transitive__Closure_Otrancl T_b V_R_2))) (exists ((B_z $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x_2) B_z)) (tptp.c_Transitive__Closure_Ortrancl T_b V_R_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_z) V_y_2)) V_R_2)))))))) (forall ((V_p_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) V_p_2) (tptp.c_Relation_OId T_b))) (not (forall ((B_x $$unsorted)) (not (= V_p_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_x_2) V_A_2))))))) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_c T_b V_f_2) V_z_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_z_2)) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_b T_c V_f_2) V_g_2) (tptp.c_Finite__Set_Ofold T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) V_f_2) V_g_2)))) (forall ((V_s_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding T_b T_c V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP V_F_2 V_A_2) V_s_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_s_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_c T_b V_f_2) V_z_2) V_A_2) (tptp.c_HOL_OThe T_b (tptp.c_Finite__Set_Ofold__graph T_c T_b V_f_2 V_z_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b _let_1 (tptp.c_Set_Oinsert T_b)) V_B_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2))) (=> (tptp.class_Lattices_Osemilattice__sup T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2))) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP _let_2 V_A_2)))))))) (forall ((V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold T_b T_c V_f_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_z_2)) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_c))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b T_c) V_A_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) _let_1) V_f_2)) V_B_2) V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one__idem T_b V_f_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP V_F_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b V_f_2) V_a_2) V_A_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (=> (tptp.class_Lattices_Osemilattice__sup T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2) V_A_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b))) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_b_2) V_A_2)))))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_a_2) V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b)) V_a_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) V_a_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_a_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Orderings_Oord__class_Omax T_b)) V_a_2) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_a_2) V_A_2)))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_a_2) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_1) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_Set_Oinsert T_b)) (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_B_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__one T_b V_f_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 (tptp.c_Finite__Set_Ofold T_b T_b V_f_2) V_x_2) V_A_2)))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b T_c) V_A_2) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_c)) V_f_2)) (tptp.c_Orderings_Obot__class_Obot T_c)) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2))) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_a_2) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (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.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b)) V_x_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) V_x_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Orderings_Oord__class_Omin T_b)) V_x_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Orderings_Oord__class_Omax T_b)) V_x_2) V_A_2)))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (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.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_x_2) V_A_2))))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c 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 _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_b) 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_c $$unsorted) (T_b $$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))) (let ((_let_3 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_c) V_A_2) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_c)) V_f_2)) (tptp.c_Orderings_Otop__class_Otop T_c)) V_A_2))))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_b_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) 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 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_b_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c (tptp.tc_fun T_b 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_c) V_a_2) V_A_2))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_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 (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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c 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_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 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c (tptp.tc_fun T_d 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_b) 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 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c (tptp.tc_fun T_d 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_b) V_a_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted)) (=> (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_I_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_A_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_I_2) V_A_2))))) (forall ((V_B_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) V_B_2) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) (tptp.hAPP (tptp.c_COMBK _let_1 T_b) V_c_2)))) (let ((_let_3 (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b 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_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_c _let_1))) (let ((_let_4 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_d _let_4) V_A_2) V_B_2)) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_3 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_3 _let_1) T_d) _let_2) V_B_2)) V_C_2)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_2))) (let ((_let_4 (tptp.tc_fun T_d _let_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_COMBC T_c _let_4 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_4 _let_2) T_c) _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_c _let_1) V_A_2) V_B_2)) V_C_2))))))) (forall ((V_M_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_u_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_u_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c T_b) V_A_2) V_M_2))) (forall ((B_x $$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 T_b) V_u_2) (tptp.hAPP V_M_2 B_x)))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (tptp.class_Orderings_Olinorder T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c T_b) V_A_2) V_f_2)) V_a_2)) (exists ((B_x $$unsorted)) (and (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 T_b) (tptp.hAPP V_f_2 B_x)) V_a_2))))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_u_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_c) V_a_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_u_2) 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 ((T_b $$unsorted)) (=> (tptp.class_Groups_Oab__semigroup__mult T_b) (tptp.c_Finite__Set_Ofun__left__comm T_b T_b (tptp.c_Groups_Otimes__class_Otimes T_b)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_b_2) (tptp.hAPP V_B_2 B_x))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.tc_prod T_c T_d))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.c_Product__Type_OPair T_c T_d))) (let ((_let_5 (tptp.c_member _let_2))) (let ((_let_6 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_fun T_d _let_6))) (let ((_let_8 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.tc_fun T_c _let_8))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_9) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_9 T_b) (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_8) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_3 _let_8) T_c) (tptp.c_COMBC T_d _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d _let_2) _let_7 T_c) (tptp.hAPP (tptp.c_COMBB _let_2 _let_6 T_d) _let_5)) _let_4)))) V_r_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_x_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_3) _let_1) V_r_2)))))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) V_B_2) (tptp.c_Orderings_Otop__class_Otop _let_1)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_c _let_1) V_A_2) V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (= (tptp.hAPP V_B_2 B_x) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (tptp.c_Finite__Set_Ofun__left__comm T_d T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c T_c) T_d) V_f_2) V_g_2)))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$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_b T_c 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_B_2 $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_I_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c (tptp.tc_fun T_d T_b)) V_A_2) V_f_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c T_b) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_d T_b) V_f_2) V_x_2))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c T_b) V_A_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d T_b) V_B_2))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d T_b) T_b T_c) _let_2) V_f_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c T_b) T_b T_d) _let_1) (tptp.hAPP (tptp.c_COMBC T_c T_d T_b) V_f_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OPow T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_3) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_3 T_c) _let_1) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_d T_c V_f_2) V_A_2)) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_1 T_d) V_B_2) V_f_2))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_d _let_1 T_c) V_B_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_1) (tptp.hAPP (tptp.c_Set_Oimage T_c T_d V_f_2) V_A_2)) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$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_c _let_1) V_C_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_2) V_C_2))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_3 V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_c) _let_1) V_B_2))))))) (forall ((T_b $$unsorted)) (= (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b tptp.tc_HOL_Obool) (tptp.c_Set_OBall T_b))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (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_d _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.hAPP V_f_2 V_y_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_1) _let_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_c T_c _let_2) _let_1)))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (T_b $$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 _let_1) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_I_2) V_A_2))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_I_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) (tptp.hAPP V_A_2 B_x)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c 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_B_2 $$unsorted) (V_J_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (let ((_let_3 (tptp.tc_fun _let_1 _let_1))) (let ((_let_4 (tptp.tc_fun T_d _let_1))) (let ((_let_5 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_1) V_J_2))) (let ((_let_6 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_I_2))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_6 V_A_2)) (tptp.hAPP _let_5 V_B_2)) (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_1 T_c) _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_4 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_4) T_c) (tptp.c_COMBB _let_1 _let_1 T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) _let_2) V_A_2))) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (T_b $$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_OINFI T_c _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_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_2) V_A_2)) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_I_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_I_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_A_2)) (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) _let_2) V_A_2))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$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_OINFI T_c _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_c) _let_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_C_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_3) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$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_c _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_c) _let_3) V_B_2)) (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_B_2))))))) (forall ((V_r_2 $$unsorted) (V_S_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_Oconverse T_c T_b))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_3) V_S_2) V_r_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_2) V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_r_2))))))) (forall ((V_M_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_c) V_A_2) (tptp.hAPP (tptp.c_COMBK T_c T_b) V_M_2)) V_M_2)))) (forall ((V_A_2 $$unsorted) (T_c $$unsorted) (V_I_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_I_2) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (let ((_let_3 (tptp.c_member T_c))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (tptp.hAPP (tptp.c_COMBC T_c _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_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2) V_B_2)) (tptp.hAPP V_B_2 V_a_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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_b _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_b 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_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) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 V_A_2)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_c) 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_Pa_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OAll T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_3 T_b) (tptp.hAPP (tptp.c_COMBS T_c 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_c) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_3 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_A_2)))) V_Pa_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_2 _let_2 T_c) _let_1) (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) V_Pa_2)))))))) (forall ((V_M_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2)) V_M_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_A_2) V_M_2)) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) V_M_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$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))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) _let_2) V_B_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_C_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_C_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) _let_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_c T_b V_r_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_3) V_A_2) V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2)))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) _let_2) V_B_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus _let_1))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) _let_2) V_B_2)))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_c))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (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)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) V_B_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OAll T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS T_c 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_c) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _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_b) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c)) (tptp.c_member T_b))) V_B_2))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (let ((_let_3 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_3) V_A_2)) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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__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_b) _let_3) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b 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_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1))) (let ((_let_4 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) _let_3) V_A_2)) V_B_2)))) (let ((_let_5 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_5 (= _let_4 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_5) (= _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_A_2)) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c 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_b _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_b 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_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) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2) V_B_2)))) (let ((_let_4 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_4 (= _let_3 V_A_2)) (=> (not _let_4) (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2)))))))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_f_2 V_x_2))) (let ((_let_2 (tptp.c_Finite__Set_Ofold T_b T_c V_f_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_z_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_z_2)) V_A_2))))))) (forall ((V_M_2 $$unsorted) (V_A_2 $$unsorted) (V_i_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_i_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b T_c) V_A_2) V_M_2)) (tptp.hAPP V_M_2 V_i_2)))))) (forall ((V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2) V_y_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_z_2) V_A_2) V_y_2)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI tptp.tc_Nat_Onat _let_16) _let_17) _let_23) _let_22) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_C_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) _let_2) V_B_2)))) (let ((_let_4 (= V_C_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (and (=> _let_4 (= _let_3 (tptp.c_Orderings_Otop__class_Otop _let_1))) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_1) V_C_2) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (T_d $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OImage T_c T_b V_r_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Relation_Osingle__valued T_b T_c (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_d tptp.tc_HOL_Obool)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2))))))))) (forall ((V_f_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_c))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b T_c) V_A_2) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c T_c) T_b) _let_1) V_f_2)) V_B_2) V_A_2)))))) (forall ((V_z_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_z_2) V_A_2)))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofold T_b T_c V_f_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_z_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_z_2)) V_A_2))))))) (forall ((V_z_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_z_2))) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP _let_1 V_A_2)))))))) (forall ((V_a_2 $$unsorted) (V_y_2 $$unsorted) (V_A_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2) V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (exists ((B_y_H $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (and (= V_y_2 (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) B_y_H)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) B_y_H))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) (tptp.c_Orderings_Otop__class_Otop T_b)) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b)) (tptp.c_Orderings_Obot__class_Obot T_b)) V_A_2))))) (forall ((V_M_2 $$unsorted) (T_b $$unsorted)) (=> (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_x) V_M_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) B_x)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Complete__Lattice_OInf__class_OInf (tptp.tc_fun T_b tptp.tc_HOL_Obool) V_M_2))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (V_X_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) 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_X_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_A_2))) (let ((_let_2 (tptp.tc_fun T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) V_A_2))) (let ((_let_2 (tptp.tc_fun T_b 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_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c (tptp.tc_fun T_d tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_A_2) V_B_2) V_b_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_c 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_c _let_1) _let_1) _let_2) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _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_c _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_A_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_B_2) V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_2))) (= (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI _let_1 _let_1) V_B_2) _let_2))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) V_C_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) B_x) V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_S_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (tptp.class_Orderings_Olinorder T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_S_2))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_a_2) B_x))))))) (forall ((V_a_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (tptp.class_Orderings_Olinorder T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_S_2)) V_a_2)) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) B_x) V_a_2))))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_b_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_A_2))) (forall ((B_x $$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_b) V_b_2) B_x))))))) (forall ((V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_A_2)) V_b_2)) (forall ((B_x $$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_Orderings_Oord__class_Oless__eq T_b) B_x) V_b_2))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_OField T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun (tptp.tc_prod T_b T_b) 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 ((V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Complete__Lattice_OSup__class_OSup (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.c_Set_OPow T_b) V_A_2)) V_A_2)) (forall ((V_S_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (let ((_let_2 (tptp.tc_fun T_c 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_b tptp.tc_HOL_Obool)) V_S_2) _let_1))))) (forall ((V_B_2 $$unsorted) (V_X_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OSigma T_b T_c))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_c) tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b 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_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) _let_2) _let_1) V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _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_c _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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_c 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_c _let_1) _let_1) _let_2) V_B_2))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2)) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_1) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_1) _let_2) V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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))) (= (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_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Relation_ORange T_c T_b))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_c T_b) tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) _let_1))))) (forall ((V_S_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Relation_ODomain T_b T_c))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_b T_c) 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_b tptp.tc_HOL_Obool)) V_S_2) _let_1))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Ovimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_c 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_b tptp.tc_HOL_Obool)) V_A_2) _let_1))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_A_2) (tptp.c_Complete__Lattice_OSup__class_OSup 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_OAll T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (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_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))) (tptp.c_Orderings_Oord__class_Oless__eq T_b))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_A_2) (tptp.c_Complete__Lattice_OInf__class_OInf 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_OAll T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (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_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))) (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless__eq T_b)))))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_B_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2)) V_B_2))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2)))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (= (tptp.c_Complete__Lattice_OSup__class_OSup (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2)) V_A_2))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_a_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_a_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_A_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) (tptp.hAPP (tptp.c_Set_OPow T_b) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$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 _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 ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop _let_1)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OAll _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fimplies) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.c_member T_b)))))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2)) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_B_2) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_B_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_A_2)))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_A_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_A_2)) V_x_2))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot T_b)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2)) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_B_2))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_a_2))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_a_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_2) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_A_2) V_C_2))) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_B_2) V_C_2))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) V_a_2))) (forall ((T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop T_b)))) (forall ((T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop T_b)))) (forall ((T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot T_b)))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_a_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_a_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_A_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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) (V_C_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_c) _let_2) V_A_2)) V_B_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_A_2) V_C_2))) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_c _let_1 V_B_2) V_C_2))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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 _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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_Oinsert _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_A_2) (tptp.hAPP (tptp.hAPP _let_2 V_B_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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 _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_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_a_2) V_b_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b (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_b tptp.tc_HOL_Obool))))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_a_2) V_b_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_B_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_B_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b V_A_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_b V_A_2)))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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_b) V_A_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_B_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_B_2) V_A_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b V_A_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_b V_A_2)))))) (forall ((V_j_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (V_C_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_C_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_d) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP V_B_2 B_x)) V_C_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_d) V_j_2) V_A_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2)))))))))) (forall ((V_k_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_b) _let_2)) (=> (forall ((B_c $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_c) V_C_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) B_c) V_k_2))) (=> (forall ((B_c1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_c1) V_C_2)) (forall ((B_c2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_c2) V_C_2)) (=> (not (= B_c1 B_c2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) B_c1) B_c2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard _let_1) V_C_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) _let_2))))))))) (forall ((V_S_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI _let_1 _let_1) V_S_2) (tptp.c_COMBI _let_1))))) (forall ((V_S_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_S_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 _let_1) V_S_2) (tptp.c_COMBI _let_1))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) V_A_2)) (=> (forall ((B_M $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_M) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) B_M)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2))))))) (forall ((V_F_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_x_2) V_F_2))) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_xa) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) V_x_2) V_F_2))) (=> (not (= B_x B_xa)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) B_x) B_xa) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_x_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_F_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_S_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (not (= V_S_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (=> (forall ((B_A $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_A) V_S_2)) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 B_A))) (tptp.c_Fun_Oinj__on T_b T_c 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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (forall ((B_X $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b 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_A_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_I_2 $$unsorted)) (=> (not (= V_I_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_i) V_I_2)) (tptp.c_Fun_Oinj__on T_c T_d V_f_2 (tptp.hAPP V_A_2 B_i)))) (tptp.c_Fun_Oinj__on T_c T_d V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_I_2) V_A_2))))) (forall ((V_v_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) V_A_2)) V_v_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2))) (not (forall ((B_y $$unsorted)) (=> (= V_v_2 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) B_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofold__graph T_b T_c V_f_2 V_z_2 V_A_2) B_y)))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$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_b) B_x) V_b_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_b_2) B_x)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_c_2) V_A_2)))))))) (forall ((V_A_2 $$unsorted) (V_C_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_X $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_X) V_C_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_A_2) B_X)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_A_2) (tptp.c_Complete__Lattice_OInf__class_OInf (tptp.tc_fun T_b tptp.tc_HOL_Obool) V_C_2))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b))) (=> (tptp.class_Lattices_Osemilattice__inf T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$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_b) V_b_2) B_x)))) (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_c_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_c_2) V_A_2)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b))) (=> (tptp.class_Lattices_Osemilattice__sup T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) B_x) V_b_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_b _let_1) V_c_2) V_A_2)) (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)))))))) (forall ((V_b $$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 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Wellfounded_Owf T_b V_r_2) (forall ((B_Q $$unsorted)) (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) B_Q))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) B_Q)) (forall ((B_y $$unsorted)) (=> (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) B_y) B_x)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_y) B_Q))))))))))) (forall ((V_P_H_2 $$unsorted) (V_Pa_2 $$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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) B_x)) (= (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (tptp.hBOOL (tptp.hAPP V_P_H_2 B_x))))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_Pa_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_P_H_2))))))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_OCollect T_b))) (let ((_let_2 (tptp.tc_fun T_b 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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Q_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_A_2) (tptp.hAPP _let_1 V_Pa_2))) (tptp.hAPP (tptp.hAPP _let_3 V_B_2) (tptp.hAPP _let_1 V_Q_2))))))))))) (forall ((V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_A_2) (tptp.c_Complete__Lattice_OSup__class_OSup (tptp.tc_fun T_b tptp.tc_HOL_Obool) V_C_2))) (not (forall ((B_X $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_A_2) B_X)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_X) V_C_2)))))))) (forall ((V_r_2 $$unsorted) (V_yx_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) V_yx_2) (tptp.hAPP (tptp.c_Relation_Oconverse T_c T_b) V_r_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (= V_yx_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) B_y) B_x)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) B_x) B_y)) V_r_2)))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_d $$unsorted) (V_xz_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_c)) V_xz_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_d T_c) V_r_2) V_s_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted) (B_z $$unsorted)) (=> (= V_xz_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) B_x) B_z)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_d)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_d) B_x) B_y)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_d T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_d T_c) B_y) B_z)) V_s_2))))))))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_r_2))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.c_Divides_Odiv__class_Odiv tptp.tc_Int_Oint)))) (let ((_let_4 (tptp.c_Nitpick_Onum T_b))) (let ((_let_5 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_6 (tptp.c_COMBB tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint))) (let ((_let_7 (tptp.hAPP _let_1 V_q_2))) (= (tptp.c_Nitpick_Oplus__frac T_b V_q_2 V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet tptp.tc_Int_Oint T_b) (tptp.c_Nitpick_Oint__lcm _let_7 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint T_b) tptp.tc_Int_Oint) (tptp.c_Nitpick_Ofrac T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Int_Oint tptp.tc_Int_Oint tptp.tc_Int_Oint) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Int_Oint (tptp.tc_fun tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint) (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint)) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_q_2))) (tptp.hAPP _let_3 _let_7)))) (tptp.hAPP (tptp.hAPP _let_6 (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_r_2))) (tptp.hAPP _let_3 _let_2))))) (tptp.c_COMBI tptp.tc_Int_Oint)))))))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_c) (=> (forall ((B_x $$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 T_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2) V_A_2)))))) (forall ((V_r $$unsorted) (V_q $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Odenom T_a))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Int_Oint))) (let ((_let_3 (tptp.c_Nitpick_Onum T_a))) (= (tptp.c_Nitpick_Otimes__frac T_a V_q V_r) (tptp.hAPP (tptp.hAPP (tptp.c_Nitpick_Ofrac T_a) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_3 V_q)) (tptp.hAPP _let_3 V_r))) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_q)) (tptp.hAPP _let_1 V_r)))))))) (forall ((V_q $$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))))))) (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)))) (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))))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Nitpick_Oless__frac T_b V_q_2 V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.hAPP (tptp.c_Nitpick_Onum T_b) (tptp.c_Nitpick_Oplus__frac T_b V_q_2 (tptp.c_Nitpick_Ouminus__frac T_b V_r_2)))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((V_r_2 $$unsorted) (V_q_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Nitpick_Oless__eq__frac T_b 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_b) (tptp.c_Nitpick_Oplus__frac T_b V_q_2 (tptp.c_Nitpick_Ouminus__frac T_b V_r_2)))) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint))))) (forall ((T_b $$unsorted)) (= (tptp.c_Nitpick_Odenom T_b) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint T_b (tptp.c_Product__Type_Osnd tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.c_Nitpick_ORep__Frac T_b)))) (forall ((T_b $$unsorted)) (= (tptp.c_Nitpick_Onum T_b) (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.tc_Int_Oint T_b (tptp.c_Product__Type_Ofst tptp.tc_Int_Oint tptp.tc_Int_Oint)) (tptp.c_Nitpick_ORep__Frac T_b)))) (forall ((T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Ointernal__split T_b T_c T_d) (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Rings_Olinordered__semidom T_c) (=> (forall ((B_x $$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_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP V_f_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.c_Groups_Ozero__class_Ozero T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2) V_A_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Ointernal__split T_c T_d T_b) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) V_a_2) V_b_2)) (tptp.hAPP (tptp.hAPP V_c_2 V_a_2) V_b_2))) (= (tptp.c_HOL_Obool_Obool__size tptp.c_fTrue) _let_12) (= (tptp.c_HOL_Obool_Obool__size tptp.c_fFalse) _let_12) (forall ((V_u_2 $$unsorted) (V_f_2 $$unsorted)) (=> (forall ((B_n $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) B_n) (tptp.hAPP V_f_2 B_n)))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Set_OCollect tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) tptp.tc_Nat_Onat) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat)) V_f_2)) V_u_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_prod tptp.tc_Nat_Onat _let_1))) (= (tptp.c_List_Olenlex T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_2 _let_1) (tptp.c_Wellfounded_Olex__prod tptp.tc_Nat_Onat _let_1 tptp.c_Wellfounded_Oless__than (tptp.c_List_Olex T_b V_r_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_1 _let_2) _let_1) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat _let_1)) (tptp.c_Nat_Osize__class_Osize _let_1))) (tptp.c_COMBI _let_1))))))) (forall ((V_ys $$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_na_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.c_List_Olexn T_b V_r_2) V_na_2))) (and (= (tptp.hAPP _let_2 V_xs_2) V_na_2) (= (tptp.hAPP _let_2 V_ys_2) V_na_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b 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_b V_r_2)))))))))))))))))))))))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (= (tptp.hAPP _let_1 (tptp.c_List_Osublist T_b 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_na_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc V_na_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_2) tptp.c_fequal) (tptp.c_Nat_Osize__class_Osize _let_2))) _let_1))) (let ((_let_4 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_6 (tptp.tc_prod _let_2 _let_2))) (let ((_let_7 (tptp.c_List_Olexn T_b V_r_2))) (let ((_let_8 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b _let_2 _let_2) (tptp.c_List_Olist_OCons T_b)))) (let ((_let_9 (tptp.tc_prod T_b _let_2))) (= (tptp.hAPP _let_7 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun _let_6 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage (tptp.tc_prod _let_9 _let_9) _let_6 (tptp.c_Product__Type_Omap__pair _let_9 _let_2 _let_9 _let_2 _let_8 _let_8)) (tptp.c_Wellfounded_Olex__prod T_b _let_2 V_r_2 (tptp.hAPP _let_7 V_na_2)))) (tptp.hAPP (tptp.c_Set_OCollect _let_6) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_2 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_5 _let_5) _let_2) (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_4 _let_2) tptp.c_fconj) _let_3))) _let_3))))))))))))))) (forall ((V_x $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 V_ys))) (not (= V_xs (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_ys)))))) (forall ((V_x $$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_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs) V_xs))) (forall ((V_list_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_list_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_b))) (= (= (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_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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olex T_b V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.c_Nat_Osize__class_Osize _let_2))) (let ((_let_6 (tptp.c_List_Olist_OCons T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_6 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_6 V_y_2) V_ys_2))) _let_1)) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2)) (= (tptp.hAPP _let_5 V_xs_2) (tptp.hAPP _let_5 V_ys_2))) (and (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) V_ys_2)) _let_1)))))))))))) (forall ((V_XS_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist 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 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_b _let_2))) (let ((_let_9 (tptp.tc_fun T_b _let_6))) (let ((_let_10 (tptp.tc_fun _let_1 _let_4))) (let ((_let_11 (tptp.tc_fun T_b _let_10))) (let ((_let_12 (tptp.tc_fun T_b _let_5))) (= (tptp.c_List_Oset__Cons T_b 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_b)) (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_b) (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_b _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_b) (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_b) (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_b)) (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_b)))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_5 T_b) (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_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 tptp.tc_HOL_Obool) (tptp.c_member T_b)) 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)))))))))))))))))))) (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_b $$unsorted)) (= (tptp.c_Random_Oselect__weight T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral T_b) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)) V_x_2)) V_xs_2)) (tptp.c_Random_Oselect__weight T_b V_xs_2))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_b_2 $$unsorted) (V_x_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_b V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.c_List_Olist_OCons T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_5 V_a_2) V_x_2)) (tptp.hAPP (tptp.hAPP _let_5 V_b_2) V_y_2))) _let_1)) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)) V_r_2)) (and (= V_a_2 V_b_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) _let_1))))))))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)))) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olex T_b V_r_2))) (and (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olexord T_b V_r_2))) (= (tptp.hAPP _let_2 V_x_2) (tptp.hAPP _let_2 V_y_2)))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_b V_r_2)))) (let ((_let_3 (tptp.c_List_Olist_OCons T_b))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_1 _let_1))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_1 _let_1)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_ys_2))) _let_2)))))))))) (forall ((V_list_2 $$unsorted) (V_a_2 $$unsorted) (V_fa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.c_List_Olist_Olist__size T_b V_fa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_a_2) V_list_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP V_fa_2 V_a_2)) (tptp.c_List_Olist_Olist__size T_b V_fa_2 V_list_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Wellfounded_Oacc _let_1 (tptp.c_List_Olistrel1 T_b V_r_2)))) (let ((_let_3 (tptp.c_member _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Wellfounded_Oacc T_b V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) _let_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) _let_2)))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel1 T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Olistrel1 T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.hAPP (tptp.c_Relation_Oconverse _let_1 _let_1) (tptp.c_List_Olistrel1 T_b V_r_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b 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 (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel1 T_b V_r_2)) (tptp.c_List_Olistrel1 T_b V_s_2)))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_b V_r_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_3 _let_3))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_3 _let_3)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 V_ys_2))) _let_1))))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_b V_r_2)))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) 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_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b V_r_2)))) (= (tptp.hAPP _let_2 V_x_2) (tptp.hAPP _let_2 V_y_2)))))) (forall ((V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Relation_Oconverse _let_1 _let_1) (tptp.c_List_Olistrel1 T_b V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_b V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (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_b))) (= (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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_xs_2))) (tptp.c_List_Olistrel1 T_b V_r_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))) (let ((_let_3 (tptp.c_Set_Oinsert _let_1))) (let ((_let_4 (tptp.c_Relation_OImage _let_1 _let_1 (tptp.c_List_Olistrel T_b V_r_2)))) (= (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) _let_2)) (tptp.c_List_Oset__Cons T_b (tptp.hAPP (tptp.c_Relation_OImage T_b T_b V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) _let_2))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_b V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_a) B_b)) V_r_2)) (= B_a B_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_b) B_a)) V_r_2)))))) (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_x_2) V_y_2)) _let_1)) (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_x_2)) _let_1))))))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel T_b V_r_2)) (tptp.c_List_Olistrel T_b V_s_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel T_b V_r_2)) (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel T_b V_r_2))) (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)))))) (forall ((V_zs_2 $$unsorted) (V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (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 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olistrel T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)) (tptp.c_Transitive__Closure_Ortrancl (tptp.tc_List_Olist T_b) (tptp.c_List_Olistrel1 T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b V_r_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olistrel T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))))))) (forall ((V_r_H_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_r_H_2)) (=> (tptp.c_Relation_Orefl__on T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_r_H_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod _let_1 _let_1) tptp.tc_HOL_Obool)) (tptp.c_List_Olistrel1 T_b V_r_2)) (tptp.c_List_Olistrel T_b V_r_H_2))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olistrel T_b V_r_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Transitive__Closure_Ortrancl _let_1 (tptp.c_List_Olistrel1 T_b V_r_2)))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel T_b V_r_2))) (let ((_let_2 (tptp.c_List_Olist_OCons T_b))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_3 _let_3))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_3 _let_3)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) V_ys_2))) _let_1)))))))))) (forall ((V_xa_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_prod T_b T_b))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun _let_3 tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun T_b _let_4))) (let ((_let_6 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.c_List_Olistrelp T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_3 _let_6) T_b) (tptp.c_COMBC T_b _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_2) _let_5 T_b) (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 T_b) (tptp.c_member _let_2))) (tptp.c_Product__Type_OPair T_b T_b)))) V_r_2) V_x_2 V_xa_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_xa_2)) (tptp.c_List_Olistrel T_b V_r_2))))))))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (V_ys_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (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_b) V_y_2) V_ys_2)) V_xs_2)) (tptp.c_List_Olistrel T_b V_r_2))) (not (forall ((B_y $$unsorted) (B_ys $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) B_y) B_ys)) (=> (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_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_b V_r_2)))))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_x_2) V_y_2)) (=> (tptp.c_List_Olistrelp T_b V_r_2 V_xs_2 V_ys_2) (tptp.c_List_Olistrelp T_b 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_a4_2 $$unsorted) (V_a2_2 $$unsorted) (V_a3_2 $$unsorted) (V_a1_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_b))) (= (tptp.c_List_Olistrelp T_b 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_b V_r_2 V_a3_2 V_a4_2))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_y_2) V_ys_2))) (tptp.c_List_Olistrel T_b V_r_2))) (not (forall ((B_x $$unsorted) (B_xs $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) B_x) B_xs)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) V_y_2)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) B_xs) V_ys_2)) (tptp.c_List_Olistrel T_b V_r_2)))))))))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olists T_b V_A_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_b) V_A_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_b) 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_b 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_b $$unsorted)) (= (tptp.c_List_Olists T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun (tptp.tc_List_Olist T_b) tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (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)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_List_Olist T_b) tptp.tc_HOL_Obool)) (tptp.c_List_Olists T_b V_A_2)) (tptp.c_List_Olists T_b V_B_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (tptp.c_Equiv__Relations_Oequiv (tptp.tc_List_Olist T_b) (tptp.c_List_Olists T_b V_A_2) (tptp.c_List_Olistrel T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (tptp.c_Relation_Orefl__on (tptp.tc_List_Olist T_b) (tptp.c_List_Olists T_b V_A_2) (tptp.c_List_Olistrel T_b V_r_2)))) (forall ((V_A_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olists T_b V_A_2))) (let ((_let_2 (tptp.c_member (tptp.tc_List_Olist T_b)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) _let_1)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member _let_1) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olists T_b (tptp.c_Wellfounded_Oacc T_b V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_Wellfounded_Oacc _let_1 (tptp.c_List_Olistrel1 T_b V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b V_r_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.c_List_Olists T_b (tptp.c_Wellfounded_Oacc T_b V_r_2)))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b V_r_2))) (and (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)) (forall ((B_n $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) B_n)) (tptp.hAPP (tptp.hAPP _let_1 V_ys_2) B_n))) V_r_2)))))))))) (forall ((V_y_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (forall ((B_x $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_x)) V_r_2)))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_y_2) V_y_2)) (tptp.c_List_Olexord T_b V_r_2))))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (= (= V_xs_2 V_ys_2) (and (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_b))) (=> (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_b)) 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_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_x)) (forall ((V_n $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_n)))) (forall ((V_na_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) V_na_2) (tptp.c_Nat_Onat_Onat__case T_b V_x_2 (tptp.hAPP _let_1 V_xs_2) V_na_2)))) (forall ((V_xs $$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_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (let ((_let_3 (tptp.hAPP _let_2 V_x_2))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_x_2) V_y_2)) (tptp.c_List_Olexord T_b V_r_2))) (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_3) (tptp.hAPP _let_2 V_y_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) _let_3) V_y_2) V_x_2)) (exists ((B_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Otake T_b) B_i))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) (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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_2 tptp.tc_Nat_Onat) tptp.c_fconj))) (let ((_let_4 (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.tc_fun tptp.tc_Nat_Onat T_b))) (let ((_let_6 (tptp.tc_fun tptp.tc_Nat_Onat _let_2))) (= (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_Osublist T_b V_xs_2 V_I_2)) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun _let_1 _let_1) T_b) _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_6 T_b) _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_5 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_5 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool tptp.tc_Nat_Onat)) tptp.c_fequal)) (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2))))) (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_1 tptp.tc_HOL_Obool) (tptp.c_member tptp.tc_Nat_Onat)) V_I_2))))))))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (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_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) 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_b $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_b))) (let ((_let_2 (tptp.c_List_Oset T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2) V_na_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (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 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_na_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.c_List_Osublist T_b V_xs_2 V_I_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_n)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs)) V_i) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_i))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))) (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Osublist T_b V_xs_2 V_I_2))))))))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_na_2) V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_I_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Osublist T_b V_xs_2 V_I_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_y_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)))) (or (= V_y_2 V_x_2) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_n) V_xs)))))) (forall ((V_xs $$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_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_na_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Osublist T_b V_l_2 (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_na_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_na_2) V_l_2))) (forall ((V_xs_2 $$unsorted) (T_b $$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_b) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2)))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (= (forall ((B_x $$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.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_i))))))) (forall ((V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$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_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) 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_b V_f_2 V_xs_2))))))) (forall ((V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_y_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP V_f_2 V_x_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Olist_Olist__size T_b V_f_2 V_xs_2))))))) (forall ((V_A_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_b)) V_xs_2) (tptp.c_List_Olists T_b V_A_2))) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) B_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_A_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (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.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_b)) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (exists ((B_i $$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_b)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_i) V_x_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_na_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_na_2)) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Olists T_b V_A_2) (tptp.hAPP (tptp.c_Set_OCollect _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_1 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) _let_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1)) (tptp.c_List_Oset T_b))) V_A_2)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat T_b))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_2 _let_1) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool tptp.tc_Nat_Onat)) tptp.c_fequal)) (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))))))))))) (forall ((V_na_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) 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_b))) 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_na_2))))))))) (forall ((V_na_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_1) (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_1) (tptp.c_Orderings_Oord__class_Oless__eq _let_2)) (tptp.c_List_Oset T_b))) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat)) (tptp.c_Nat_Osize__class_Osize _let_1))) V_na_2))))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel T_b V_r_2))) (and (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (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_b _let_4))) (let ((_let_6 (tptp.tc_fun T_b 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_b T_b) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_3 _let_6) T_b) (tptp.c_COMBC T_b _let_3 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) _let_5 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 T_b) _let_2)) (tptp.c_Product__Type_OPair T_b T_b)))) V_r_2)) B_x))))))))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (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_b T_c) 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_c _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c _let_1) tptp.tc_Nat_Onat) (tptp.c_Product__Type_OPair T_b T_c)) (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2))) (tptp.hAPP (tptp.c_List_Onth T_c) 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_b)) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_ys_2))))))))))))) (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 $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_y) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_a T_b) V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_ys))))) (forall ((T_b $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (= (= V_xs_2 V_ys_2) (and (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_b) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) B_x))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) V_ys_2)))) (not (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_c) V_ys_2))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_c) V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_b) V_xs_2) V_xs_2)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (= V_a_2 V_b_2))))) (forall ((V_ys $$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_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 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) (tptp.hAPP (tptp.c_List_Onth T_b) 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_b)) V_xs_2)))))) (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_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_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 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Int_Oint V_i_2 V_j_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) (tptp.c_List_Oupto (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)) V_j_2)))) (forall ((V_l_2 $$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)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.c_List_Olist_Olist__size T_b V_f_2 V_xs_2)) (tptp.c_List_Olist_Olist__size T_b V_g_2 V_xs_2))))) (forall ((V_n $$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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) V_i)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_x) V_xs))) (forall ((V_x $$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_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_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_y $$unsorted) (V_x $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_1 V_x)) V_i) V_y) (tptp.hAPP _let_1 V_y)))) (forall ((V_x_H $$unsorted) (V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted) (V_i_H $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Olist__update T_a V_xs))) (=> (not (= V_i V_i_H)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP (tptp.hAPP _let_1 V_i) V_x)) V_i_H) V_x_H) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP (tptp.hAPP _let_1 V_i_H) V_x_H)) V_i) V_x))))) (forall ((V_v_2 $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_List_Olist_OCons T_b))) (let ((_let_3 (tptp.hAPP _let_2 V_x_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b (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_b _let_1) (tptp.c_List_Olist__update T_b 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_y $$unsorted) (V_i $$unsorted) (V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a (tptp.hAPP _let_1 V_xs)) (tptp.hAPP tptp.c_Nat_OSuc V_i)) V_y) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_a V_xs) V_i) V_y))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_b 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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_i_2) V_x_2))) V_A_2))))))) (forall ((V_x_2 $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_i_2) V_x_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_x $$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_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_i_2 $$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_b)) V_xs_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_i_2) V_x_2) V_xs_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_i_2) V_x_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_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 $$unsorted) (T_b $$unsorted) (V_na_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_na_2) V_x_2)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Obutlast T_a V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) V_xs))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) V_ys_2)) (tptp.c_List_Olistrel1 T_b V_r_2))) (exists ((B_y $$unsorted) (B_n $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_n)) B_y)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) B_n) B_y))))))) (forall ((V_i_2 $$unsorted) (V_a_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_i_2)) (tptp.c_Orderings_Obot__class_Obot _let_2)))))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_i_2) V_a_2)))))))) (forall ((V_j $$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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Osublist T_b V_xs_2 V_I_2)))))) (forall ((V_ys $$unsorted) (T_b $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs) V_ys))))) (forall ((V_xs $$unsorted) (T_b $$unsorted) (V_ys $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_ys)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_prod T_b T_a)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_a) V_xs) V_ys))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (and (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.hBOOL (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (forall ((B_j $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_b) 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_b)) V_xs_2))) (=> (not (= B_i B_j)) (not (= (tptp.hAPP _let_1 B_i) (tptp.hAPP _let_1 B_j))))))))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2))) (let ((_let_2 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) 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_b) 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 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_b) (tptp.c_COMBI T_b)) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Ocard_H T_b V_A_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (let ((_let_4 (tptp.tc_fun T_b 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_b))) V_A_2))) (tptp.c_List_Odistinct T_b)))))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))))) (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 $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_a))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_k)) (= (tptp.hAPP (tptp.hAPP _let_1 V_xs) B_i) (tptp.hAPP (tptp.hAPP _let_1 V_ys) B_i))))) (= (tptp.hAPP _let_1 V_xs) (tptp.hAPP _let_1 V_ys))))))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (= (tptp.hAPP _let_2 (tptp.c_List_Oremove1 T_b 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_b) 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 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_b)) (tptp.c_List_Olist_ONil T_b)) (tptp.c_List_Olists T_b V_A_2)))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) V_xs_2) _let_1)) (tptp.c_List_Olistrel T_b V_r_2))) (= V_xs_2 _let_1))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.tc_List_Olist 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) _let_1) V_xs_2)) (tptp.c_List_Olistrel T_b V_r_2))) (= V_xs_2 _let_1))))) (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_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 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Obutlast T_a _let_1) _let_1))) (forall ((V_x $$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 ((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_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_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_na_2) V_xs_2) _let_1) (or (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_xs_2 _let_1))))) (forall ((V_xs $$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 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) _let_1) _let_1))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (tptp.c_List_Osublist T_b _let_1 V_A_2) _let_1))) (forall ((V_zs $$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 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Oremove1 T_a V_x _let_1) _let_1))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (tptp.c_List_Olistrelp T_b V_r_2 _let_1 _let_1))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (tptp.c_List_Olistrelp T_b V_r_2 _let_1 _let_1))) (forall ((V_ys $$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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_a T_b) V_xs) (tptp.c_List_Olist_ONil T_b)) (tptp.c_List_Olist_ONil (tptp.tc_prod T_a T_b)))) (forall ((T_a $$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_xs_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.c_List_Olist_ONil T_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_Olist_ONil T_b)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_k_2) V_x_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_fa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist_Olist__size T_b V_fa_2 (tptp.c_List_Olist_ONil T_b)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (= (tptp.c_List_Oremove1 T_b V_x_2 V_xs_2) V_xs_2))) (forall ((V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))) (not (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_b V_y_2 V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_a_2))) (=> (not (= V_a_2 V_b_2)) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_b V_b_2 V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Osublist T_b V_xs_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (tptp.c_List_Olist_ONil T_b))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (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 (tptp.c_List_Oremove1 T_b V_x_2 V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$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_b)) V_xs_2))) (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))))) (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b))) (tptp.c_List_Olistrel1 T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.c_List_Olist_ONil T_b)) V_xs_2)) (tptp.c_List_Olistrel1 T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_1) _let_1)) (tptp.c_List_Olistrel T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b))) (tptp.c_List_Olexord T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) V_xs_2) (tptp.c_List_Olist_ONil T_b))) (tptp.c_List_Olex T_b V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b)) V_ys_2)) (tptp.c_List_Olex T_b V_r_2)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_na_2) (tptp.hAPP _let_3 V_xs_2)) (tptp.c_Nat_Onat_Onat__case _let_2 (tptp.c_List_Olist_ONil T_b) (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_na_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 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) _let_1))) (let ((_let_3 (tptp.c_List_Osublist T_b _let_2 V_A_2))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_A_2)))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 _let_1)))))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Olists T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_List_Olist_ONil T_b)) (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_List_Olist_ONil T_b)) (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_b V_r_2)) _let_2) _let_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Oset__Cons T_b 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_b _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.c_List_Olist_OCons T_b)) _let_1)) V_A_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_2 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_3 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_b V_x_2 V_xs_2)))) (let ((_let_4 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))) (=> (not _let_4) (= _let_3 _let_2)))))))) (forall ((V_j_2 $$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_b $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Product__Type_Ofst T_b (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP (tptp.c_Random_Oselect T_b V_xs_2) V_s_2))) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Olistset T_b (tptp.c_List_Olist_ONil (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert _let_1) (tptp.c_List_Olist_ONil T_b)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_b T_c))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (let ((_let_3 (tptp.tc_List_Olist _let_2))) (let ((_let_4 (tptp.tc_List_Olist T_b))) (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_c) V_y_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_5 (tptp.tc_fun _let_6 _let_6) T_b) (tptp.c_COMBB _let_3 _let_3 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 T_b) (tptp.c_List_Olist_OCons _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c _let_2) (tptp.c_Product__Type_OPair T_b T_c)) V_y_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_4 (tptp.tc_List_Olist T_c) _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_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Olist_Olist__case T_b T_c V_f1_2 V_f2_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_c) 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_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Olist_Olist__case T_b T_c V_f1_2 V_f2_2) (tptp.c_List_Olist_ONil T_c)) V_f1_2)) (forall ((V_As_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olistset T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_As_2)) (tptp.c_List_Oset__Cons T_b V_A_2 (tptp.c_List_Olistset T_b V_As_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_b T_c))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (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_c))) (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_b) V_x_2) V_xs_2)) V_ys_2) (tptp.hAPP (tptp.c_List_Olist_Olist__case _let_3 T_c (tptp.c_List_Olist_ONil _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_6 _let_6) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_6 _let_6) T_c) (tptp.c_COMBB _let_3 _let_3 _let_5)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 T_c) (tptp.c_List_Olist_OCons _let_2)) (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2)))) (tptp.hAPP _let_1 V_xs_2))) V_ys_2))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_1) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (tptp.c_List_Oremove1 T_b V_x_2 (tptp.c_List_Olinorder__class_Osorted__list__of__set T_b V_A_2))))))) (forall ((V_Pa_2 $$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_Pa_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_Pa_2 B_i) B_j))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_a0_2) V_a1_2))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_List_Olist_ONil T_b)))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Oanamorph T_c T_b V_f_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 T_c))) (let ((_let_4 (tptp.tc_prod _let_2 T_c))) (let ((_let_5 (tptp.tc_fun T_c _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_b)) 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.c_Option_Ooption_Ooption__case _let_4 (tptp.tc_prod T_b T_c) _let_8 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_7 (tptp.tc_fun _let_5 _let_5) T_b) (tptp.c_COMBB _let_4 _let_4 T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 _let_7 T_b) (tptp.c_Product__Type_Oprod_Oprod__case _let_2 T_c _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_2 _let_2) _let_6 T_b) (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_2) _let_3)) (tptp.c_List_Olist_OCons T_b))))) (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 $$unsorted) (T_b $$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_b))) (let ((_let_4 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2))) (= (tptp.c_List_Osublist T_b (tptp.hAPP _let_4 V_l_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_If (tptp.tc_List_Olist T_b)) (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_b 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_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_ys_2) V_zs_2) _let_1) (or (and (= V_ys_2 (tptp.c_List_Olist_ONil T_b)) (= V_zs_2 _let_1)) (exists ((B_ys_H $$unsorted)) (and (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) B_ys_H)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_ys_2) V_zs_2)) (or (and (= V_ys_2 (tptp.c_List_Olist_ONil T_b)) (= _let_1 V_zs_2)) (exists ((B_ys_H $$unsorted)) (and (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) B_ys_H) V_ys_2) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) B_ys_H) V_zs_2)))))))) (forall ((V_y_2 $$unsorted) (V_ys_2 $$unsorted) (V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.c_List_Olist_OCons T_b))) (let ((_let_3 (tptp.c_List_Oappend T_b))) (= (= (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1)) (tptp.hAPP (tptp.hAPP _let_3 V_ys_2) (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1))) (and (= V_xs_2 V_ys_2) (= V_x_2 V_y_2))))))) (forall ((V_ys $$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 $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (=> (= (tptp.hAPP _let_2 V_xs1) V_ys) (=> (= V_xs (tptp.hAPP (tptp.hAPP _let_1 V_xs1) V_zs)) (= (tptp.hAPP _let_2 V_xs) (tptp.hAPP (tptp.hAPP _let_1 V_ys) V_zs))))))) (forall ((V_A_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olists T_b V_A_2))) (let ((_let_2 (tptp.c_member (tptp.tc_List_Olist T_b)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) _let_1)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) _let_1)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_ys_2) _let_1))))))) (forall ((V_us $$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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_b))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_ys_2) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_zs_2) V_xs_2)) (= V_ys_2 V_zs_2)))) (forall ((V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2))) (= (= (tptp.hAPP _let_1 V_ys_2) (tptp.hAPP _let_1 V_zs_2)) (= V_ys_2 V_zs_2)))) (forall ((V_ts_2 $$unsorted) (V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_b))) (= (= (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 $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_b))) (let ((_let_2 (tptp.hAPP _let_1 B_us))) (or (and (= V_xs_2 (tptp.hAPP (tptp.hAPP _let_1 V_zs_2) B_us)) (= (tptp.hAPP _let_2 V_ys_2) V_ts_2)) (and (= (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) B_us) V_zs_2) (= V_ys_2 (tptp.hAPP _let_2 V_ts_2)))))))))) (forall ((V_zs $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_xs))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_ys)) V_zs) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_ys) V_zs)))))) (forall ((V_ys $$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_ys $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 V_xs)) (tptp.hAPP _let_1 V_ys))))) (forall ((V_vs $$unsorted) (V_ys $$unsorted) (V_us $$unsorted) (T_b $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_a T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (=> (= (tptp.hAPP _let_3 V_xs) (tptp.hAPP _let_2 V_us)) (=> (= (tptp.hAPP _let_3 V_ys) (tptp.hAPP _let_2 V_vs)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_us) V_vs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP _let_1 V_xs) V_us)) (tptp.hAPP (tptp.hAPP _let_1 V_ys) V_vs))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (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_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))))) (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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (and (= V_xs_2 _let_1) (= V_ys_2 _let_1))))) (forall ((V_xs $$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_b $$unsorted) (V_xs_2 $$unsorted)) (= (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (= V_ys_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_ys_2 $$unsorted)) (= (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) 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_b $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2) V_xs_2) (= V_ys_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2) V_ys_2) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((T_a $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted)) (=> (= V_xs V_ys) (= V_xs (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.c_List_Olist_ONil T_a)) V_ys)))) (forall ((V_xs $$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 $$unsorted) (V_x $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_3 (tptp.c_List_Oremove1 T_b V_x_2 (tptp.hAPP _let_2 V_ys_2)))) (let ((_let_4 (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))))) (and (=> _let_4 (= _let_3 (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_b V_x_2 V_xs_2)) V_ys_2))) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 (tptp.c_List_Oremove1 T_b V_x_2 V_ys_2)))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (or (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_b V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_b V_ys_2))))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_b V_r_2))) (let ((_let_2 (tptp.c_List_Oappend T_b))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_3 _let_3))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_3 _let_3)))) (=> (or (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_xs_2) V_ys_2)) _let_1)) (= V_us_2 V_vs_2)) (and (= V_xs_2 V_ys_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 V_us_2) V_vs_2)) _let_1)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) V_us_2)) (tptp.hAPP (tptp.hAPP _let_2 V_ys_2) V_vs_2))) _let_1))))))))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_v_2 $$unsorted) (V_u_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_b V_r_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Oappend T_b) V_x_2))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (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_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_b))) (let ((_let_3 (tptp.c_List_Odistinct T_b))) (= (tptp.hBOOL (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2))) (and (tptp.hBOOL (tptp.hAPP _let_3 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_3 V_ys_2)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 V_ys_2)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_ys $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oappend T_b))) (= (tptp.c_List_Osublist T_b (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_b V_l_2 V_A_2)) (tptp.c_List_Osublist T_b 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_b)) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_List_Olist_OCons T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Oappend T_b) V_us_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_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_b 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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Oappend T_b) V_u_2))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_3 _let_3)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_3 _let_3) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_x_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_y_2)))) (tptp.c_List_Olexord T_b V_r_2)))))))) (forall ((V_xs $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistrel1 T_b V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (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_b))) (let ((_let_6 (tptp.c_List_Olist_OCons T_b))) (let ((_let_7 (tptp.c_List_Oappend T_b))) (= (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_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) V_r_2))))))))))))) (forall ((V_r_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (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_b V_r_2))) (not (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (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) B_x) B_y)) V_r_2)) (forall ((B_us $$unsorted) (B_vs $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Oappend T_b) 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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_b V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (let ((_let_5 (tptp.hAPP (tptp.c_List_Oappend T_b) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_5 V_u_2)) (tptp.hAPP _let_5 V_v_2))) _let_1)) (=> (forall ((B_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_List_Orotate1 T_b) V_xs_2) (tptp.hAPP (tptp.c_List_Olist_Olist__case _let_2 T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 _let_2) (tptp.c_List_Oappend T_b))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.c_List_Olist_OCons T_b)) _let_1))) V_xs_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 $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) V_n) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs)) V_xs)) (forall ((V_xs $$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 $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_y) V_ys)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) V_n) V_y))) (forall ((V_i $$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_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate1 T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))) (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_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 $$unsorted) (V_m $$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) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs)) (tptp.hAPP _let_1 V_xs)))) (forall ((V_xs $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate1 T_b) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_na_2) V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_na_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) _let_1) _let_1))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.hAPP (tptp.c_List_Orotate1 T_b) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_m_2 $$unsorted) (V_na_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_b))) (let ((_let_2 (tptp.c_List_Oset T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_na_2) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_na_2) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_na_2) V_xs_2) (tptp.c_List_Olist_ONil T_b)) (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_b)) V_xs_2)) V_na_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_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2) V_zs_2) (and (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) _let_1) V_zs_2)) (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) _let_1) V_zs_2)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (= (tptp.hAPP (tptp.hAPP _let_1 V_na_2) _let_3) (tptp.c_Nat_Onat_Onat__case _let_2 _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_2 _let_2) _let_1) V_xs_2) V_na_2)))))) (forall ((V_xs $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_2 (tptp.hAPP _let_1 V_ys_092_060_094isub_0621_2))) (let ((_let_3 (tptp.c_List_Odrop T_b))) (let ((_let_4 (tptp.c_List_Oappend T_b))) (let ((_let_5 (tptp.c_List_Otake T_b))) (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 $$unsorted) (T_b $$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) (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 $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_i) V_j))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) _let_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs))) (= (tptp.hAPP (tptp.hAPP _let_1 _let_2) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.hAPP _let_1 V_i) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 V_j) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_i) V_xs)))))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs) V_xs))) (forall ((V_zs $$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_zs $$unsorted) (V_ys $$unsorted) (V_xs $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_ys))) (let ((_let_2 (tptp.c_List_Ozip T_a T_b))) (= (tptp.hAPP (tptp.hAPP _let_2 V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_ys) V_zs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend (tptp.tc_prod T_a T_b)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_a) _let_1) V_xs)) V_ys)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) _let_1) V_xs)) V_zs)))))) (forall ((V_xs $$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_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))) (=> (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 ((V_ys $$unsorted) (T_a $$unsorted) (V_xs $$unsorted)) (let ((_let_1 (tptp.c_List_Ohd T_a))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)))) (let ((_let_3 (= V_xs (tptp.c_List_Olist_ONil T_a)))) (and (=> _let_3 (= _let_2 (tptp.hAPP _let_1 V_ys))) (=> (not _let_3) (= _let_2 (tptp.hAPP _let_1 V_xs)))))))) (forall ((T_b $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_List_Ohd T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))))) (forall ((T_a $$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 $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (=> (not (= V_xs _let_1)) (= (tptp.hAPP (tptp.c_List_Orotate1 T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) (tptp.hAPP (tptp.c_List_Otl T_a) V_xs)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs)) _let_1)))))) (forall ((V_n $$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 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Odrop T_a) V_n))) (let ((_let_2 (tptp.c_List_Otl T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)))))) (forall ((V_xs $$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 $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Otl T_a) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x) V_xs)) V_xs)) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Otl T_a) V_xs)))))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_b V_na_2) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Orotate T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_na_2)) (tptp.hAPP (tptp.c_Fun_Ocomp _let_1 _let_1 _let_1 (tptp.c_List_Orotate T_b V_m_2)) (tptp.c_List_Orotate T_b V_na_2))))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_m) (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs)) (tptp.hAPP (tptp.c_List_Orotate T_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m) V_n)) V_xs))) (forall ((T_b $$unsorted)) (= (tptp.c_List_Orotate T_b (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_Fun_Oid (tptp.tc_List_Olist T_b)))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs)) (tptp.hAPP _let_1 V_xs)))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_b V_na_2) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.c_List_Otl T_a) _let_1) _let_1))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.hAPP (tptp.c_List_Orotate T_b V_na_2) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_xs $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orotate1 T_a))) (let ((_let_2 (tptp.c_List_Orotate T_a V_n))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs)))))) (forall ((V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Orotate T_b V_na_2) (tptp.hAPP (tptp.hAPP (tptp.c_Nat_Ocompow _let_1 _let_1) V_na_2) (tptp.c_List_Orotate1 T_b))))) (forall ((V_ys $$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)) (= (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 $$unsorted)) (let ((_let_1 (tptp.c_List_Odrop T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_xs) (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.c_List_Otl T_a) V_xs))))) (forall ((V_xs $$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 ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_List_Otl T_b))) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2)) (tptp.hAPP (tptp.c_List_Olist_Olist__case _let_2 T_b (tptp.hAPP _let_3 V_ys_2) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_2 _let_2) T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 _let_2) _let_1) V_ys_2))) V_xs_2)))))) (forall ((V_xs $$unsorted) (T_a $$unsorted) (V_n $$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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_List_Orotate T_a V_n) V_xs) V_xs))) (forall ((V_xs $$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 $$unsorted)) (let ((_let_1 (tptp.c_List_Otake T_a))) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP tptp.c_Nat_OSuc V_n)) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_a) (tptp.hAPP (tptp.c_List_Ohd T_a) V_xs)) (tptp.hAPP (tptp.hAPP _let_1 V_n) (tptp.hAPP (tptp.c_List_Otl T_a) V_xs))))))) (forall ((V_xs $$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))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Lattices_Oab__semigroup__idem__mult T_b) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))) (= (tptp.hAPP (tptp.c_Finite__Set_Ofold1 T_b _let_1) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 (tptp.hAPP (tptp.c_List_Ohd T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Otl T_b) V_xs_2))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (exists ((B_b $$unsorted) (B_z $$unsorted)) (= V_y_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) 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_b) V_x_2) V_y_2))) (tptp.c_List_Olexord T_b V_r_2)))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c V_f_2 V_a_2) (tptp.c_List_Olist_ONil T_c)) V_a_2)) (forall ((V_ns_2 $$unsorted) (V_na_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m_2))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_na_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofoldl tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_na_2) V_ns_2)))))) (forall ((V_zs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Omonoid__add T_b) (= (tptp.hAPP (tptp.hAPP _let_1 V_x_2) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 (tptp.c_Groups_Ozero__class_Ozero T_b)) V_zs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 V_x_2) V_zs_2))))) (forall ((V_zs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (=> (tptp.class_Groups_Osemigroup__add T_b) (= (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 (tptp.hAPP _let_2 V_y_2)) V_zs_2) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 V_y_2) V_zs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c V_f_2 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_c) V_x_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP V_f_2 V_a_2) V_x_2)) V_xs_2))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofoldl T_b T_c V_f_2 V_a_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_c) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c V_f_2 (tptp.hAPP _let_1 V_xs_2)) V_ys_2)))) (forall ((T_b $$unsorted)) (=> (tptp.class_Groups_Omonoid__add T_b) (= (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.c_List_Ofoldl T_b T_b (tptp.c_Groups_Oplus__class_Oplus T_b) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_ns_2 $$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_2 $$unsorted) (V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm__idem T_b T_c V_f_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_c T_b (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) V_f_2) V_y_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) (tptp.c_Orderings_Obot__class_Obot T_b)) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.c_Complete__Lattice_OInf__class_OInf T_b (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) (tptp.c_Orderings_Otop__class_Otop T_b)) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.c_Big__Operators_Olattice__class_OSup__fin T_b (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Lattices_Olattice T_b) (= (tptp.c_Big__Operators_Olattice__class_OInf__fin T_b (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olinorder__class_OMin T_b (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b (tptp.c_Orderings_Oord__class_Omin T_b) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olinorder__class_OMax T_b (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b (tptp.c_Orderings_Oord__class_Omax T_b) V_x_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omin T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olattice_OInf__fin T_b _let_1 (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 V_x_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_Big__Operators_Olattice_OSup__fin T_b _let_1 (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 V_x_2) V_xs_2))))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c T_b) (tptp.hAPP (tptp.c_List_Oset T_c) V_xs_2)) V_f_2) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c (tptp.hAPP (tptp.c_COMBC T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_b)) V_f_2)) (tptp.c_Orderings_Obot__class_Obot T_b)) V_xs_2)))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c T_b) (tptp.hAPP (tptp.c_List_Oset T_c) V_xs_2)) V_f_2) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c (tptp.hAPP (tptp.c_COMBC T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) (tptp.c_Lattices_Osemilattice__inf__class_Oinf T_b)) V_f_2)) (tptp.c_Orderings_Otop__class_Otop T_b)) V_xs_2)))) (forall ((V_ns_2 $$unsorted) (V_na_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) V_ns_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_List_Ofoldl tptp.tc_Nat_Onat tptp.tc_Nat_Onat (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_ns_2))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$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.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_b (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b (tptp.c_COMBI T_b)) V_x_2) (tptp.c_List_Olinorder__class_Osorted__list__of__set T_b (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_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.c_List_Olist_ONil T_b)) V_y_2)) (tptp.c_List_Olexord T_b V_r_2))) (exists ((B_a $$unsorted) (B_x $$unsorted)) (= V_y_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) B_a) B_x)))))) (forall ((V_ys_2 $$unsorted) (T_c $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_c T_b V_f_2) V_x_2))) (let ((_let_2 (tptp.c_List_Olist_OCons T_c))) (let ((_let_3 (tptp.hAPP _let_2 V_y_2))) (let ((_let_4 (tptp.hAPP _let_3 V_ys_2))) (let ((_let_5 (tptp.hAPP _let_1 _let_4))) (let ((_let_6 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))))) (=> (tptp.class_Orderings_Olinorder T_b) (and (=> _let_6 (= _let_5 (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_4))) (=> (not _let_6) (= _let_5 (tptp.hAPP _let_3 (tptp.hAPP _let_1 V_ys_2))))))))))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_c V_f_2) V_x_2) _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) _let_1))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_c V_f_2) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_c V_f_2) V_x_2) V_xs_2)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP _let_1 V_xs_2)))))) (forall ((T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (tptp.c_Finite__Set_Ofun__left__comm T_b (tptp.tc_List_Olist T_b) (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b (tptp.c_COMBI T_b))))) (forall ((V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b (tptp.c_COMBI T_b)))) (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_b) (= (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 $$unsorted) (T_c $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olinorder__class_Oinsort__key T_c T_b 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_b) (=> (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 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Oremove1 T_b V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b (tptp.c_COMBI T_b)) V_x_2) V_xs_2)) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_c) (not (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_c V_f_2) V_a_2) V_xs_2) (tptp.c_List_Olist_ONil T_b))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_c V_f_2) V_x_2) V_xs_2))) (and (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (T_c $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (not (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_c T_b V_f_2) (tptp.hAPP (tptp.c_List_Oset T_c) V_xs_2))))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_c T_b V_f_2 V_x_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_c T_b V_f_2) V_x_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_COMBI T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_b _let_1 V_x_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b _let_1) V_x_2) V_xs_2)))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (=> (tptp.class_Orderings_Olinorder T_c) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_c V_f_2 V_x_2 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_b (tptp.c_COMBI T_b) V_x_2 V_xs_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_b (tptp.c_COMBI T_b) V_x_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_c $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.hAPP (tptp.c_List_Oset T_c) V_xs_2)))) (= (tptp.c_List_Olinorder__class_Oinsort__insert__key T_c T_b V_f_2 V_x_2 V_xs_2) V_xs_2)))) (forall ((V_x $$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 $$unsorted)) (=> (not (= V_xs (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Olast T_a V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) V_xs) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) V_xs)) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)))))) (forall ((V_x $$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 ((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)) (=> (= 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_xs $$unsorted) (T_a $$unsorted) (V_ys $$unsorted)) (let ((_let_1 (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)))) (let ((_let_2 (= V_ys (tptp.c_List_Olist_ONil T_a)))) (and (=> _let_2 (= _let_1 (tptp.c_List_Olast T_a V_xs))) (=> (not _let_2) (= _let_1 (tptp.c_List_Olast T_a V_ys))))))) (forall ((V_xs $$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 $$unsorted)) (=> (= V_ys (tptp.c_List_Olist_ONil T_a)) (= (tptp.c_List_Olast T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) V_xs) V_ys)) (tptp.c_List_Olast T_a V_xs)))) (forall ((T_b $$unsorted) (V_as_2 $$unsorted)) (=> (not (= V_as_2 (tptp.c_List_Olist_ONil T_b))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.c_List_Olast T_b V_as_2)) (tptp.hAPP (tptp.c_List_Oset T_b) V_as_2))))) (forall ((V_x $$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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) _let_1)) V_ys_2) (and (not (= V_ys_2 _let_1)) (= (tptp.c_List_Obutlast T_b V_ys_2) V_xs_2) (= (tptp.c_List_Olast T_b V_ys_2) V_x_2))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_b)) V_a_2) (tptp.c_List_Olists T_b V_A_2))) (or (= V_a_2 (tptp.c_List_Olist_ONil T_b)) (exists ((B_a $$unsorted) (B_l $$unsorted)) (and (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) B_a) B_l)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_a) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_b)) B_l) (tptp.c_List_Olists T_b V_A_2)))))))) (forall ((T_b $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))) (= (tptp.c_Random_Oselect__weight T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_b)) (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral T_b) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral))) V_xs_2)) (tptp.c_Random_Oselect T_b V_xs_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_c) (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))) (= (tptp.c_List_Olast T_c (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) V_xs_2)) (tptp.hAPP V_f_2 (tptp.c_List_Olast T_b V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_a_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d T_b))) (let ((_let_2 (tptp.tc_fun T_d T_c))) (= (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c V_g_2 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_d (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _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_b) (tptp.c_COMBB T_c T_b T_d)) V_g_2)) V_f_2) V_a_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP (tptp.c_List_Orotate T_b V_na_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orotate T_c V_na_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Otl T_c) V_xs_2)) (tptp.hAPP (tptp.c_List_Otl T_b) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_ys_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_xs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun T_d _let_1))) (let ((_let_3 (tptp.tc_fun T_d T_c))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_f_2) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_b T_d) _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_d _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_COMBB T_c _let_1 T_d)) (tptp.c_Product__Type_OPair T_b T_c))) V_f_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_d) V_xs_2) V_ys_2))))))) (forall ((V_ys_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_d))) (let ((_let_2 (tptp.tc_fun T_e _let_1))) (let ((_let_3 (tptp.tc_fun T_e T_d))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_c T_d) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_e T_d) V_g_2) V_ys_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_c T_e) T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_e T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun T_e T_b) T_c) (tptp.hAPP (tptp.c_COMBB _let_1 T_b T_e) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_d _let_1) (tptp.tc_fun _let_3 _let_2) T_c) (tptp.c_COMBB T_d _let_1 T_e)) (tptp.c_Product__Type_OPair T_c T_d))) V_g_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_c T_e) V_xs_2) V_ys_2))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_d))) (let ((_let_2 (tptp.tc_fun T_d _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_c T_d) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_e T_c) V_g_2) V_xs_2)) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_e T_d) T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_e T_d T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun T_d T_b) T_e) (tptp.hAPP (tptp.c_COMBB _let_1 T_b T_d) V_f_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c _let_2 T_e) (tptp.c_Product__Type_OPair T_c T_d)) V_g_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_e T_d) V_xs_2) V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) V_f_2) V_xs_2)) V_ys_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_d T_c) _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_c _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c _let_1) T_d) (tptp.c_Product__Type_OPair T_b T_c)) V_f_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_d T_c) V_xs_2) V_ys_2))))) (forall ((V_ys_2 $$unsorted) (V_g_2 $$unsorted) (T_e $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun T_c _let_1))) (let ((_let_3 (tptp.tc_fun T_e _let_1))) (let ((_let_4 (tptp.tc_fun T_e T_c))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) V_f_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_e T_c) V_g_2) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_d T_e) _let_1) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_d T_e _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_d _let_4 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_4 _let_3) T_d) (tptp.c_COMBB T_c _let_1 T_e)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_2 T_d) (tptp.c_Product__Type_OPair T_b T_c)) V_f_2))) V_g_2))) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_d T_e) V_xs_2) V_ys_2)))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_b))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_b) V_xs_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b _let_1) (tptp.c_Product__Type_OPair T_b T_b)) (tptp.c_COMBI T_b))) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2) (tptp.c_List_Olist_ONil T_b)) (= V_xs_2 (tptp.c_List_Olist_ONil T_c)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) (tptp.c_List_Olist_ONil T_c)) (tptp.c_List_Olist_ONil T_b))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Olist_ONil T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2)) (= V_xs_2 (tptp.c_List_Olist_ONil T_c)))) (forall ((V_y_2 $$unsorted) (V_k_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_c V_xs_2) V_k_2) V_y_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b (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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_c T_b))) (= (= (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_xs_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) (tptp.hAPP (tptp.c_List_Oset T_c) V_xs_2))) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_xs_2))) (forall ((V_ys_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) V_g_2) V_ys_2)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_d)) V_ys_2)))) (forall ((V_xs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Groups_Omonoid__add T_b) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) (tptp.hAPP (tptp.c_COMBK T_b T_c) _let_1)) V_xs_2)) _let_1)))) (forall ((V_xs_2 $$unsorted) (V_c_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_c T_b))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (let ((_let_3 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Osemiring__0 T_b) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _let_3) V_f_2)) V_c_2)) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))) V_c_2))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (V_c_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_c T_b))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (let ((_let_3 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Osemiring__0 T_b) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b T_b T_c) _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 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_c T_b))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus T_b))) (=> (tptp.class_Groups_Oab__group__add T_b) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _let_3) V_f_2)) V_g_2)) V_xs_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_c T_b))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (let ((_let_3 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_b T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b T_b) T_c) _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_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (=> (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (=> (tptp.c_Fun_Oinj__on T_c T_b V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= V_xs_2 V_ys_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (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_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_b) (tptp.c_COMBI T_b)) V_xs_2) V_xs_2)) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_na_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_c) V_na_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP _let_1 (tptp.c_List_Obutlast T_c V_xs_2)) (tptp.c_List_Obutlast T_b (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.hAPP (tptp.c_List_Oset T_c) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_c) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_c) V_xs_2) V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 V_ys_2))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))) (= (tptp.hAPP (tptp.c_List_Ohd T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) V_xs_2)) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_List_Ohd T_b) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_na_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_c) V_na_2) V_xs_2))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_List_Olist T_d) (tptp.tc_List_Olist T_c) (tptp.tc_List_Olist T_b) (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_f_2)) (tptp.hAPP (tptp.c_List_Omap T_b T_d) V_g_2)) (tptp.hAPP (tptp.c_List_Omap T_b T_c) (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_f_2) V_g_2)))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (T_d $$unsorted)) (= (tptp.hAPP (tptp.c_Fun_Ocomp (tptp.tc_List_Olist T_d) (tptp.tc_List_Olist T_c) (tptp.tc_List_Olist T_b) (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_f_2)) (tptp.hAPP (tptp.c_List_Omap T_b T_d) V_g_2)) (tptp.hAPP (tptp.c_List_Omap T_b T_c) (tptp.hAPP (tptp.c_Fun_Ocomp T_d T_c T_b V_f_2) V_g_2)))) (forall ((V_list_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_g_2) V_list_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_d V_f_2) V_g_2)) V_list_2))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_g_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_d V_f_2) V_g_2)) V_xs_2))) (forall ((T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Omap T_b T_b) (tptp.c_COMBI T_b)) (tptp.c_Fun_Oid (tptp.tc_List_Olist T_b)))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist_Olist__size T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_g_2) V_xs_2)) (tptp.c_List_Olist_Olist__size T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b tptp.tc_Nat_Onat T_c V_f_2) V_g_2) V_xs_2))) (forall ((T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Omap T_b T_b) (tptp.c_Fun_Oid T_b)) (tptp.c_Fun_Oid (tptp.tc_List_Olist T_b)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2))) (let ((_let_2 (tptp.c_List_Oset T_b))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_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_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_c))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (=> (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_2 V_ys_2)) (=> (tptp.c_Fun_Oinj__on T_c T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 V_ys_2))) (= V_xs_2 V_ys_2)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_na_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) V_xs_2)) V_na_2) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_na_2))))) (forall ((V_v_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_b T_c))) (=> (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_Fun_Ofun__upd T_b T_c 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_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_c) V_xs_2)) (tptp.c_Fun_Oinj__on T_c T_b V_f_2 (tptp.hAPP (tptp.c_List_Oset T_c) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_b))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (=> (tptp.class_Groups_Oordered__ab__group__add__abs T_b) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_b) _let_1) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ouminus__class_Ouminus T_b))) (let ((_let_2 (tptp.c_List_Omap T_c T_b))) (let ((_let_3 (tptp.c_List_Omonoid__add__class_Olistsum T_b))) (=> (tptp.class_Groups_Oab__group__add T_b) (= (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_b T_b T_c _let_1) V_f_2)) V_xs_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Oinj__on T_b T_c 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_b)) V_A_2))) (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_c) (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) V_A_2))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_c) (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.c_Fun_Oinj__on _let_1 (tptp.tc_List_Olist T_c) (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Rings_Osemiring__1 T_b) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) (tptp.hAPP (tptp.c_COMBK T_b T_c) V_r_2)) V_xs_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_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_xs_2))) V_r_2)))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_c))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_c) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_b V_x_2 V_xs_2)))))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Nitpick_Osetsum_H T_b T_c V_f_2 V_A_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (let ((_let_4 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (and (=> _let_2 (= _let_1 (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) (tptp.c_Hilbert__Choice_OEps _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_3 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_3) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 _let_4 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_4 tptp.tc_HOL_Obool) _let_3) tptp.c_fequal) (tptp.c_List_Oset T_b))) V_A_2))) (tptp.c_List_Odistinct T_b))))))) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero T_c)))))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_l_2 $$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_b)) V_xs_2))) (= (tptp.hAPP (tptp.c_Random_Opick T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_b)) (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Code__Numeral_Ocode__numeral T_b) (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_b) V_xs_2) V_l_2)))) (= _let_21 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat _let_12)) (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)))) (= tptp.c_Code__Numeral_Oof__nat (tptp.c_Nat_Osemiring__1__class_Oof__nat tptp.tc_Code__Numeral_Ocode__numeral)) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (= (= (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_na_2) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m_2)) (= V_na_2 V_m_2))) (= _let_20 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat _let_19)) (forall ((V_m_2 $$unsorted) (V_na_2 $$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_na_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_na_2) V_m_2)))) (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_ys_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (= (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)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_b T_c) T_b) (tptp.c_Product__Type_Ofst T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) V_ys_2)) V_xs_2))) (forall ((V_ys_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (= (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)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_prod T_b T_c) T_c) (tptp.c_Product__Type_Osnd T_b T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) V_ys_2)) V_ys_2))) (= _let_18 (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 ((V_zs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.c_Product__Type_Ofst T_b T_c)) V_zs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_c) (tptp.c_Product__Type_Osnd T_b T_c)) V_zs_2)) V_zs_2))) (forall ((V_m $$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_na_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_na_2)) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat V_m_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_na_2) V_m_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_b))) (=> (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_b)) V_xs_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Random_Opick T_b V_xs_2) V_i_2)) (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.c_Product__Type_Osnd tptp.tc_Code__Numeral_Ocode__numeral T_b)) V_xs_2))))))) (forall ((V_zs_2 $$unsorted) (V_ys_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_ys_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2) V_ys_2) V_zs_2) (and (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.c_Product__Type_Ofst T_b T_c)) V_zs_2) V_xs_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_c) (tptp.c_Product__Type_Osnd T_b T_c)) V_zs_2) V_ys_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist_Olist__size T_b 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_b tptp.tc_Nat_Onat) 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_i_2 $$unsorted)) (= (not (= V_i_2 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)) V_i_2)))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oupto V_i_2 V_j_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_b) V_f_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) _let_1)) (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Int_Oint T_b) V_f_2) _let_1)))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oupto V_k_2 V_l_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Int_Oint T_b) V_f_2) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Int_Oint T_b) V_f_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Int_Oint) _let_1)))))) (forall ((V_s_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_b))) (=> (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_b)) V_xs_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) (tptp.hAPP (tptp.c_Product__Type_Ofst T_b (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP (tptp.c_Random_Oselect__weight T_b V_xs_2) V_s_2))) (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.c_Product__Type_Osnd tptp.tc_Code__Numeral_Ocode__numeral T_b)) V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (T_b $$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_b _let_1))) (= (tptp.c_Random_Oselect__weight T_b 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_b) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_Ofst tptp.tc_Code__Numeral_Ocode__numeral T_b)) V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun _let_1 _let_2) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_OPair T_b _let_1)) (tptp.c_Random_Opick T_b V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (T_b $$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_b _let_1))) (= (tptp.c_Random_Oselect T_b 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_b)) V_xs_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun _let_1 _let_2) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Product__Type_OPair T_b _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat T_b tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2)) tptp.c_Code__Numeral_Onat__of))))))) (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)))) (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 $$unsorted)) (= (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_x_2) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_y_2)) (= V_x_2 V_y_2))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Code__Numeral_Ocode__numeral) V_n) V_m) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_n)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m))))) (tptp.c_Typedef_Otype__definition tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Nat_Onat tptp.c_Code__Numeral_Onat__of tptp.c_Code__Numeral_Oof__nat _let_17) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) V_na_2) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_na_2)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m_2))))) (forall ((V_x_2 $$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 $$unsorted)) (= (tptp.hAPP tptp.c_Code__Numeral_Onat__of (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral) V_k)) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Nat_Onat) V_k))) (= tptp.c_Code__Numeral_Oint__of (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_Nat_Onat tptp.tc_Int_Oint tptp.tc_Code__Numeral_Ocode__numeral _let_14) tptp.c_Code__Numeral_Onat__of)) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) V_na_2) V_m_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_na_2)) (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_m_2))))) (forall ((V_i $$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)))) (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))) (forall ((V_k $$unsorted)) (= (tptp.c_Code__Numeral_OSuc__code__numeral V_k) (tptp.hAPP tptp.c_Code__Numeral_Oof__nat (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_k))))) (forall ((V_m $$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))))) (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))) (= tptp.c_Code__Numeral_Osubtract__code__numeral _let_15) (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 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Code__Numeral_Onat__of V_k))) (let ((_let_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (=> (not (= V_k (tptp.hAPP tptp.c_Code__Numeral_Oof__nat _let_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_1) (tptp.hAPP tptp.c_Nat_OSuc _let_2))) _let_1)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Omap T_b T_c))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_c))) (=> (and (tptp.class_Groups_Omonoid__add T_c) (tptp.class_Groups_Oordered__ab__semigroup__add T_c)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP V_f_2 B_x)) (tptp.hAPP V_g_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_f_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_g_2) V_xs_2))))))))) (forall ((V_i_2 $$unsorted) (V_xq_2 $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Opos__not__seq V_xq_2) V_i_2) (tptp.c_Lazy__Sequence_Ohb__not__seq (tptp.hAPP V_xq_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Code__Numeral_Ocode__numeral) (tptp.hAPP (tptp.c_Int_Onumber__class_Onumber__of tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_Int_OBit1 (tptp.c_Int_OBit1 tptp.c_Int_OPls)))) V_i_2))))) (forall ((V_no_2 $$unsorted) (V_yes_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (=> (= (tptp.c_List_Opartition T_b V_Pa_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_b 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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)))) (let ((_let_4 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_5 (tptp.hAPP (tptp.c_COMBC T_b _let_4 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (= (tptp.c_List_Olistsp T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_4) (tptp.hAPP _let_5 V_A_2)) (tptp.hAPP _let_5 V_B_2))) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) (tptp.hAPP _let_3 (tptp.c_List_Olists T_b V_A_2))) (tptp.hAPP _let_3 (tptp.c_List_Olists T_b V_B_2)))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b V_A_2) (tptp.c_List_Olist_ONil T_b)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b V_A_2) (tptp.c_List_Olist_ONil T_b)))) (forall ((V_xs_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b V_A_2) V_xs_2)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_A_2 B_x)))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) V_A_2)) V_B_2)) V_x_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b V_A_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b V_B_2) V_x_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olistsp T_b (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 (tptp.tc_List_Olist T_b) tptp.tc_HOL_Obool)) (tptp.c_List_Olistsp T_b V_A_2)) (tptp.c_List_Olistsp T_b V_B_2)))) (forall ((V_B_2 $$unsorted) (V_l_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b V_A_2) V_l_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b V_B_2) V_l_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b (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)) V_l_2))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistsp T_b V_A_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) 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_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olistsp T_b V_A_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) 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_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_List_Olist T_b) tptp.tc_HOL_Obool)) (tptp.c_List_Olistsp T_b V_A_2)) (tptp.c_List_Olistsp T_b V_B_2))))) (forall ((V_no_2 $$unsorted) (V_yes_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (= (tptp.c_List_Opartition T_b V_Pa_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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_yes_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_no_2))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Opartition T_b V_Pa_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_2 _let_2) _let_1) _let_1))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_b)) V_x_2) (tptp.c_List_Olists T_b V_A_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.tc_prod _let_2 _let_2))) (let ((_let_5 (tptp.tc_fun _let_2 _let_4))) (let ((_let_6 (tptp.tc_fun _let_2 _let_2))) (let ((_let_7 (tptp.tc_fun _let_4 _let_4))) (let ((_let_8 (tptp.tc_fun _let_2 _let_7))) (= (tptp.c_List_Opartition T_b V_Pa_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case _let_2 _let_2 _let_4) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_2 _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_8 (tptp.tc_fun _let_5 _let_5) _let_2) (tptp.c_COMBS _let_2 _let_4 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_8 _let_2) (tptp.hAPP (tptp.c_COMBB _let_4 _let_7 _let_2) (tptp.hAPP (tptp.c_If _let_4) (tptp.hAPP V_Pa_2 V_x_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_5 _let_2) _let_3) _let_1)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_6 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 (tptp.tc_fun _let_6 _let_5) _let_2) (tptp.c_COMBB _let_2 _let_4 _let_2)) _let_3)) _let_1))) (tptp.c_List_Opartition T_b V_Pa_2 V_xs_2)))))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_l_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)))) (let ((_let_3 (tptp.hAPP (tptp.c_member (tptp.tc_List_Olist T_b)) V_l_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olists T_b V_A_2))) (=> (tptp.hBOOL (tptp.hAPP _let_3 (tptp.c_List_Olists T_b V_B_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Olistsp T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 V_A_2)) (tptp.hAPP _let_2 V_B_2))) V_l_2)))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (= (tptp.c_Enum_Oproduct T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) 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_c _let_1) (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_x_2)) V_ys_2)) (tptp.c_Enum_Oproduct T_b T_c V_xs_2 V_ys_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2) (and (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_ys_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (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_b T_c) V_xs_2) V_ys_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) V_Pa_2) B_x)))))))) (forall ((V_bs_2 $$unsorted) (V_as_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_b) V_f_2) V_as_2) V_bs_2) (tptp.c_List_Olist__all2 T_d T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_d) V_Pa_2) V_f_2) V_as_2 V_bs_2))) (forall ((V_bs_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_as_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_d T_c))) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_as_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_f_2) V_bs_2)) (tptp.c_List_Olist__all2 T_b T_d (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) (tptp.tc_fun _let_2 _let_1) T_b) (tptp.c_COMBB T_c tptp.tc_HOL_Obool T_d)) V_Pa_2)) V_f_2) V_as_2 V_bs_2))))) (forall ((V_na_2 $$unsorted) (V_bs_2 $$unsorted) (V_as_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_as_2 V_bs_2) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) V_na_2) V_as_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_c) V_na_2) V_bs_2)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_a_2 V_b_2) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_c_2 V_d_2) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_a_2) V_c_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_c) V_b_2) V_d_2))))) (forall ((V_vs_2 $$unsorted) (V_us_2 $$unsorted) (V_Pa_2 $$unsorted) (V_ys_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (= (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)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_us_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_c) V_ys_2) V_vs_2)) (and (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_us_2 V_vs_2))))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_c) V_y_2) V_ys_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_x_2) V_y_2)) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2)))) (forall ((V_na_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_na_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_c) V_na_2) V_ys_2)))) (forall ((T_b $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted)) (= (= V_xs_2 V_ys_2) (tptp.c_List_Olist__all2 T_b T_b tptp.c_fequal V_xs_2 V_ys_2))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_ys_2)))) (forall ((V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.c_List_Olist_ONil T_b) V_ys_2) (= V_ys_2 (tptp.c_List_Olist_ONil T_c)))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 (tptp.c_List_Olist_ONil T_c)) (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_uu $$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_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2) (and (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_ys_2)) (forall ((B_i $$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_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_i)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_c) V_ys_2) B_i)))))))) (forall ((V_p_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_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_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_p_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_c) V_ys_2) V_p_2)))))) (forall ((V_p_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_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_c)) V_ys_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_p_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_c) V_ys_2) V_p_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_i_2 $$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_b)) V_xs_2))) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_x_2) V_y_2)) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_i_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_c V_ys_2) V_i_2) V_y_2)))))) (forall ((V_i_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_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_c)) V_ys_2))) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_b V_xs_2) V_i_2) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist__update T_c V_ys_2) V_i_2) V_y_2)))))) (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_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset (tptp.tc_prod T_b T_c)) (tptp.c_Enum_Oproduct T_b T_c V_xs_2 V_ys_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OSigma T_b T_c) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) (tptp.hAPP (tptp.c_List_Oset T_c) V_ys_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_Set_Oimage _let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool) _let_1) (tptp.hAPP (tptp.c_List_Oset _let_2) (tptp.c_Enum_Osublists T_b V_xs_2))) (tptp.hAPP (tptp.c_Set_OPow T_b) (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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_List_Olist _let_1))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2))) (= (tptp.c_Enum_Osublists T_b (tptp.hAPP _let_3 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_HOL_OLet _let_2 _let_2) (tptp.c_Enum_Osublists T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) 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_b) _let_1) (tptp.c_List_Oset T_b)) (tptp.c_Enum_Osublists T_b V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_List_Oset _let_3) (tptp.c_Enum_On__lists T_b V_na_2 V_xs_2)) (tptp.hAPP (tptp.c_Set_OCollect _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS _let_3 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_3) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 tptp.tc_Nat_Onat tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_3) tptp.c_fequal) (tptp.c_Nat_Osize__class_Osize _let_3))) V_na_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_3 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_3) (tptp.c_Orderings_Oord__class_Oless__eq _let_2)) _let_1)) (tptp.hAPP _let_1 V_xs_2))))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Enum_Oenum__the T_b V_Pa_2) (tptp.c_HOL_OThe T_b V_Pa_2))) (forall ((V_n $$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 ((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 $$unsorted) (T_a $$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 ((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_na_2 $$unsorted) (V_ys_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) V_ys_2) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.c_Enum_On__lists T_b V_na_2 V_xs_2)))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) V_ys_2) V_na_2)))) (forall ((V_Pa_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_a_2) V_b_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c tptp.tc_HOL_Obool) V_Pa_2) B_x))))) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_a_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_b_2)) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_a_2 V_b_2)))) (forall ((V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_i $$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_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_i))))) (=> (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.hBOOL (tptp.hAPP V_Pa_2 V_x_2))))) (forall ((V_xs_2 $$unsorted) (V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofun__left__comm T_b T_c V_f_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold T_b T_c V_f_2) V_y_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofoldl T_c T_b (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) V_f_2) V_y_2) (tptp.c_List_Oremdups T_b V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_OremoveAll T_b V_x_2) V_xs_2)) (tptp.hAPP (tptp.c_List_OremoveAll T_c (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 $$unsorted) (T_a $$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 $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_OremoveAll T_a V_x) V_xs)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Oremdups T_b V_xs_2) V_xs_2) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_1 (tptp.c_List_Oremdups T_b V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oremdups T_b V_xs_2))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (= (= (tptp.hAPP _let_2 _let_1) (tptp.hAPP _let_2 V_xs_2)) (= _let_1 V_xs_2))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oremdups T_a V_xs))) (= (tptp.c_List_Oremdups T_a _let_1) _let_1))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.c_List_Oremdups T_b V_x_2) _let_1) (= V_x_2 _let_1)))) (forall ((V_x $$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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= _let_1 (tptp.c_List_Oremdups T_b V_x_2)) (= V_x_2 _let_1)))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Oremdups T_a _let_1) _let_1))) (forall ((V_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_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 ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.c_List_Oremdups T_b (tptp.hAPP _let_1 (tptp.c_List_Oremdups T_c V_xs_2))) (tptp.c_List_Oremdups T_b (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_x $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_a) V_xs)) (= (tptp.c_List_Oremove1 T_a V_x (tptp.c_List_Oremdups T_a V_xs)) (tptp.c_List_Oremdups T_a (tptp.c_List_Oremove1 T_a V_x V_xs))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (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.c_List_OremoveAll T_b V_x_2) V_xs_2) V_xs_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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oremdups T_b V_xs_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2))) (let ((_let_3 (tptp.c_List_Oremdups T_b (tptp.hAPP _let_2 V_xs_2)))) (let ((_let_4 (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))))) (and (=> _let_4 (= _let_3 _let_1)) (=> (not _let_4) (= _let_3 (tptp.hAPP _let_2 _let_1))))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) (tptp.c_List_Oremdups T_b V_xs_2)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_OremoveAll T_b V_x_2) V_xs_2)) (tptp.hAPP (tptp.c_List_OremoveAll T_c (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_OremoveAll T_b 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_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot _let_1))))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) (tptp.c_List_Oremdups T_b (tptp.c_List_Oconcat T_b V_xss_2))) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xss_2)) (tptp.c_List_Oset T_b)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted__list__of__set T_b (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2)) (tptp.c_List_Olinorder__class_Osort__key T_b T_b (tptp.c_COMBI T_b) (tptp.c_List_Oremdups T_b V_xs_2))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Oconcat T_b V_xss_2) (tptp.hAPP (tptp.c_List_Ofoldl _let_1 _let_1 (tptp.c_List_Oappend T_b) (tptp.c_List_Olist_ONil T_b)) V_xss_2)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 _let_1) _let_1)))) (forall ((V_xss_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (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_b V_xss_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 V_xs_2)) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 V_xs_2)) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)))))) (forall ((T_a $$unsorted)) (= (tptp.c_List_Oconcat T_a (tptp.c_List_Olist_ONil (tptp.tc_List_Olist T_a))) (tptp.c_List_Olist_ONil T_a))) (forall ((V_xs $$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_xss_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Oconcat T_b V_xss_2) (tptp.c_List_Olist_ONil T_b)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xss_2))) (= B_x (tptp.c_List_Olist_ONil T_b))))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Olist_ONil T_b) (tptp.c_List_Oconcat T_b V_xss_2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (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_b))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP _let_1 (tptp.c_List_Oconcat T_c V_xs_2)) (tptp.c_List_Oconcat T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap (tptp.tc_List_Olist T_c) (tptp.tc_List_Olist T_b)) _let_1) V_xs_2))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (= (tptp.hAPP _let_2 (tptp.c_List_Oconcat T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP _let_1 (tptp.c_List_Oconcat T_b V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR _let_2 (tptp.tc_fun T_b 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 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_c V_f_2) V_x_2) (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 V_xs_2))))) (forall ((V_ys $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_COMBI T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osort__key T_b T_b _let_1 V_xs_2) (tptp.hAPP (tptp.c_List_Ofoldl _let_2 T_b (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b _let_1)) (tptp.c_List_Olist_ONil T_b)) V_xs_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist (tptp.tc_List_Olist T_b))))) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b V_xs_2) (tptp.c_List_Oconcat T_b V_ys_2)) (= V_xs_2 V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist (tptp.tc_List_Olist T_b))))) (=> (= (tptp.c_List_Oconcat T_b V_xs_2) (tptp.c_List_Oconcat T_b V_ys_2)) (=> (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (=> (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Oconcat T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun _let_1 _let_1) T_c) (tptp.c_List_Olist_OCons T_b)) V_f_2)) (tptp.c_List_Olist_ONil T_b))) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.tc_List_Olist _let_1))) (= (tptp.c_Enum_On__lists T_b (tptp.hAPP tptp.c_Nat_OSuc V_na_2) V_xs_2) (tptp.c_List_Oconcat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b _let_1) (tptp.tc_fun _let_1 _let_2) _let_1) (tptp.c_List_Omap T_b _let_1)) (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_List_Olist_OCons T_b)))) V_xs_2)) (tptp.c_Enum_On__lists T_b V_na_2 V_xs_2))))))) (forall ((V_xss_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_List_Olist_ONil _let_1))) (let ((_let_3 (tptp.c_List_Olist_OCons _let_1))) (let ((_let_4 (tptp.tc_List_Olist _let_1))) (let ((_let_5 (tptp.c_List_Olist_ONil T_b))) (let ((_let_6 (tptp.c_List_Olist_OCons T_b))) (let ((_let_7 (tptp.hAPP _let_6 V_x_2))) (= (tptp.c_List_Otranspose T_b (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_7 V_xs_2)) V_xss_2)) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_7 (tptp.c_List_Oconcat T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_1) (tptp.c_List_Olist_Olist__case _let_1 T_b _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBK _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) _let_6) _let_5)))) V_xss_2)))) (tptp.c_List_Otranspose T_b (tptp.hAPP (tptp.hAPP _let_3 V_xs_2) (tptp.c_List_Oconcat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_4) (tptp.c_List_Olist_Olist__case _let_4 T_b _let_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 _let_4) T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_4 _let_4) _let_3) _let_2)))) V_xss_2)))))))))))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Oconcat T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 _let_2) (tptp.c_List_Olist_Olist__case _let_2 T_b _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 (tptp.tc_fun _let_2 _let_2) T_b) (tptp.c_COMBK _let_2 _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_2) (tptp.c_List_Olist_OCons T_b)) _let_1)))) V_xss_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 T_b) (tptp.c_List_Ohd T_b)) (tptp.hAPP (tptp.c_List_Ofilter _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_2) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 tptp.tc_HOL_Obool) tptp.c_fequal) _let_1))) V_xss_2)))))) (forall ((V_xs_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_p_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP _let_1 (tptp.c_List_Oconcat T_b V_xs_2)) (tptp.c_List_Oconcat T_b (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_Pa_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_Pa_2))) (=> (tptp.class_Orderings_Olinorder T_c) (= (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 V_xs_2)) (tptp.c_List_Olinorder__class_Osort__key T_b T_c V_f_2 (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) V_Pa_2)) V_xs_2))) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (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 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2) V_xs_2) (forall ((B_x $$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.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Opartition T_b 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_b V_f_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b tptp.c_fNot) V_f_2)) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_Product__Type_Osnd _let_1 _let_1) (tptp.c_List_Opartition T_b V_Pa_2 V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b tptp.c_fNot) V_Pa_2)) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_Product__Type_Ofst _let_1 _let_1) (tptp.c_List_Opartition T_b V_Pa_2 V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) (tptp.hAPP (tptp.c_List_Ofilter T_b V_Q_2) V_xs_2)) (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (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) V_Q_2)) V_Pa_2)) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_Q_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_Q_2))) (= (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_b V_x_2 V_xs_2)) (tptp.c_List_Oremove1 T_b V_x_2 (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))) (= (tptp.c_List_Oremove1 T_b V_x_2 _let_1) _let_1)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_Pa_2))) (=> (tptp.class_Orderings_Olinorder T_c) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_c V_f_2) V_x_2) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b tptp.tc_HOL_Obool T_c V_Pa_2) V_f_2)) V_xs_2))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) _let_1) _let_1))) (forall ((T_a $$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_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_Pa_2))) (let ((_let_2 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_3 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) 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_Pa_2 V_x_2)))) (and (=> _let_5 (= _let_4 (tptp.hAPP _let_3 _let_2))) (=> (not _let_5) (= _let_4 _let_2))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_Pa_2))) (let ((_let_2 (tptp.c_List_Oappend T_b))) (= (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 ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2) (tptp.c_List_Olist_ONil T_b)) (forall ((B_x $$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))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2))) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) (tptp.hAPP (tptp.c_List_Ofilter T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b tptp.tc_HOL_Obool T_c V_Pa_2) V_f_2)) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_b V_Pa_2))) (= (tptp.c_List_Oremdups T_b (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_Oremdups T_b V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))) (= (tptp.hAPP (tptp.c_List_OremoveAll T_b V_x_2) _let_1) _let_1)))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_OremoveAll T_b V_x_2) (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.c_fequal V_x_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 $$unsorted) (T_b $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2))) (let ((_let_3 (tptp.c_List_Otranspose T_b V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) _let_3))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_1) _let_3) V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat T_b) (tptp.c_List_Onth T_b)) V_i_2)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_1) _let_2) (tptp.c_Nat_Osize__class_Osize _let_1))) V_xs_2)))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap (tptp.tc_List_Olist T_c) (tptp.tc_List_Olist T_b)) (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2)))) (= (tptp.c_List_Otranspose T_b (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_Otranspose T_c V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2)) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (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 (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) (tptp.hAPP _let_1 V_xs_2)))) V_Pa_2))))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (=> (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))) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_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_b V_Pa_2) V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))))) (forall ((V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus _let_2) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_y_2) (tptp.c_Orderings_Obot__class_Obot _let_2))) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_y_2))) V_xs_2)))))) (forall ((V_s_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Osublist T_b V_xs_2 V_s_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (= (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) (tptp.hAPP (tptp.c_List_Oset T_b) _let_1))) V_xs_2) _let_1)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Otranspose T_b V_xs_2) (tptp.c_List_Olist_ONil (tptp.tc_List_Olist T_b))) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xs_2))) (= B_x (tptp.c_List_Olist_ONil T_b))))))) (forall ((V_xs_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b 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_b tptp.tc_HOL_Obool tptp.tc_Nat_Onat) V_p_2) (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2)))))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b _let_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 _let_3) T_b) (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_b)) (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_b)))) V_xss_2))))))) (forall ((V_xss_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b))) (let ((_let_6 (tptp.c_List_Olist_OCons T_b))) (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_b)) _let_8)) (= (tptp.c_List_Otranspose T_b _let_8) (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_7 (tptp.c_List_Oconcat T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 _let_1) (tptp.c_List_Olist_Olist__case _let_1 T_b _let_5 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 (tptp.tc_fun _let_1 _let_1) T_b) (tptp.c_COMBK _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) _let_6) _let_5)))) V_xss_2)))) (tptp.c_List_Otranspose T_b (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_b _let_2 (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 _let_4) T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_4 _let_4) _let_3) _let_2)))) V_xss_2)))))))))))))))) (forall ((V_xss_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons _let_1) (tptp.c_List_Olist_ONil T_b)) V_xss_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_List_Olist _let_1) (tptp.c_List_Otranspose__rel T_b)) _let_2)) (= (tptp.c_List_Otranspose T_b _let_2) (tptp.c_List_Otranspose T_b V_xss_2)))))) (forall ((V_is_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ozip T_b tptp.tc_Nat_Onat) V_xs_2))) (let ((_let_2 (tptp.c_Product__Type_Osnd T_b tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_prod T_b tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_HOL_Obool _let_3) V_Pa_2))) (let ((_let_5 (tptp.hAPP (tptp.c_List_Omap _let_3 T_b) (tptp.c_Product__Type_Ofst T_b 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 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Random_Oselect__weight T_b (tptp.hAPP (tptp.c_List_Ofilter (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Code__Numeral_Ocode__numeral T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_COMBK tptp.tc_HOL_Obool T_b)) (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_b V_xs_2))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Random_Opick T_b (tptp.hAPP (tptp.c_List_Ofilter (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral T_b) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case tptp.tc_Code__Numeral_Ocode__numeral T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_Code__Numeral_Ocode__numeral) (tptp.c_COMBK tptp.tc_HOL_Obool T_b)) (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_b V_xs_2))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_List_Olist_ONil _let_1))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_List_Olist _let_1) (tptp.c_List_Otranspose__rel T_b)) _let_2)) (= (tptp.c_List_Otranspose T_b _let_2) _let_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.c_List_Ofoldr _let_1 tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_1) (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat)) (tptp.c_Nat_Osize__class_Osize _let_1)) (tptp.c_List_Otranspose T_b V_xs_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_1) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_1 tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olist_ONil T_b)))) V_xs_2))))) (forall ((V_xss_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist T_c))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize _let_2))) (let ((_let_4 (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_2))) (let ((_let_5 (tptp.c_Orderings_Oord__class_Omax tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.hAPP _let_4 _let_5))) (let ((_let_7 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (= (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP tptp.c_Nat_OSuc _let_7)) (tptp.c_List_Ofoldr _let_2 tptp.tc_Nat_Onat (tptp.hAPP _let_6 _let_3) V_xss_2 _let_1)) (tptp.hAPP tptp.c_Nat_OSuc (tptp.hAPP (tptp.hAPP _let_5 _let_7) (tptp.c_List_Ofoldr _let_2 tptp.tc_Nat_Onat (tptp.hAPP _let_6 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP _let_4 (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat)) _let_3)) (tptp.hAPP tptp.c_Nat_OSuc _let_1))) (tptp.hAPP (tptp.c_List_Ofilter _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool _let_2) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_2 _let_2 tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olist_ONil T_c)))) V_xss_2) _let_1)))))))))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Ofoldr T_c T_b V_f_2 (tptp.c_List_Olist_ONil T_c) V_a_2) V_a_2)) (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Ofoldr T_c T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_c) V_x_2) V_xs_2) V_a_2) (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) (tptp.c_List_Ofoldr T_c T_b V_f_2 V_xs_2 V_a_2)))) (forall ((V_a_2 $$unsorted) (V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Ofoldr T_c T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_c) V_xs_2) V_ys_2) V_a_2) (tptp.c_List_Ofoldr T_c T_b V_f_2 V_xs_2 (tptp.c_List_Ofoldr T_c T_b V_f_2 V_ys_2 V_a_2)))) (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Oab__semigroup__add T_b) (= (tptp.c_List_Ofoldr T_b T_b _let_1 V_xs_2 V_a_2) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 V_a_2) V_xs_2))))) (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_g_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Ofoldr T_c T_b V_g_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_d T_c) V_f_2) V_xs_2) V_a_2) (tptp.c_List_Ofoldr T_d T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_c (tptp.tc_fun T_b T_b) T_d V_g_2) V_f_2) V_xs_2 V_a_2))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_b))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Omonoid__add T_b) (= (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_2 _let_1) V_xs_2) (tptp.c_List_Ofoldr T_b T_b _let_2 V_xs_2 _let_1)))))) (forall ((V_xs_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (=> (tptp.class_Groups_Omonoid__add T_b) (= (tptp.hAPP (tptp.c_List_Ofoldl T_b T_b _let_1 V_a_2) V_xs_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) (tptp.c_List_Ofoldr T_b T_b _let_1 V_xs_2 (tptp.c_Groups_Ozero__class_Ozero T_b))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) (tptp.c_List_Otranspose T_b 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))))) (forall ((V_A_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (= (tptp.c_List_Osublist T_b V_xs_2 V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_1 T_b) (tptp.c_Product__Type_Ofst T_b tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_List_Ofilter _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) _let_1) (tptp.c_member tptp.tc_Nat_Onat)) (tptp.c_Product__Type_Osnd T_b tptp.tc_Nat_Onat))) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Ozip T_b tptp.tc_Nat_Onat) V_xs_2) (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))))))))) (forall ((V_i_2 $$unsorted) (V_xs_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Ozip T_b tptp.tc_Nat_Onat) V_xs_2))) (let ((_let_3 (tptp.c_Product__Type_Osnd T_b tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_5 (tptp.tc_prod T_b tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun _let_6 tptp.tc_HOL_Obool) _let_5) (tptp.c_member tptp.tc_Nat_Onat)))) (let ((_let_8 (tptp.c_COMBC _let_5 _let_6 tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.hAPP (tptp.c_List_Omap _let_5 T_b) (tptp.c_Product__Type_Ofst T_b tptp.tc_Nat_Onat)))) (= (tptp.hAPP _let_9 (tptp.hAPP (tptp.c_List_Ofilter _let_5 (tptp.hAPP (tptp.hAPP _let_8 (tptp.hAPP _let_7 _let_3)) V_A_2)) (tptp.hAPP _let_2 (tptp.c_List_Oupt V_i_2 (tptp.hAPP (tptp.hAPP _let_4 V_i_2) _let_1))))) (tptp.hAPP _let_9 (tptp.hAPP (tptp.c_List_Ofilter _let_5 (tptp.hAPP (tptp.hAPP _let_8 (tptp.hAPP _let_7 (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_5 tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat) _let_5) _let_4) _let_3)) V_i_2))) V_A_2)) (tptp.hAPP _let_2 (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) _let_1))))))))))))))) (forall ((V_j_2 $$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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.c_List_Oupt V_i (tptp.hAPP tptp.c_Nat_OSuc V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) V_j) (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat)))))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.c_List_Oupt V_i (tptp.hAPP tptp.c_Nat_OSuc V_j)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_i) V_j)))) (and (=> _let_3 (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons tptp.tc_Nat_Onat) V_j) _let_1)))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_k $$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)))))) (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 $$unsorted)) (= (tptp.c_List_Oupt V_i (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.c_List_Olist_ONil tptp.tc_Nat_Onat))) (forall ((V_n $$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_na_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_na_2)) (tptp.c_List_Oupt V_m_2 V_na_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.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 $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_i V_j)))) (forall ((V_j $$unsorted) (V_i $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist tptp.tc_Nat_Onat)) (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j) V_i))) (forall ((V_n $$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 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_j)) (= (tptp.c_List_Olast tptp.tc_Nat_Onat (tptp.c_List_Oupt V_i V_j)) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_j) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.c_Nat_OSuc) (tptp.c_List_Oupt V_m_2 V_na_2)) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_m_2) (tptp.hAPP tptp.c_Nat_OSuc V_na_2)))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat V_na_2 V_m_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt V_na_2 (tptp.hAPP tptp.c_Nat_OSuc V_m_2))))) (forall ((V_na_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OlessThan tptp.tc_Nat_Onat) V_na_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_na_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_na_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanAtMost tptp.tc_Nat_Onat V_na_2 V_m_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_na_2) (tptp.hAPP tptp.c_Nat_OSuc V_m_2))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted)) (= (tptp.c_SetInterval_Oord__class_OgreaterThanLessThan tptp.tc_Nat_Onat V_na_2 V_m_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) (tptp.c_List_Oupt (tptp.hAPP tptp.c_Nat_OSuc V_na_2) V_m_2)))) (forall ((V_na_2 $$unsorted)) (= (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatMost tptp.tc_Nat_Onat) V_na_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_na_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_b) (tptp.hAPP (tptp.c_List_Onth T_b) 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_b)) V_xs_2))) V_xs_2)) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oupt V_m_2 V_na_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) 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_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_b) V_f_2) _let_1)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oupt V_m_2 V_na_2))) (=> (tptp.class_Groups_Ocomm__monoid__add T_b) (= (tptp.hAPP (tptp.c_List_Omonoid__add__class_Olistsum T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_b) V_f_2) _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum tptp.tc_Nat_Onat T_b) V_f_2) (tptp.hAPP (tptp.c_List_Oset tptp.tc_Nat_Onat) _let_1)))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_m_2 $$unsorted) (V_na_2 $$unsorted) (V_i_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_na_2) V_m_2))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_b) V_f_2) (tptp.c_List_Oupt V_m_2 V_na_2))) V_i_2) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_m_2) V_i_2))))) (forall ((V_na_2 $$unsorted) (T_b $$unsorted) (V_xs_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat T_b))) (let ((_let_4 (tptp.tc_List_Olist tptp.tc_Nat_Onat))) (=> (=> (= V_xs_2 (tptp.c_List_Olist_ONil _let_2)) (= V_na_2 _let_1)) (=> (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_1)) V_xs_2))) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_1) V_xs_2) B_i)) V_na_2)))) (= (tptp.c_List_Otranspose T_b V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat _let_4 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_4 _let_2) tptp.tc_Nat_Onat) (tptp.c_List_Omap tptp.tc_Nat_Onat T_b)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_3 tptp.tc_Nat_Onat) (tptp.c_List_Onth T_b)) (tptp.hAPP (tptp.c_List_Onth _let_2) V_xs_2))))) (tptp.c_List_Oupt _let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_2)) V_xs_2)))) (tptp.c_List_Oupt _let_1 V_na_2)))))))))) (forall ((V_f_2 $$unsorted) (V_xs_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (=> (tptp.c_List_Olinorder__class_Osorted T_c (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2) V_xs_2)) (=> (= (tptp.hAPP (tptp.c_List_Ohd T_b) (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c tptp.tc_HOL_Obool T_b) (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_b T_c V_f_2) V_a_2) (tptp.c_List_Oremove1 T_b 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) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.c_List_Olinorder__class_Osort__key T_b T_b (tptp.c_COMBI T_b) V_xs_2)))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.c_List_Olinorder__class_Oinsort__insert__key T_b T_b (tptp.c_COMBI T_b) V_x_2 V_xs_2))))) (forall ((V_n $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_a) V_n) V_xs))))) (forall ((V_j $$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_a $$unsorted) (V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Oremove1 T_a V_a V_xs))))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_List_Olinorder__class_Osorted T_a V_xs) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.c_List_Otl T_a) V_xs))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b (tptp.c_COMBI T_b)) V_x_2) V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_b 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 $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.c_List_Olist_ONil T_a)))) (forall ((V_zs_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_b))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_zs_2))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP _let_1 V_x_2) _let_2)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) V_y_2)) (tptp.c_List_Olinorder__class_Osorted T_b _let_2))))))) (forall ((V_zs $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_OCons T_a))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_y) V_zs))) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y)) (=> (tptp.c_List_Olinorder__class_Osorted T_a _let_2) (tptp.c_List_Olinorder__class_Osorted T_a (tptp.hAPP (tptp.hAPP _let_1 V_x) _let_2)))))))) (forall ((V_l $$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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.c_List_Odistinct T_b))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_xs_2)) (=> (tptp.c_List_Olinorder__class_Osorted T_b 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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) (tptp.c_List_Olinorder__class_Osort__key T_c T_b V_f_2 V_xs_2))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 (tptp.c_List_Olinorder__class_Oinsort__insert__key T_c T_b V_f_2 V_x_2 V_xs_2))))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 (tptp.c_List_Oremove1 T_c V_x_2 V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_c T_b V_f_2) V_x_2) V_xs_2))) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 V_xs_2)) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_c V_Pa_2) V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) (tptp.hAPP (tptp.c_List_Ofilter T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_b tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) T_c) tptp.c_fequal) V_f_2)) (tptp.hAPP V_g_2 V_xs_2))) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_g_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.hAPP V_g_2 V_xs_2))) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (and (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (forall ((B_x $$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.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) B_x)))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (and (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_b V_ys_2) (forall ((B_x $$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))) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) (tptp.hAPP (tptp.c_List_Oset T_b) V_ys_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) B_x) B_xa)))))))))) (forall ((V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ofilter T_c V_Pa_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_c T_b V_f_2) V_x_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))))))))) (forall ((V_a1_2 $$unsorted) (V_a2_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_a2_2) V_a1_2)) (and (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_a1_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_a2_2) B_x)))) (tptp.c_List_Olinorder__class_Osorted T_b V_a1_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olinorder__class_Osorted__list__of__set T_b V_A_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (and (= (tptp.hAPP (tptp.c_List_Oset T_b) _let_1) V_A_2) (tptp.c_List_Olinorder__class_Osorted T_b _let_1) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) _let_1))))))) (forall ((V_xs_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (=> (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olinorder__class_Oinsort__key T_b T_b (tptp.c_COMBI T_b)) V_a_2) (tptp.c_List_Oremove1 T_b V_a_2 V_xs_2)) V_xs_2))))) (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_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (forall ((B_j $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_j) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (forall ((B_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Onth T_b) 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_b) (tptp.hAPP _let_1 B_i)) (tptp.hAPP _let_1 B_j))))))))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_ys_2))) (let ((_let_3 (tptp.hAPP _let_1 V_xs_2))) (let ((_let_4 (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2))) (let ((_let_5 (tptp.hAPP _let_4 V_ys_2))) (let ((_let_6 (tptp.c_List_Odistinct T_c))) (let ((_let_7 (tptp.hAPP _let_4 V_xs_2))) (=> (tptp.class_Orderings_Olinorder T_c) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool)) _let_3) _let_2)) (=> (tptp.c_List_Olinorder__class_Osorted T_c _let_7) (=> (tptp.hBOOL (tptp.hAPP _let_6 _let_7)) (=> (tptp.c_List_Olinorder__class_Osorted T_c _let_5) (=> (tptp.hBOOL (tptp.hAPP _let_6 _let_5)) (=> (= _let_3 _let_2) (= V_xs_2 V_ys_2)))))))))))))))) (forall ((V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 tptp.tc_Nat_Onat T_b) (tptp.c_List_Onth T_b)) 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_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_List_Onth _let_1))) (let ((_let_3 (tptp.c_List_Onth T_b))) (let ((_let_4 (tptp.c_List_Otranspose T_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (tptp.c_Fun_Oinj__on _let_1 _let_1 (tptp.c_List_Orev T_b) V_A_2))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (= _let_1 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2)) (= V_xs_2 _let_1)))) (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) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) (tptp.c_List_Olist_ONil T_b)))) (= (= _let_1 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2)) (= V_xs_2 _let_1)))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) (tptp.c_List_Olist_ONil T_b)))) (= (= (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2) _let_1) (= V_xs_2 _let_1)))) (forall ((V_ys $$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 ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_List_Orev T_b))) (= (tptp.hAPP _let_2 (tptp.c_List_Oconcat T_b V_xs_2)) (tptp.c_List_Oconcat T_b (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_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.hAPP (tptp.c_List_Orev T_b) (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_c) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_b))) (let ((_let_2 (tptp.c_List_Ofilter T_b V_Pa_2))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))) (forall ((V_ys $$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_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2)) (tptp.hAPP _let_1 V_xs_2)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2) (tptp.hAPP (tptp.c_List_Orev T_c) V_ys_2)) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 V_ys_2))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2) V_ys_2) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_xs_2 (tptp.hAPP (tptp.c_List_Orev T_c) V_ys_2)))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_a))) (= (tptp.hAPP _let_1 (tptp.hAPP _let_1 V_xs)) V_xs))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_b))) (= (= (tptp.hAPP _let_1 V_xs_2) V_ys_2) (= V_xs_2 (tptp.hAPP _let_1 V_ys_2))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Orev T_b))) (= (= (tptp.hAPP _let_1 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= V_xs_2 V_ys_2)))) (forall ((V_xs $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Omonoid__add__class_Olistsum T_a))) (=> (tptp.class_Groups_Ocomm__monoid__add T_a) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Orev T_a) V_xs)) (tptp.hAPP _let_1 V_xs))))) (forall ((V_xs $$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_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c V_f_2 V_a_2) V_xs_2) (tptp.c_List_Ofoldr T_c T_b (tptp.hAPP (tptp.c_COMBC T_b T_c T_b) V_f_2) (tptp.hAPP (tptp.c_List_Orev T_c) V_xs_2) V_a_2))) (forall ((V_a_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Ofoldr T_c T_b V_f_2 V_xs_2 V_a_2) (tptp.hAPP (tptp.c_List_Ofoldl T_b T_c (tptp.hAPP (tptp.c_COMBC T_c T_b T_b) V_f_2) V_a_2) (tptp.hAPP (tptp.c_List_Orev T_c) V_xs_2)))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_y_2))) (let ((_let_2 (tptp.c_List_Orev T_b))) (= (= (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP _let_1 V_ys_2)) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) (tptp.hAPP _let_2 V_ys_2)) (tptp.hAPP _let_1 (tptp.c_List_Olist_ONil T_b)))))))) (forall ((V_xs $$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 ((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 ((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 ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (= (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2) (tptp.hAPP (tptp.c_List_Ofoldl _let_1 T_b (tptp.hAPP (tptp.c_COMBC T_b _let_1 _let_1) (tptp.c_List_Olist_OCons T_b)) (tptp.c_List_Olist_ONil T_b)) 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_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Oord__class_Omax T_b))) (let ((_let_2 (tptp.c_List_Ofoldr T_b T_b _let_1 V_xs_2 V_y_2))) (let ((_let_3 (= V_xs_2 (tptp.c_List_Olist_ONil T_b)))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2)) (and (=> _let_3 (= _let_2 V_y_2)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_y_2)))))))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize _let_2))) (let ((_let_4 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist _let_2)) (tptp.c_List_Otranspose T_b V_xs_2)))) (let ((_let_5 (= V_xs_2 (tptp.c_List_Olist_ONil _let_2)))) (=> (tptp.c_List_Olinorder__class_Osorted tptp.tc_Nat_Onat (tptp.hAPP (tptp.c_List_Orev tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap _let_2 tptp.tc_Nat_Onat) _let_3) V_xs_2))) (and (=> _let_5 (= _let_4 _let_1)) (=> (not _let_5) (= _let_4 (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth _let_2) V_xs_2) _let_1)))))))))))) (forall ((V_i_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (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_b 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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (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_b (tptp.c_List_Otranspose T_b 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_b))) 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_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (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_b V_Pa_2 V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2) V_xs_2) (forall ((B_x $$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.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_member T_b) V_x_2))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)))) (and (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))))))) (forall ((V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ozip T_b T_c) V_xs_2))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_c V_Pa_2 V_ys_2)) (tptp.c_List_OtakeWhile _let_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_c tptp.tc_HOL_Obool _let_2 V_Pa_2) (tptp.c_Product__Type_Osnd T_b T_c)) (tptp.hAPP _let_1 V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_b T_c))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_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_Pa_2) (tptp.c_Product__Type_Ofst T_b T_c)) (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2)))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.c_List_OtakeWhile T_b V_Pa_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b tptp.tc_HOL_Obool T_c V_Pa_2) V_f_2) V_xs_2))))) (forall ((V_l_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))) (= (tptp.c_List_OtakeWhile T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_l_2))) (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2))) (let ((_let_2 (tptp.c_List_OtakeWhile T_b V_Pa_2 (tptp.hAPP _let_1 V_xs_2)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)))) (and (=> _let_3 (= _let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)))) (=> (not _let_3) (= _let_2 (tptp.c_List_Olist_ONil T_b)))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (tptp.c_List_OtakeWhile T_b V_Pa_2 _let_1) _let_1))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2))) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) _let_1)) V_xs_2)))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)))))) (forall ((V_l_2 $$unsorted)) (= (tptp.c_List_Oreturn__list V_l_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Int_Oint tptp.tc_Nat_Onat) tptp.c_Int_Onat) V_l_2))) (forall ((V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (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))) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))) (= (tptp.c_List_OtakeWhile T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Onth T_b))) (let ((_let_2 (tptp.c_List_OtakeWhile T_b V_Pa_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_b)) _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 ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) _let_2) (tptp.hAPP _let_1 V_xs_2))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) _let_2)))))))) (forall ((T_b $$unsorted) (V_xs_2 $$unsorted)) (=> (not (= V_xs_2 (tptp.c_List_Olist_ONil T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (= (tptp.c_List_OtakeWhile T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) (tptp.c_List_Olast T_b V_xs_2))) V_xs_2) (tptp.c_List_Obutlast T_b V_xs_2))))) (forall ((V_t_2 $$unsorted) (V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_c) (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) V_t_2)) V_f_2))) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.c_List_Orev T_b) (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) V_xs_2))) (= (tptp.hAPP (tptp.c_List_Ofilter T_c _let_1) V_xs_2) (tptp.c_List_OtakeWhile T_c _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 $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Nat_Onat) V_l) V_m) (tptp.c_List_Oreturn__list (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend tptp.tc_Int_Oint) (tptp.c_List_Oembed__list V_l)) (tptp.c_List_Oembed__list V_m))))) (forall ((V_l_2 $$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))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_x_2)))) (let ((_let_2 (tptp.c_List_Orev T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (= (tptp.c_List_OtakeWhile T_b _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_List_Otl T_b) (tptp.c_List_OdropWhile T_b _let_1 V_xs_2))))))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (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 T_b tptp.tc_HOL_Obool) tptp.c_fequal) V_x_2)))) (let ((_let_2 (tptp.c_List_Orev T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) V_xs_2)) (=> (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.c_List_OdropWhile T_b _let_1 (tptp.hAPP _let_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) (tptp.hAPP _let_2 (tptp.c_List_OtakeWhile T_b _let_1 V_xs_2))))))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)) (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2)) V_xs_2)) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2))) (tptp.hAPP _let_1 V_xs_2))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2) (tptp.c_List_Olist_ONil T_b)) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))) (let ((_let_2 (tptp.c_List_OdropWhile T_b V_Pa_2 _let_1))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)))) (and (=> _let_3 (= _let_2 (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2))) (=> (not _let_3) (= _let_2 _let_1))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_b))) (= (tptp.c_List_OdropWhile T_b V_Pa_2 _let_1) _let_1))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Odistinct T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_xs_2)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2))) (=> (not (= _let_1 (tptp.c_List_Olist_ONil T_b))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_List_Ohd T_b) _let_1))))))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2))) (= (tptp.c_List_OdropWhile T_b V_Pa_2 (tptp.hAPP _let_1 V_xs_2)) (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b tptp.tc_HOL_Obool T_c V_Pa_2) V_f_2) V_xs_2))))) (forall ((V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oappend T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))) (= (tptp.c_List_OdropWhile T_b V_Pa_2 (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2)) (tptp.hAPP (tptp.hAPP _let_1 (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2)) V_ys_2)))))) (forall ((V_ys_2 $$unsorted) (V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_y_2) V_ys_2))) (= (= (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2) _let_1) (and (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)) _let_1)) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 V_y_2))))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Odrop T_b) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2))) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_2 (tptp.c_List_Onth T_b))) (let ((_let_3 (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_j_2) (tptp.hAPP _let_1 _let_3))) (= (tptp.hAPP (tptp.hAPP _let_2 _let_3) V_j_2) (tptp.hAPP (tptp.hAPP _let_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_j_2) (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2)))))))))) (forall ((V_f_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Omap T_b T_c) V_f_2))) (let ((_let_2 (tptp.c_List_Omonoid__add__class_Olistsum T_c))) (=> (tptp.class_Groups_Omonoid__add T_c) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (=> (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c))))) (= (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2))) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2)))))))) (forall ((V_xs $$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_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_OdropWhile T_b V_Pa_2 V_xs_2)))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))) (= (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2) (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_xs_2)))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_na_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_na_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_i))))))) (=> (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_na_2))))) (= (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) V_na_2) V_xs_2))))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted) (V_Pa_2 $$unsorted) (V_j_2 $$unsorted)) (let ((_let_1 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (let ((_let_2 (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_j_2))) (=> (forall ((B_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_i) V_j_2)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_i))))) (=> (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_xs_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_xs_2))))))))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (= (tptp.c_List_Olinorder__class_Osorted T_b V_a_2) (or (= V_a_2 (tptp.c_List_Olist_ONil T_b)) (exists ((B_xs $$unsorted) (B_x $$unsorted)) (and (= V_a_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) B_x) B_xs)) (forall ((B_xa $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) (tptp.hAPP (tptp.c_List_Oset T_b) B_xs))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) B_x) B_xa)))) (tptp.c_List_Olinorder__class_Osorted T_b B_xs))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oinsert T_b V_x_2 V_xs_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))))) (and (=> _let_2 (= _let_1 V_xs_2)) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (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.c_List_Oinsert T_b V_x_2 V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)))) (forall ((V_xs $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oinsert T_a V_x (tptp.c_List_Oremdups T_a V_xs)) (tptp.c_List_Oremdups T_a (tptp.c_List_Oinsert T_a V_x V_xs)))) (forall ((V_x $$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_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (= (tptp.c_List_Oinsert T_b V_x_2 V_xs_2) V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Oset T_b))) (= (tptp.hAPP _let_1 (tptp.c_List_Oinsert T_b V_x_2 V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) 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)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Omaps T_c T_b V_f_2 V_xs_2) (tptp.c_List_Oconcat T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c (tptp.tc_List_Olist T_b)) V_f_2) V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Oconcat T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c (tptp.tc_List_Olist T_b)) V_f_2) V_xs_2)) (tptp.c_List_Omaps T_c T_b V_f_2 V_xs_2))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Omaps T_c T_b V_f_2 (tptp.c_List_Olist_ONil T_c)) (tptp.c_List_Olist_ONil T_b))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_List_Omaps T_c T_b V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_c) V_x_2) V_xs_2)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.c_List_Omaps T_c T_b V_f_2 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct (tptp.tc_List_Olist T_b)) V_xs_2)) (=> (forall ((B_ys $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_ys) (tptp.hAPP (tptp.c_List_Oset _let_1) V_xs_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) B_ys))))) (=> (forall ((B_ys $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (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 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_List_Oset T_b))) (let ((_let_3 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_3) B_zs) (tptp.hAPP (tptp.c_List_Oset _let_3) V_xs_2))) (=> (not (= B_ys B_zs)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP _let_2 B_ys)) (tptp.hAPP _let_2 B_zs)) (tptp.c_Orderings_Obot__class_Obot _let_1))))))))))) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) (tptp.c_List_Oconcat T_b V_xs_2))))))) (forall ((V_fs_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) 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_b V_fs_2))) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_List_Omeasures T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_b tptp.tc_Nat_Onat)) V_f_2) V_fs_2)))))))) (forall ((V_fs_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Wellfounded_Owf T_b (tptp.c_List_Omeasures T_b V_fs_2))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) (tptp.c_List_Omeasures T_b (tptp.c_List_Olist_ONil (tptp.tc_fun T_b tptp.tc_Nat_Onat))))))) (forall ((V_fs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_List_Olist tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_List_Olist _let_1))) (= (tptp.c_List_Omeasures T_b V_fs_2) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image _let_2 T_b) (tptp.c_List_Olex tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_Nat_Onat) (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_List_Omap _let_1 tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_COMBC _let_1 T_b tptp.tc_Nat_Onat) (tptp.c_COMBI _let_1)))) V_fs_2))))))) (forall ((V_fs_2 $$unsorted) (V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (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_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_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_b 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_b V_fs_2)))))))))) (forall ((V_fs_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) V_x_2) V_y_2)) (tptp.c_List_Omeasures T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons (tptp.tc_fun T_b tptp.tc_Nat_Onat)) V_f_2) V_fs_2)))))) (forall ((V_s_2 $$unsorted) (V_g_2 $$unsorted) (V_h_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_d $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$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 (tptp.c_Fun_Ocomp T_d T_d T_c (tptp.hAPP V_f_2 B_x)) V_h_2) (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_c V_h_2) (tptp.hAPP V_g_2 B_x))))) (= (tptp.hAPP (tptp.c_List_Ofoldl T_d T_b (tptp.hAPP (tptp.c_COMBC T_b T_d T_d) 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_c T_b (tptp.hAPP (tptp.c_COMBC T_b T_c T_c) V_g_2) V_s_2) V_xs_2))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b T_c))) (=> (and (tptp.class_Orderings_Oorder T_c) (tptp.class_Enum_Oenum 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 (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_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_COMBS T_b 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_b) tptp.c_fequal) V_f_2)) V_g_2))))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (and (tptp.class_Enum_Oenum T_c) (tptp.class_Enum_Oenum T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex _let_1) V_Pa_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_Enum_Oenum__class_Oenum__ex T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_2 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c) V_Pa_2)) (tptp.c_Product__Type_OPair T_b T_c)))))))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Enum_Oenum T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_b) V_Pa_2)) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1)))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Enum_Oenum T_b) (= (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__ex T_b) V_Pa_2))))) (forall ((V_Pa_2 $$unsorted) (V_ys_2 $$unsorted) (T_c $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Ozip T_b T_c))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)))) (let ((_let_3 (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)))) (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_c) _let_4) V_ys_2))) (let ((_let_6 (tptp.hAPP (tptp.hAPP (tptp.c_List_Otake T_b) _let_4) V_xs_2))) (=> (=> (= (tptp.hAPP _let_3 _let_6) (tptp.hAPP _let_2 _let_5)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP _let_1 _let_6) _let_5)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_ys_2))))))))))) (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)))) (forall ((V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (= (tptp.c_List_OdropWhile T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (tptp.c_List_OdropWhile T_b V_Pa_2 V_ys_2)))) (forall ((V_Pa_2 $$unsorted) (V_b_2 $$unsorted) (T_c $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_a_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_b_2)) (=> (forall ((B_n $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) B_n) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_a_2) B_n)) (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_c) V_b_2) B_n))))) (tptp.c_List_Olist__all2 T_b T_c V_Pa_2 V_a_2 V_b_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (exists ((B_x $$unsorted)) (and (= (tptp.hAPP (tptp.c_List_Oset T_b) B_x) V_A_2) (tptp.c_List_Olinorder__class_Osorted T_b B_x) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) B_x)) (forall ((B_y $$unsorted)) (=> (and (= (tptp.hAPP (tptp.c_List_Oset T_b) B_y) V_A_2) (tptp.c_List_Olinorder__class_Osorted T_b B_y) (tptp.hBOOL (tptp.hAPP (tptp.c_List_Odistinct T_b) B_y))) (= B_y B_x)))))))) (forall ((V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2))) (=> (forall ((B_x $$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.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (= (tptp.c_List_OtakeWhile T_b V_Pa_2 (tptp.hAPP _let_1 V_ys_2)) (tptp.hAPP _let_1 (tptp.c_List_OtakeWhile T_b V_Pa_2 V_ys_2)))))) (forall ((V_a_2 $$unsorted) (V_f_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_c) (=> (forall ((B_x $$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.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_c) (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_b T_c V_f_2) V_a_2) V_xs_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_a_2) V_xs_2))))) (forall ((V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2) (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_ys_2)) (exists ((B_us $$unsorted) (B_vs $$unsorted)) (and (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) B_us) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) B_vs))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) B_us))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (= V_xs_2 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) B_vs)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_ys_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) V_ys_2) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (exists ((B_us $$unsorted) (B_vs $$unsorted)) (and (= V_ys_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) B_us) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) B_vs))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) B_us))) (not (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (= V_xs_2 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) B_vs)))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (and (tptp.class_Orderings_Oorder T_c) (tptp.class_Enum_Oenum T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b T_c)) V_f_2) V_g_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b 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_b) (tptp.c_Orderings_Oord__class_Oless__eq T_c)) V_f_2)) V_g_2)))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Enum_Oenum T_b) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_b) V_Pa_2))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Enum_Oenum T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_b) V_Pa_2)) (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1)))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (and (tptp.class_Enum_Oenum T_c) (tptp.class_Enum_Oenum T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all _let_1) V_Pa_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Enum_Oenum__class_Oenum__all T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_Enum_Oenum__class_Oenum__all T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_2 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c) V_Pa_2)) (tptp.c_Product__Type_OPair T_b T_c)))))))))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted) (V_na_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_na_2) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) V_xs_2))) (=> (forall ((B_x $$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.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) V_na_2)))))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_b) (=> (forall ((B_x $$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.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_b) V_x_2) B_x)))) (=> (tptp.c_List_Olinorder__class_Osorted T_b V_xs_2) (tptp.c_List_Olinorder__class_Osorted T_b (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex T_b V_Pa_2 V_xs_2) (exists ((B_n $$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_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_n))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (exists ((B_ys $$unsorted) (B_zs $$unsorted)) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) B_ys) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) B_zs)))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (not (tptp.c_List_Olist__ex T_b V_Pa_2 (tptp.c_List_Olist_ONil T_b)))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (or (tptp.c_List_Olist__ex T_b V_Pa_2 V_xs_2) (tptp.c_List_Olist__ex T_b V_Pa_2 V_ys_2)))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex T_b V_Pa_2 V_xs_2) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex T_b V_Pa_2 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2)) (tptp.c_List_Olist__ex T_b V_Pa_2 V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (or (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (tptp.c_List_Olist__ex T_b V_Pa_2 V_xs_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (exists ((B_ys $$unsorted)) (and (exists ((B_zs $$unsorted)) (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) B_ys) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) B_zs)))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) B_ys)))))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (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))) (exists ((B_ys $$unsorted) (B_zs $$unsorted)) (and (= V_xs_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) B_ys) (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) B_zs))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) B_zs)))))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all T_b V_Pa_2 V_xs_2) (forall ((B_n $$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_b)) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_b) V_xs_2) B_n))))))) (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_Pa_2 $$unsorted) (T_b $$unsorted)) (tptp.c_List_Olist__all T_b V_Pa_2 (tptp.c_List_Olist_ONil T_b))) (forall ((V_ys_2 $$unsorted) (V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_b) V_xs_2) V_ys_2)) (and (tptp.c_List_Olist__all T_b V_Pa_2 V_xs_2) (tptp.c_List_Olist__all T_b V_Pa_2 V_ys_2)))) (forall ((T_b $$unsorted)) (=> (tptp.class_Nat_Osize T_b) (tptp.c_FunDef_Ois__measure T_b (tptp.c_Nat_Osize__class_Osize T_b)))) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (tptp.c_FunDef_Ois__measure T_b V_a_2)) (forall ((V_a_2 $$unsorted) (T_b $$unsorted)) (tptp.c_FunDef_Ois__measure T_b V_a_2)) (forall ((V_f_2 $$unsorted) (T_b $$unsorted)) (tptp.c_FunDef_Ois__measure T_b V_f_2)) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all T_b V_Pa_2 V_xs_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (=> (tptp.c_FunDef_Ois__measure T_b V_f_2) (tptp.c_FunDef_Ois__measure _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b tptp.tc_Nat_Onat _let_1) V_f_2) (tptp.c_Product__Type_Ofst T_b T_c)))))) (forall ((T_c $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_c T_b))) (=> (tptp.c_FunDef_Ois__measure T_b V_f_2) (tptp.c_FunDef_Ois__measure _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b tptp.tc_Nat_Onat _let_1) V_f_2) (tptp.c_Product__Type_Osnd T_c T_b)))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all T_b V_Pa_2 (tptp.hAPP (tptp.c_List_Orev T_b) V_xs_2)) (tptp.c_List_Olist__all T_b V_Pa_2 V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (and (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)) (tptp.c_List_Olist__all T_b V_Pa_2 V_xs_2)))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (= (forall ((B_x $$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.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (tptp.c_List_Olist__all T_b V_Pa_2 V_xs_2))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__all T_b V_Pa_2 V_xs_2) (forall ((B_x $$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.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)))) (= (tptp.c_List_Olist__ex1 T_b V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) (and (=> _let_1 (tptp.c_List_Olist__all T_b (tptp.hAPP (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_fdisj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot) V_Pa_2))) (tptp.hAPP tptp.c_fequal V_x_2)) V_xs_2)) (=> (not _let_1) (tptp.c_List_Olist__ex1 T_b V_Pa_2 V_xs_2)))))) (tptp.c_Nat__Transfer_Otransfer__morphism tptp.tc_Nat_Onat tptp.tc_Int_Oint _let_14 (tptp.hAPP (tptp.c_COMBK tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fTrue)) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (not (tptp.c_List_Olist__ex1 T_b V_Pa_2 (tptp.c_List_Olist_ONil T_b)))) (= (tptp.hAPP _let_13 tptp.c_fFalse) _let_12) (= (tptp.hAPP _let_13 tptp.c_fTrue) _let_12) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex1 T_b V_Pa_2 V_xs_2) (exists ((B_x $$unsorted)) (and (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.hBOOL (tptp.hAPP V_Pa_2 B_x)) (forall ((B_y $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_y) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_y))) (= B_y B_x))))))) (forall ((V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_New__DSequence_Oneg__decr__bind T_c T_b 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_b))) (=> (not _let_2) (= _let_1 (tptp.c_Lazy__Sequence_Ohb__bind T_c T_b (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_c tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_b))) V_f_2) V_i_2)))))))) (forall ((V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_New__DSequence_Opos__decr__bind T_c T_b V_x_2 V_f_2) V_i_2))) (let ((_let_2 (= V_i_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)))) (and (=> _let_2 (= _let_1 (tptp.c_Lazy__Sequence_Oempty T_b))) (=> (not _let_2) (= _let_1 (tptp.hAPP (tptp.c_Lazy__Sequence_Obind T_c T_b (tptp.hAPP V_x_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)) V_f_2) V_i_2)))))))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_b))))) (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_c T_b V_R_2 V_f_2 V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Oneg__decr__bind T_c T_b (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_c _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c 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_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (= (tptp.c_New__Random__Sequence_Opos__decr__bind T_c T_b V_R_2 V_f_2 V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Opos__decr__bind T_c T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_nrandom_2) V_size_2) V_seed_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) V_f_2) V_nrandom_2)) V_size_2)) V_seed_2))))))) (forall ((V_i_2 $$unsorted) (V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Oneg__bind T_c T_b V_x_2 V_f_2) V_i_2) (tptp.c_Lazy__Sequence_Ohb__bind T_c T_b (tptp.hAPP V_x_2 V_i_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_b))) 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))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_New__Random__Sequence_Opos__empty T_b V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Opos__empty T_b))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence (tptp.tc_Option_Ooption T_b))))) (let ((_let_2 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_3 (tptp.tc_fun _let_2 _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_New__Random__Sequence_Oneg__bind T_c T_b V_R_2 V_f_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Oneg__bind T_c T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R_2 V_nrandom_2) V_size_2) V_seed_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral _let_3)) V_f_2) V_nrandom_2)) V_size_2)) V_seed_2))))))) (forall ((V_i_2 $$unsorted) (V_f_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Opos__bind T_c T_b V_x_2 V_f_2) V_i_2) (tptp.hAPP (tptp.c_Lazy__Sequence_Obind T_c T_b (tptp.hAPP V_x_2 V_i_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)) V_f_2) V_i_2)))) (forall ((V_Pa_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_New__Random__Sequence_Oneg__map T_c T_b V_f_2 V_Pa_2) (tptp.c_New__Random__Sequence_Oneg__bind T_c T_b V_Pa_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b (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_b)))))) T_c (tptp.c_New__Random__Sequence_Oneg__single T_b)) V_f_2)))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_New__Random__Sequence_Oneg__single T_b) V_x_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Oneg__single T_b V_x_2))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)))) (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_c T_b V_R_2 V_f_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Opos__bind T_c T_b (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_c _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c tptp.tc_Code__Numeral_Ocode__numeral _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c 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 $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Oneg__single T_a V_x) V_i_1) (tptp.c_Lazy__Sequence_Ohb__single T_a V_x))) (forall ((V_Pa_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_New__Random__Sequence_Opos__map T_c T_b V_f_2 V_Pa_2) (tptp.c_New__Random__Sequence_Opos__bind T_c T_b V_Pa_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral) (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_Lazy__Sequence_Olazy__sequence T_b))))) T_c (tptp.c_New__Random__Sequence_Opos__single T_b)) V_f_2)))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_New__Random__Sequence_Opos__single T_b) V_x_2) V_nrandom_2) V_size_2) V_seed_2) (tptp.c_New__DSequence_Opos__single T_b 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_ys $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Osplice T_a (tptp.c_List_Olist_ONil T_a) V_ys) 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_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 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)))) (= (tptp.c_Wellfounded_Oacyclic T_b (tptp.hAPP _let_2 (tptp.hAPP _let_1 (tptp.c_Predicate_Oconversep T_b T_b V_r_2)))) (tptp.c_Wellfounded_Oacyclic T_b (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_r_2))))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_Predicate_Oconversep T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun T_c (tptp.tc_fun T_b 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_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Predicate_Oconversep T_c T_b V_r_2)) (tptp.c_Predicate_Oconversep T_c T_b V_s_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_Predicate_Oconversep T_c T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_c (tptp.tc_fun T_b 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_b (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Predicate_Oconversep T_c T_b V_r_2)) (tptp.c_Predicate_Oconversep T_c T_b V_s_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_Predicate_Oconversep T_c T_b (tptp.c_Predicate_Opred__comp T_c T_d T_b V_r_2 V_s_2)) (tptp.c_Predicate_Opred__comp T_b T_d T_c (tptp.c_Predicate_Oconversep T_d T_b V_s_2) (tptp.c_Predicate_Oconversep T_c T_d V_r_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_b T_c 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 ((T_c $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_b T_c V_r_2) V_b_2) V_a_2)))) (forall ((V_a1_2 $$unsorted) (V_a2_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_b T_c 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 ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_b T_c V_r_2) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_r_2 V_b_2) V_a_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_Predicate_Oconversep T_c T_b (tptp.c_Predicate_Oconversep T_b T_c V_r_2)) V_r_2)) (forall ((T_b $$unsorted)) (= (tptp.c_Predicate_Oconversep T_b T_b tptp.c_fequal) tptp.c_fequal)) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_b T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool T_b) tptp.c_fNot)) tptp.c_fequal)) V_x_2) V_y_2)) (not (= V_x_2 V_y_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_2 tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun T_c _let_3))) (let ((_let_5 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Predicate_Oconversep T_b T_c (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_2 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_5) T_b) (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_4 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 T_c) (tptp.c_member _let_1))) (tptp.c_Product__Type_OPair T_b T_c)))) V_r_2)) V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_b) V_x_2) V_y_2)) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_c) V_r_2)))))))))) (forall ((V_s2_2 $$unsorted) (V_s1_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.tc_Lazy__Sequence_Olazy__sequence _let_1))) (let ((_let_3 (tptp.tc_fun T_c _let_2))) (= (tptp.c_Lazy__Sequence_Oproduct T_b T_c V_s1_2 V_s2_2) (tptp.hAPP (tptp.c_Lazy__Sequence_Obind T_b _let_1 V_s1_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_b) (tptp.c_Lazy__Sequence_Obind T_c _let_1 V_s2_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c _let_1) _let_3 T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_2 T_c) (tptp.c_Lazy__Sequence_Osingle _let_1))) (tptp.c_Product__Type_OPair T_b T_c))))))))) (forall ((V_x $$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_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_List_Ofilter T_b (tptp.hAPP tptp.c_fequal V_x_2)) V_xs_2))) (= (tptp.c_List_Oreplicate T_b (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_b)) _let_1) V_x_2) _let_1))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_a)) (tptp.c_List_Oreplicate T_a V_n V_x)) V_n)) (forall ((V_lst_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) (tptp.hAPP (tptp.c_COMBK T_b T_c) V_k_2)) V_lst_2) (tptp.c_List_Oreplicate T_b (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_List_Olist T_c)) V_lst_2) V_k_2))) (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 $$unsorted) (V_k $$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) (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_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_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_Pa_2 $$unsorted) (V_a_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_Oreplicate T_b V_na_2 V_a_2)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (and (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)) (not (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_Pa_2 $$unsorted) (V_a_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_Oreplicate T_b V_na_2 V_a_2)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (or (tptp.hBOOL (tptp.hAPP V_Pa_2 V_a_2)) (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))) (forall ((V_y_2 $$unsorted) (V_na_2 $$unsorted) (V_x_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Oreplicate T_b V_m_2 V_x_2) (tptp.c_List_Oreplicate T_b V_na_2 V_y_2)) (and (= V_m_2 V_na_2) (=> (not (= V_m_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= V_x_2 V_y_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted) (V_n $$unsorted) (V_i $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) V_i) V_n)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Onth T_a) (tptp.c_List_Oreplicate T_a V_n V_x)) V_i) V_x))) (forall ((V_x $$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 $$unsorted)) (let ((_let_1 (tptp.c_List_Oreplicate T_a V_n V_x))) (let ((_let_2 (tptp.c_List_Oreplicate T_a V_k V_x))) (let ((_let_3 (tptp.c_List_Oappend T_a))) (= (tptp.hAPP (tptp.hAPP _let_3 _let_1) _let_2) (tptp.hAPP (tptp.hAPP _let_3 _let_2) _let_1)))))) (forall ((V_na_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_List_Oreplicate T_b V_na_2 V_x_2))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Ofilter T_b V_Pa_2) _let_1))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.c_List_Olist_ONil T_b)))))))) (forall ((V_i $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Olist_ONil T_a))) (= (tptp.c_List_Oconcat T_a (tptp.c_List_Oreplicate (tptp.tc_List_Olist T_a) V_i _let_1)) _let_1))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (= (tptp.c_List_Oreplicate T_a (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_x) (tptp.c_List_Olist_ONil T_a))) (forall ((V_x_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Olist_ONil T_b) (tptp.c_List_Oreplicate T_b V_na_2 V_x_2)) (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_x_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_List_Oreplicate T_b V_na_2 V_x_2) (tptp.c_List_Olist_ONil T_b)) (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (forall ((V_x_2 $$unsorted) (V_na_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap T_c T_b) V_f_2) (tptp.c_List_Oreplicate T_c V_na_2 V_x_2)) (tptp.c_List_Oreplicate T_b V_na_2 (tptp.hAPP V_f_2 V_x_2)))) (forall ((V_y_2 $$unsorted) (V_na_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_Oreplicate T_b V_na_2 V_y_2)))) (and (= V_x_2 V_y_2) (not (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))))) (forall ((V_x $$unsorted) (V_i $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_List_Oreplicate T_a V_i V_x))) (let ((_let_2 (tptp.hAPP (tptp.c_List_Olist_OCons T_a) V_x))) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Oappend T_a) _let_1) (tptp.hAPP _let_2 (tptp.c_List_Olist_ONil T_a))) (tptp.hAPP _let_2 _let_1))))) (forall ((V_i_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_List_Omap tptp.tc_Nat_Onat T_b) (tptp.hAPP (tptp.c_COMBK T_b tptp.tc_Nat_Onat) V_x_2)) (tptp.c_List_Oupt (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) V_i_2)) (tptp.c_List_Oreplicate T_b V_i_2 V_x_2))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (V_na_2 $$unsorted)) (=> (not (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (= (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_Oreplicate T_b V_na_2 V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (V_na_2 $$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_List_Oset T_b) (tptp.c_List_Oreplicate T_b V_na_2 V_x_2)))) (let ((_let_3 (= V_na_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (and (=> _let_3 (= _let_2 _let_1)) (=> (not _let_3) (= _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) _let_1)))))))) (forall ((V_x_2 $$unsorted) (V_na_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_List_Oset T_b) (tptp.c_List_Oreplicate T_b (tptp.hAPP tptp.c_Nat_OSuc V_na_2) V_x_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert T_b) V_x_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_d_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (let ((_let_2 (tptp.c_Lazy__Sequence_Osingle _let_1))) (let ((_let_3 (tptp.tc_Lazy__Sequence_Olazy__sequence _let_1))) (let ((_let_4 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Int_Oint) V_d_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint)))) (=> (tptp.class_Lazy__Sequence_Osmall__lazy T_b) (= (tptp.c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy _let_1 V_d_2) (tptp.c_Lazy__Sequence_Oappend _let_1 (tptp.hAPP _let_2 (tptp.c_List_Olist_ONil T_b)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_If _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Int_Oint)) V_d_2)) (tptp.hAPP (tptp.c_Lazy__Sequence_Obind (tptp.tc_prod T_b _let_1) _let_1 (tptp.c_Lazy__Sequence_Oproduct T_b _let_1 (tptp.c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy T_b _let_4) (tptp.c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy _let_1 _let_4))) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b _let_1 _let_3) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 _let_1) (tptp.tc_fun _let_1 _let_3) T_b) (tptp.hAPP (tptp.c_COMBB _let_1 _let_3 _let_1) _let_2)) (tptp.c_List_Olist_OCons T_b))))) (tptp.c_Lazy__Sequence_Oempty _let_1)))))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Opred__comp T_b T_b T_b tptp.c_fequal (tptp.c_Predicate_Opred__comp T_b T_b T_b V_R_2 tptp.c_fequal)) V_R_2)) (forall ((V_d $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (and (tptp.class_Lazy__Sequence_Osmall__lazy T_a) (tptp.class_Lazy__Sequence_Osmall__lazy T_b)) (= (tptp.c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy (tptp.tc_prod T_a T_b) V_d) (tptp.c_Lazy__Sequence_Oproduct T_a T_b (tptp.c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy T_a V_d) (tptp.c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy T_b V_d))))) (forall ((V_i_2 $$unsorted) (V_yq_2 $$unsorted) (V_xq_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_New__DSequence_Opos__union T_b V_xq_2 V_yq_2) V_i_2) (tptp.c_Lazy__Sequence_Oappend T_b (tptp.hAPP V_xq_2 V_i_2) (tptp.hAPP V_yq_2 V_i_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Fields_Olinordered__field T_a) (=> (forall ((B_e $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_a) (tptp.c_Groups_Ozero__class_Ozero T_a)) B_e)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_y) B_e))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq T_a) V_x) V_y))))) (forall ((V_seed_2 $$unsorted) (V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_R2_2 $$unsorted) (V_R1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_New__Random__Sequence_Opos__union T_b V_R1_2 V_R2_2 V_nrandom_2 V_size_2 V_seed_2) (tptp.c_New__DSequence_Opos__union T_b (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R1_2 V_nrandom_2) V_size_2) V_seed_2) (tptp.hAPP (tptp.hAPP (tptp.hAPP V_R2_2 V_nrandom_2) V_size_2) V_seed_2)))) (forall ((V_i $$unsorted) (V_d $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Osmall__lazy_H V_d V_i))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_d) V_i)))) (and (=> _let_2 (= _let_1 (tptp.c_Lazy__Sequence_Oempty tptp.tc_Int_Oint))) (=> (not _let_2) (= _let_1 (tptp.c_Lazy__Sequence_Oappend tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Lazy__Sequence_Osingle tptp.tc_Int_Oint) V_i) (tptp.c_Lazy__Sequence_Osmall__lazy_H V_d (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)))))))))) (forall ((V_d $$unsorted)) (= (tptp.c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy tptp.tc_Int_Oint V_d) (tptp.c_Lazy__Sequence_Osmall__lazy_H V_d (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus tptp.tc_Int_Oint) V_d)))) (forall ((V_i_2 $$unsorted) (V_d_2 $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Osmall__lazy_H V_d_2 V_i_2))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Int_Oint) V_d_2) V_i_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Lazy__Sequence_Osmall__lazy_H__rel) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair tptp.tc_Int_Oint tptp.tc_Int_Oint) V_d_2) V_i_2))) (and (=> _let_2 (= _let_1 (tptp.c_Lazy__Sequence_Oempty tptp.tc_Int_Oint))) (=> (not _let_2) (= _let_1 (tptp.c_Lazy__Sequence_Oappend tptp.tc_Int_Oint (tptp.hAPP (tptp.c_Lazy__Sequence_Osingle tptp.tc_Int_Oint) V_i_2) (tptp.c_Lazy__Sequence_Osmall__lazy_H V_d_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Int_Oint) V_i_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Int_Oint))))))))))) (forall ((V_lazy__sequence $$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_yq $$unsorted) (V_xq $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (= (tptp.c_Lazy__Sequence_Oappend T_a (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_x V_xq) V_yq) (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_x (tptp.c_Lazy__Sequence_Oappend T_a V_xq V_yq)))) (forall ((V_lazy__sequence_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_lazy__sequence_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_b V_a_2 V_lazy__sequence_2) (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_b V_a_H_2 V_lazy__sequence_H_2)) (and (= V_a_2 V_a_H_2) (= V_lazy__sequence_2 V_lazy__sequence_H_2)))) (forall ((V_lazy__sequence_2 $$unsorted) (V_a_2 $$unsorted) (V_fa_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size T_b V_fa_2))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (= (tptp.hAPP _let_1 (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_b V_a_2 V_lazy__sequence_2)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP V_fa_2 V_a_2)) (tptp.hAPP _let_1 V_lazy__sequence_2))) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))))))) (forall ((V_Pa_2 $$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_Lazy__Sequence_Osmall__lazy_H__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_d $$unsorted) (B_i $$unsorted)) (let ((_let_1 (tptp.hAPP V_Pa_2 B_d))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Wellfounded_Oaccp (tptp.tc_prod tptp.tc_Int_Oint tptp.tc_Int_Oint) tptp.c_Lazy__Sequence_Osmall__lazy_H__rel) (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.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 V_Pa_2 V_a0_2) V_a1_2))))) (forall ((V_xq_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Lazy__Sequence_Olazy__sequence T_b))) (let ((_let_2 (tptp.c_Nat_Osize__class_Osize _let_1))) (= (tptp.hAPP _let_2 V_xq_2) (tptp.c_Option_Ooption_Ooption__case tptp.tc_Nat_Onat (tptp.tc_prod T_b _let_1) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b _let_1 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBK (tptp.tc_fun _let_1 tptp.tc_Nat_Onat) T_b) (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_b) V_xq_2)))))) (forall ((V_xq_2 $$unsorted) (V_s_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size T_b V_s_2))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_4 (tptp.tc_Lazy__Sequence_Olazy__sequence T_b))) (let ((_let_5 (tptp.tc_fun _let_4 tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.tc_fun _let_4 _let_3))) (= (tptp.hAPP _let_1 V_xq_2) (tptp.c_Option_Ooption_Ooption__case tptp.tc_Nat_Onat (tptp.tc_prod T_b _let_4) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b tptp.tc_Nat_Onat _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_6 (tptp.tc_fun tptp.tc_Nat_Onat _let_5) T_b) (tptp.c_COMBC _let_4 tptp.tc_Nat_Onat tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_5 _let_6 T_b) (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_3 _let_4) _let_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_5 _let_5) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 (tptp.tc_fun _let_5 _let_5) T_b) (tptp.c_COMBB tptp.tc_Nat_Onat tptp.tc_Nat_Onat _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_Nat_Onat _let_3 T_b) _let_2) V_s_2))) _let_1)))) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (tptp.hAPP (tptp.c_Lazy__Sequence_Oyield T_b) V_xq_2)))))))))) (forall ((V_xq_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_Lazy__Sequence_Olazy__sequence T_c))) (= (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case T_b T_c V_f_2 V_g_2 V_xq_2) (tptp.c_Option_Ooption_Ooption__case T_b (tptp.tc_prod T_c _let_1) V_f_2 (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c _let_1 T_b) V_g_2) (tptp.hAPP (tptp.c_Lazy__Sequence_Oyield T_c) V_xq_2))))) (forall ((T_b $$unsorted)) (= (tptp.c_Lazy__Sequence_Oyieldn T_b) (tptp.c_Lazy__Sequence_Oanamorph (tptp.tc_Lazy__Sequence_Olazy__sequence T_b) T_b (tptp.c_Lazy__Sequence_Oyield T_b)))) (forall ((V_lazy__sequence_2 $$unsorted) (V_a_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case T_b T_c V_f1_2 V_f2_2 (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_c V_a_2 V_lazy__sequence_2)) (tptp.hAPP (tptp.hAPP V_f2_2 V_a_2) V_lazy__sequence_2))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (tptp.c_Relation_Orefl__on T_c (tptp.hAPP V_A_2 B_x) (tptp.hAPP V_r_2 B_x)))) (tptp.c_Relation_Orefl__on T_c (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_S_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun (tptp.tc_prod T_c T_c) tptp.tc_HOL_Obool)) V_S_2) V_r_2)))) (forall ((V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (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.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_List_Omember T_b) V_xs_2) V_x_2)))) (forall ((V_y_2 $$unsorted) (V_xs_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Omember T_b))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_List_Olist_OCons T_b) V_x_2) V_xs_2)) V_y_2)) (or (= V_x_2 V_y_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_xs_2) V_y_2)))))) (forall ((T_b $$unsorted)) (= (tptp.c_List_Omember T_b) (tptp.c_List_Oset T_b))) (forall ((V_y $$unsorted) (T_a $$unsorted)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_List_Omember T_a) (tptp.c_List_Olist_ONil T_a)) V_y)))) (forall ((V_x_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_List_Omember T_b) V_xs_2) V_x_2)) (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))))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex1 T_b V_Pa_2 V_xs_2) (exists ((B_x $$unsorted)) (and (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.hBOOL (tptp.hAPP V_Pa_2 B_x)) (forall ((B_y $$unsorted)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_y) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_y))) (= B_y B_x))))))) (forall ((V_f_2 $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum T_b T_c) V_f_2))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__add T_c) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite _let_2) V_S_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_x) V_S_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) B_x)))) (=> (forall ((B_T1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_T1) V_S_2)) (forall ((B_T2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_T2) V_S_2)) (=> (not (= B_T1 B_T2)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_member T_b) B_x))) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_T1)) (=> (tptp.hBOOL (tptp.hAPP _let_1 B_T2)) (= (tptp.hAPP V_f_2 B_x) (tptp.c_Groups_Ozero__class_Ozero T_c))))))))))) (= (tptp.hAPP _let_1 (tptp.c_Complete__Lattice_OSup__class_OSup _let_2 V_S_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__add__class_Osetsum _let_2 T_c) _let_1) V_S_2))))))))) (forall ((V_d_2 $$unsorted) (T_b $$unsorted) (V_Pa_2 $$unsorted)) (let ((_let_1 (tptp.c_FunDef_OTHE__default T_b V_d_2 V_Pa_2))) (and (=> (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_y)) (= B_y B_x))))) (= _let_1 (tptp.c_HOL_OThe T_b V_Pa_2))) (=> (not (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)) (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Pa_2 B_y)) (= B_y B_x)))))) (= _let_1 V_d_2))))) (forall ((T_d $$unsorted) (T_c $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP V_B_2 B_x) V_b_2) V_c_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b (tptp.tc_fun T_c (tptp.tc_fun T_d tptp.tc_HOL_Obool))) V_A_2) V_B_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_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c (tptp.tc_fun T_d tptp.tc_HOL_Obool))) V_A_2) V_B_2) V_b_2) V_c_2)) (not (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) 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_M_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite (tptp.tc_List_Olist T_b)) V_M_2)) (exists ((B_n $$unsorted)) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.tc_List_Olist T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_x) V_M_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Nat_Osize__class_Osize _let_1) B_x)) B_n)))))))) (forall ((V_fa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size T_b V_fa_2) (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_b)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((T_a $$unsorted)) (= (tptp.hAPP (tptp.c_Nat_Osize__class_Osize (tptp.tc_Lazy__Sequence_Olazy__sequence T_a)) (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_a)) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) (forall ((V_yq $$unsorted) (T_a $$unsorted)) (= (tptp.c_Lazy__Sequence_Oappend T_a (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_a) V_yq) V_yq)) (forall ((V_lazy__sequence_H $$unsorted) (V_a_H $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_a_H V_lazy__sequence_H) (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_a)))) (forall ((V_lazy__sequence_H $$unsorted) (V_a_H $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_a) (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_a V_a_H V_lazy__sequence_H)))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case T_b T_c V_f1_2 V_f2_2 (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_c)) V_f1_2)) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.c_List_Olist__ex tptp.tc_Int_Oint V_Pa_2 (tptp.c_List_Oupto V_i_2 V_j_2)) (not (tptp.c_List_Oall__interval__int (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Int_Oint tptp.c_fNot) V_Pa_2) V_i_2 V_j_2)))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.c_List_Olist__all tptp.tc_Int_Oint V_Pa_2 (tptp.c_List_Oupto V_i_2 V_j_2)) (tptp.c_List_Oall__interval__int V_Pa_2 V_i_2 V_j_2))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.c_List_Oall__interval__int V_Pa_2 V_i_2 V_j_2) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Int_Oint) B_x) (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Int_Oint V_i_2 V_j_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_code__numeral_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec T_b V_f1_2 V_f2_2 (tptp.c_Code__Numeral_OSuc__code__numeral V_code__numeral_2)) (tptp.hAPP (tptp.hAPP V_f2_2 V_code__numeral_2) (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec T_b V_f1_2 V_f2_2 V_code__numeral_2)))) (forall ((V_size_2 $$unsorted) (V_g_2 $$unsorted) (T_b $$unsorted) (V_nrandom_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Product__Type_Ounit tptp.tc_Code__Evaluation_Oterm))) (let ((_let_2 (tptp.tc_prod T_b _let_1))) (let ((_let_3 (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_HOL_Obool (tptp.tc_Option_Ooption (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)))))) (let ((_let_4 (tptp.tc_fun _let_3 _let_3))) (let ((_let_5 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (let ((_let_6 (tptp.c_Product__Type_OPair _let_3 _let_5))) (let ((_let_7 (tptp.tc_prod _let_3 _let_5))) (let ((_let_8 (tptp.tc_fun _let_5 _let_7))) (let ((_let_9 (tptp.tc_fun _let_3 _let_8))) (let ((_let_10 (tptp.c_Random__Sequence_ORandom T_b V_g_2 V_nrandom_2 V_size_2))) (let ((_let_11 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Code__Numeral_Ocode__numeral) V_nrandom_2) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))))) (and (=> _let_11 (= _let_10 (tptp.hAPP _let_6 (tptp.c_DSequence_Oempty T_b)))) (=> (not _let_11) (= _let_10 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oscomp _let_5 _let_2 _let_5 _let_7) (tptp.hAPP V_g_2 V_size_2)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_9 _let_8 _let_2) (tptp.hAPP (tptp.c_Product__Type_Oscomp _let_5 _let_3 _let_5 _let_7) (tptp.c_Random__Sequence_ORandom T_b V_g_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Code__Numeral_Ocode__numeral) V_nrandom_2) (tptp.c_Groups_Oone__class_Oone tptp.tc_Code__Numeral_Ocode__numeral)) V_size_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 _let_9 _let_2) (tptp.hAPP (tptp.c_COMBB _let_3 _let_8 _let_3) _let_6)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_4 _let_2) (tptp.c_DSequence_Ounion T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b _let_3 _let_2) (tptp.c_DSequence_Osingle T_b)) (tptp.c_Product__Type_Ofst T_b _let_1))))))))))))))))))))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec T_b V_f1_2 V_f2_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral)) V_f1_2)) (forall ((V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Random__Sequence_Oempty T_b V_nrandom_2 V_size_2) (tptp.hAPP (tptp.c_Product__Type_OPair (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_HOL_Obool (tptp.tc_Option_Ooption (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)))) (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.c_DSequence_Oempty T_b)))) (forall ((V_size_2 $$unsorted) (V_nrandom_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Random__Sequence_Osingle T_b) V_x_2) V_nrandom_2) V_size_2) (tptp.hAPP (tptp.c_Product__Type_OPair (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_HOL_Obool (tptp.tc_Option_Ooption (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)))) (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral)) (tptp.hAPP (tptp.c_DSequence_Osingle T_b) V_x_2)))) (forall ((V_Pa_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Code__Numeral_Ocode__numeral tptp.tc_Code__Numeral_Ocode__numeral))) (= (tptp.c_Random__Sequence_Omap T_c T_b V_f_2 V_Pa_2) (tptp.c_Random__Sequence_Obind T_c T_b V_Pa_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun _let_1 (tptp.tc_prod (tptp.tc_fun tptp.tc_Code__Numeral_Ocode__numeral (tptp.tc_fun tptp.tc_HOL_Obool (tptp.tc_Option_Ooption (tptp.tc_Lazy__Sequence_Olazy__sequence T_b)))) _let_1)))) T_c (tptp.c_Random__Sequence_Osingle T_b)) V_f_2))))) (forall ((T_d $$unsorted) (V_f_2 $$unsorted) (V_Q_2 $$unsorted) (V_p_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted) (V_Pa_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_Product__Type_Osnd T_c T_b) V_p_2)) (tptp.hAPP (tptp.c_Product__Type_Ofst T_c T_b) V_p_2))) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_y) B_x)) (tptp.hBOOL (tptp.hAPP V_Q_2 (tptp.hAPP (tptp.hAPP V_f_2 B_x) B_y))))) (tptp.hBOOL (tptp.hAPP V_Q_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_b T_d) V_f_2) V_p_2)))))) (forall ((V_A_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Relation_OImage T_c T_b (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_c T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_c T_b tptp.tc_HOL_Obool) V_Pa_2))) V_A_2) (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_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_b) (tptp.hAPP (tptp.c_COMBS T_c 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_c) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_A_2)))) (tptp.hAPP (tptp.c_COMBC T_c T_b tptp.tc_HOL_Obool) V_Pa_2))))))) (forall ((V_z_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_List_Olexord T_b V_r_2))) (let ((_let_2 (tptp.tc_List_Olist T_b))) (let ((_let_3 (tptp.c_Product__Type_OPair _let_2 _let_2))) (let ((_let_4 (tptp.hAPP _let_3 V_x_2))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_2 _let_2)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_y_2)) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_3 V_y_2) V_z_2)) _let_1)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP _let_4 V_z_2)) _let_1))))))))))) (tptp.c_Relation_Otrans tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than) (forall ((V_R2_2 $$unsorted) (T_c $$unsorted) (V_R1_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_R1_2) (=> (tptp.c_Relation_Otrans T_c V_R2_2) (tptp.c_Relation_Otrans (tptp.tc_prod T_b T_c) (tptp.c_Wellfounded_Olex__prod T_b T_c V_R1_2 V_R2_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Otrans T_b V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_x) B_y)) V_r_2)) (forall ((B_z $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_y) B_z)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_z)) V_r_2)))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_a_2))) (let ((_let_3 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.c_Relation_Otrans T_b V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_b_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 (tptp.hAPP _let_2 V_c_2)) V_r_2))))))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (= (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.c_member T_b)))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.c_Relation_Otrans T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.c_Relation_Otrans T_c (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_b T_c) V_r_2) V_f_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) V_r_2) V_r_2)) V_r_2)))) (forall ((T_b $$unsorted)) (= (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b tptp.tc_HOL_Obool) (tptp.c_Set_OBex T_b))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (=> (tptp.c_Relation_Otrans T_b V_s_2) (tptp.c_Relation_Otrans T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1))))) (forall ((V_Q_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 T_c) (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_1 tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2)))) V_Q_2))))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_b tptp.tc_HOL_Obool) V_Q_2) B_x)))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.c_Relation_Otrans (tptp.tc_List_Olist T_b) (tptp.c_List_Olistrel T_b V_r_2)))) (forall ((V_Pa_2 $$unsorted) (T_b $$unsorted)) (not (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Otrans T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.c_Relation_Otrans T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.c_Relation_Otrans (tptp.tc_List_Olist T_b) (tptp.c_List_Olexord T_b V_r_2)))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Otrans (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Ofinite__psubset T_b))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Otrans T_b (tptp.c_Relation_OId T_b))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Otrans T_b (tptp.c_Relation_OId__on T_b V_A_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Otrans T_b (tptp.c_Transitive__Closure_Otrancl T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (= (tptp.c_Transitive__Closure_Otrancl T_b V_r_2) V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Otrans T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2))) (forall ((V_xs_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_List_Olist__ex T_b V_Pa_2 V_xs_2) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_Pa_2 $$unsorted) (V_xs_2 $$unsorted) (T_b $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.hAPP (tptp.c_List_Oset T_b) V_xs_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (tptp.c_List_Olist__ex T_b V_Pa_2 V_xs_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_c _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool _let_1) (tptp.c_HOL_OEx T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 _let_1) (tptp.hAPP (tptp.c_COMBS T_c tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_c) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) _let_1) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c)) tptp.c_fequal)) V_B_2)))))))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_c _let_1))) (= (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_c _let_1) V_A_2) V_B_2) (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 (tptp.hAPP (tptp.c_Set_OCollect _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool _let_1) (tptp.c_HOL_OEx T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 _let_1) (tptp.hAPP (tptp.c_COMBS T_c tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_c) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_2 tptp.tc_HOL_Obool) (tptp.c_member T_c)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) _let_1) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_c)) tptp.c_fequal)) V_B_2)))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_1 T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2) V_x_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool _let_1)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 T_c T_b) (tptp.c_COMBI _let_1)) V_x_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_1 T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.c_Complete__Lattice_OSup__class_OSup _let_1 V_A_2) V_x_2) (tptp.c_Complete__Lattice_OSup__class_OSup T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool _let_1)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 T_c T_b) (tptp.c_COMBI _let_1)) V_x_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_1 T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2) V_x_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool _let_1)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 T_c T_b) (tptp.c_COMBI _let_1)) V_x_2)))))))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c T_b))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_1 T_b))) (=> (tptp.class_Complete__Lattice_Ocomplete__lattice T_b) (= (tptp.hAPP (tptp.c_Complete__Lattice_OInf__class_OInf _let_1 V_A_2) V_x_2) (tptp.c_Complete__Lattice_OInf__class_OInf T_b (tptp.hAPP (tptp.c_Set_OCollect T_b) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_b) (tptp.c_HOL_OEx _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_b) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) T_b) (tptp.c_COMBB T_b tptp.tc_HOL_Obool _let_1)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 T_c T_b) (tptp.c_COMBI _let_1)) V_x_2)))))))))))) (forall ((V_R_2 $$unsorted) (V_Y_2 $$unsorted) (V_X_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_X_2)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_Y_2)) (=> (not (= V_Y_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_X_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_Y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_xa)))))) (tptp.c_Wellfounded_Omax__extp T_b V_R_2 V_X_2 V_Y_2))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Otransp T_b V_r_2) (tptp.c_Relation_Otrans T_b (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) V_r_2))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_r_2 V_x_2))) (=> (tptp.c_Predicate_Otransp T_b 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_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (and (tptp.c_Predicate_Oreflp T_b V_R_2) (tptp.c_Predicate_Osymp T_b V_R_2) (tptp.c_Predicate_Otransp T_b V_R_2)))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Predicate_Oreflp T_b V_R_2) (=> (tptp.c_Predicate_Osymp T_b V_R_2) (=> (tptp.c_Predicate_Otransp T_b V_R_2) (tptp.c_Equiv__Relations_Oequivp T_b V_R_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Predicate_Osymp T_b 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_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (not (=> (tptp.c_Predicate_Oreflp T_b V_R_2) (=> (tptp.c_Predicate_Osymp T_b V_R_2) (not (tptp.c_Predicate_Otransp T_b V_R_2))))))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Finite__Set_Ofinite T_b))) (= (tptp.c_Wellfounded_Omax__extp T_b 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_b tptp.tc_HOL_Obool)))) (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_a1_2)) (exists ((B_xa $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_a2_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_xa)))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Otrans T_b V_r_2) (=> (tptp.c_Relation_Oantisym T_b V_r_2) (tptp.c_Relation_Otrans T_b (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b)))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Oantisym T_b (tptp.c_Relation_OId__on T_b V_A_2))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2)) (=> (tptp.c_Relation_Oantisym T_b V_s_2) (tptp.c_Relation_Oantisym T_b V_r_2)))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Oantisym T_b (tptp.c_Relation_OId T_b))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Oantisym T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Oantisym T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.c_Relation_Oantisym T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Oantisym T_b V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_y) B_x)) V_r_2)) (= B_x B_y)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.c_Relation_Oantisym T_b V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)) V_r_2)) (= V_a_2 V_b_2))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Oantisym T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.c_Relation_OId T_b))) (tptp.c_Relation_Oantisym T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Wellfounded_Oacyclic T_b V_r_2) (tptp.c_Relation_Oantisym T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_L_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.tc_fun T_d T_c))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun _let_1 T_c))) (= (tptp.c_Partial__Function_Ofun__lub T_c T_b T_d V_L_2 V_A_2 V_x_2) (tptp.hAPP V_L_2 (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 tptp.tc_HOL_Obool T_c) (tptp.c_HOL_OEx _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 T_c) (tptp.hAPP (tptp.c_COMBS _let_1 tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) _let_1) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 tptp.tc_HOL_Obool) (tptp.c_member _let_1)) V_A_2)))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c _let_3 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun T_c tptp.tc_HOL_Obool) (tptp.tc_fun _let_3 _let_2) T_c) (tptp.c_COMBB T_c tptp.tc_HOL_Obool _let_1)) tptp.c_fequal)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 T_d T_c) (tptp.c_COMBI _let_1)) V_x_2))))))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osym T_b V_r_2) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Orel__comp T_b T_b T_b) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) V_r_2)) V_r_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Osym T_b V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) B_y)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_y) B_x)) V_r_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_b))) (let ((_let_2 (tptp.c_member (tptp.tc_prod T_b T_b)))) (=> (tptp.c_Relation_Osym T_b V_r_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) V_r_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)) V_r_2))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osym T_b V_r_2) (tptp.c_Relation_Osym T_c (tptp.hAPP (tptp.hAPP (tptp.c_Relation_Oinv__image T_b T_c) V_r_2) V_f_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Osym T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osym T_b V_r_2) (=> (tptp.c_Relation_Osym T_b V_s_2) (tptp.c_Relation_Osym T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osym T_b V_r_2) (tptp.c_Relation_Osym (tptp.tc_List_Olist T_b) (tptp.c_List_Olistrel T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Osym T_b (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)) (tptp.c_Relation_Osym T_b V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_Osym T_b V_r_2) (= (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2) V_r_2))) (forall ((T_b $$unsorted)) (tptp.c_Relation_Osym T_b (tptp.c_Relation_OId T_b))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osym T_b V_r_2) (=> (tptp.c_Relation_Osym T_b V_s_2) (tptp.c_Relation_Osym T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) V_s_2))))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Osym T_b (tptp.c_Relation_OId__on T_b V_A_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osym T_b V_r_2) (tptp.c_Relation_Osym T_b (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Osym T_b V_r_2) (tptp.c_Relation_Osym T_b (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Relation_Osym T_b (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf (tptp.tc_fun (tptp.tc_prod T_b T_b) tptp.tc_HOL_Obool)) V_r_2) (tptp.hAPP (tptp.c_Relation_Oconverse T_b T_b) V_r_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (and (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (tptp.c_Relation_Osym T_b V_r_2) (tptp.c_Relation_Otrans T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (=> (tptp.c_Relation_Osym T_b V_r_2) (=> (tptp.c_Relation_Otrans T_b V_r_2) (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (not (=> (tptp.c_Relation_Orefl__on T_b V_A_2 V_r_2) (=> (tptp.c_Relation_Osym T_b V_r_2) (not (tptp.c_Relation_Otrans T_b V_r_2))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Predicate_Osymp T_b V_r_2) (tptp.c_Relation_Osym T_b (tptp.hAPP (tptp.c_Set_OCollect (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_b tptp.tc_HOL_Obool) V_r_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Equiv__Relations_Opart__equivp T_b V_R_2) (and (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_x))) (tptp.c_Predicate_Osymp T_b V_R_2) (tptp.c_Predicate_Otransp T_b V_R_2)))) (forall ((T_b $$unsorted) (V_R_2 $$unsorted)) (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_x))) (=> (tptp.c_Predicate_Osymp T_b V_R_2) (=> (tptp.c_Predicate_Otransp T_b V_R_2) (tptp.c_Equiv__Relations_Opart__equivp T_b V_R_2))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (tptp.c_Equiv__Relations_Opart__equivp T_b V_R_2))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Equiv__Relations_Opart__equivp T_b V_R_2) (and (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_x))) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.hAPP V_R_2 B_y))) (let ((_let_2 (tptp.hAPP V_R_2 B_x))) (= (tptp.hBOOL (tptp.hAPP _let_2 B_y)) (and (tptp.hBOOL (tptp.hAPP _let_2 B_x)) (tptp.hBOOL (tptp.hAPP _let_1 B_y)) (= _let_2 _let_1))))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Opart__equivp T_b 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_b $$unsorted)) (let ((_let_1 (tptp.hAPP V_R_2 V_x_2))) (=> (tptp.c_Equiv__Relations_Opart__equivp T_b V_R_2) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_y_2) V_z_2)) (tptp.hBOOL (tptp.hAPP _let_1 V_z_2))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Opart__equivp T_b V_R_2) (not (=> (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 B_x) B_x))) (=> (tptp.c_Predicate_Osymp T_b V_R_2) (not (tptp.c_Predicate_Otransp T_b V_R_2))))))) (forall ((V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Opart__equivp T_b V_R_2) (exists ((B_d $$unsorted)) (let ((_let_1 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b _let_1))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member _let_1) B_d) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool _let_1) (tptp.c_HOL_OEx T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 _let_1) (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b T_b tptp.tc_HOL_Obool) V_R_2) (tptp.c_COMBI T_b))))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC _let_1 _let_2 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) (tptp.tc_fun _let_2 _let_1) _let_1) (tptp.c_COMBB _let_1 tptp.tc_HOL_Obool T_b)) tptp.c_fequal)) V_R_2)))))))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (= (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) V_f_2) (tptp.c_Fun_Oid T_b))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) V_f_2) (tptp.c_Fun_Oid T_b))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (=> (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_c T_b V_f_2) V_A_2))) (tptp.c_Fun_Oinj__on T_b T_c (tptp.c_Hilbert__Choice_Oinv__into T_c T_b V_A_2 V_f_2) V_B_2))) (forall ((V_S_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_S_2) V_A_2)) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c V_A_2 V_f_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_S_2)) V_S_2)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) _let_2) _let_1) (tptp.c_Fun_Oinj__on T_b T_c (tptp.c_Hilbert__Choice_Oinv__into T_c T_b _let_2 V_f_2) _let_1))))) (forall ((V_X_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_X_2)) V_X_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_A_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) _let_1)))) (forall ((V_B_H_2 $$unsorted) (V_A_H_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_c T_b V_f_2))) (=> (= (tptp.hAPP _let_1 V_A_2) V_A_H_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_H_2) V_A_H_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c (tptp.c_Hilbert__Choice_Oinv__into T_c T_b V_A_2 V_f_2)) V_B_H_2)) V_B_H_2))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (forall ((B_x $$unsorted)) (= (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_c T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_f_2) B_x)) B_x)))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_c T_b _let_1 V_f_2) V_y_2)) V_y_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Set_Oimage T_c T_b V_f_2))) (=> (= (tptp.hAPP _let_2 _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP _let_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c (tptp.c_Hilbert__Choice_Oinv__into T_c T_b _let_1 V_f_2)) V_A_2)) V_A_2))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_y_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_y_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2))) (= (tptp.hAPP V_f_2 (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_c T_b V_A_2 V_f_2) V_y_2)) V_y_2))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) V_A_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_c T_b V_A_2 V_f_2) V_x_2)) V_A_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2))) (let ((_let_2 (tptp.c_member T_c))) (let ((_let_3 (tptp.c_Hilbert__Choice_Oinv__into T_b T_c V_A_2 V_f_2))) (=> (= (tptp.hAPP _let_3 V_x_2) (tptp.hAPP _let_3 V_y_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x_2) _let_1)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_y_2) _let_1)) (= V_x_2 V_y_2)))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_A_2)) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_b T_c 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_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 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_x_2) V_y_2) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_b T_c V_A_2 V_f_2) V_y_2) V_x_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (=> (= (tptp.hAPP V_f_2 V_x_2) V_y_2) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2) V_y_2) V_x_2))))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2) (tptp.hAPP V_f_2 V_x_2)) V_x_2)))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.c_Fun_Oid T_b))) (= (tptp.c_Hilbert__Choice_Oinv__into T_b T_b (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) _let_1) _let_1))) (forall ((V_x_2 $$unsorted) (V_f_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_b T_c V_A_2 V_f_2) V_x_2) (tptp.c_Hilbert__Choice_OEps T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_b tptp.tc_HOL_Obool tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool) T_b) tptp.c_fconj) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b (tptp.tc_fun T_b tptp.tc_HOL_Obool) tptp.tc_HOL_Obool) (tptp.c_member T_b)) V_A_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_f_2)) V_x_2))))) (forall ((V_y_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_b T_c (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_f_2) V_y_2) (tptp.c_Hilbert__Choice_OEps T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_b T_c tptp.tc_HOL_Obool) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c (tptp.tc_fun T_c tptp.tc_HOL_Obool) T_b) tptp.c_fequal) V_f_2)) V_y_2)))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_d T_c V_g_2) (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2))) V_f_2) V_g_2)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Set_Oimage T_d T_b V_g_2) V_A_2))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1) (=> (tptp.c_Fun_Oinj__on T_d T_b V_g_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) V_x_2) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) _let_1))) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_d T_c V_A_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_d V_f_2) V_g_2)) V_x_2) (tptp.hAPP (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_d T_c (tptp.c_Hilbert__Choice_Oinv__into T_d T_b V_A_2 V_g_2)) (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) V_x_2))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (= (= (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_f_2) _let_1) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_b T_b V_f_2) (tptp.c_Hilbert__Choice_Oinv__into T_c T_b _let_1 V_f_2)) (tptp.c_Fun_Oid T_b))))) (forall ((V_x_2 $$unsorted) (V_Pa_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_y) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) _let_1))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2) B_y)))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2))))) (forall ((V_g_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_I_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone T_d))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_d))) (let ((_let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_d T_c _let_2) V_g_2) _let_1))) (let ((_let_4 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.class_Groups_Ocomm__monoid__mult T_d) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_I_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_I_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) (tptp.hAPP V_A_2 B_x))))) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_I_2)) (forall ((B_xa $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_xa) V_I_2)) (=> (not (= B_x B_xa)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) (tptp.hAPP V_A_2 B_x)) (tptp.hAPP V_A_2 B_xa)) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (= (tptp.hAPP _let_3 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b _let_4) V_I_2) V_A_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_Finite__Set_Ofold__image T_d T_b _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 T_d T_b) _let_3) V_A_2)) _let_1) V_I_2))))))))))) (forall ((V_Pa_2 $$unsorted) (V_y_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_y_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (=> (forall ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP tptp.c_Code__Numeral_Onat__of B_x)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_y_2))))) (forall ((V_y_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_y_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (not (forall ((B_x $$unsorted)) (not (= V_y_2 (tptp.hAPP tptp.c_Code__Numeral_Onat__of B_x))))))) (forall ((V_x_2 $$unsorted) (V_Pa_2 $$unsorted)) (=> (forall ((B_y $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_y) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat B_y))))) (tptp.hBOOL (tptp.hAPP V_Pa_2 V_x_2)))) (forall ((V_x_2 $$unsorted)) (not (forall ((B_y $$unsorted)) (=> (= V_x_2 (tptp.hAPP tptp.c_Code__Numeral_Oof__nat B_y)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) B_y) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))))))) (forall ((V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec T_b T_c V_f1_2 V_f2_2 (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_c)) V_f1_2)) (forall ((V_l $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Quickcheck_Obeyond V_k V_l))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Code__Numeral_Ocode__numeral) V_k) V_l)))) (and (=> _let_2 (= _let_1 V_l)) (=> (not _let_2) (= _let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))))))) (forall ((V_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Code__Numeral_Ocode__numeral))) (= (tptp.c_Quickcheck_Obeyond V_k _let_1) _let_1))) (forall ((V_lazy__sequence_2 $$unsorted) (V_a_2 $$unsorted) (V_f2_2 $$unsorted) (V_f1_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec T_b T_c V_f1_2 V_f2_2 (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_c V_a_2 V_lazy__sequence_2)) (tptp.hAPP (tptp.hAPP (tptp.hAPP V_f2_2 V_a_2) V_lazy__sequence_2) (tptp.c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec T_b T_c V_f1_2 V_f2_2 V_lazy__sequence_2)))) (forall ((V_Pa_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) (tptp.hAPP (tptp.c_Set_OCollect T_b) V_Pa_2)) (tptp.hAPP (tptp.c_Set_OCollect T_c) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_b tptp.tc_HOL_Obool T_c) V_Pa_2) (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Ocurry T_c T_d T_b V_c_2) V_x_2) V_y_2) (tptp.hAPP V_c_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) V_x_2) V_y_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Ocurry T_b T_c tptp.tc_HOL_Obool V_f_2) V_a_2) V_b_2)))) (forall ((V_A_2 $$unsorted) (T_b $$unsorted)) (tptp.c_Fun_Obij__betw T_b T_b (tptp.c_Fun_Oid T_b) V_A_2 V_A_2)) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_1 _let_2) (tptp.c_Fun_Obij__betw T_c T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2) _let_2 _let_1))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_1 _let_2) (= (tptp.c_Hilbert__Choice_Oinv__into T_c T_b _let_2 (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) V_f_2))))) (forall ((V_a_2 $$unsorted) (V_A_H_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_A_H_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_a_2) V_A_2)) (= (tptp.hAPP (tptp.c_Hilbert__Choice_Oinv__into T_c T_b V_A_H_2 (tptp.c_Hilbert__Choice_Oinv__into T_b T_c V_A_2 V_f_2)) V_a_2) (tptp.hAPP V_f_2 V_a_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (tptp.c_Fun_Obij__betw T_c T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c V_A_2 V_f_2) V_B_2 V_A_2))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_d tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Obij__betw T_d T_b V_g_2 _let_2 _let_1) (= (tptp.c_Hilbert__Choice_Oinv__into T_d T_c _let_2 (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_d V_f_2) V_g_2)) (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_d T_c (tptp.c_Hilbert__Choice_Oinv__into T_d T_b _let_2 V_g_2)) (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_d $$unsorted) (V_A_H_H_2 $$unsorted) (V_A_H_2 $$unsorted) (V_f_H_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_H_2 V_A_H_2 V_A_H_H_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_d T_b V_f_2) V_A_2)) V_A_H_2)) (= (tptp.c_Fun_Obij__betw T_d T_b V_f_2 V_A_2 V_A_H_2) (tptp.c_Fun_Obij__betw T_d T_c (tptp.hAPP (tptp.c_Fun_Ocomp T_b T_c T_d V_f_H_2) V_f_2) V_A_2 V_A_H_H_2))))) (forall ((V_g_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_d 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_c tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_2 _let_3) (=> (tptp.c_Fun_Obij__betw T_c T_d V_g_2 _let_3 _let_1) (tptp.c_Fun_Obij__betw T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_g_2) V_f_2) _let_2 _let_1))))))) (forall ((V_A_H_H_2 $$unsorted) (V_f_H_2 $$unsorted) (T_d $$unsorted) (V_A_H_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_A_H_2) (= (tptp.c_Fun_Obij__betw T_c T_d V_f_H_2 V_A_H_2 V_A_H_H_2) (tptp.c_Fun_Obij__betw T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_f_H_2) V_f_2) V_A_2 V_A_H_H_2)))) (forall ((V_C_2 $$unsorted) (V_g_2 $$unsorted) (T_d $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (=> (tptp.c_Fun_Obij__betw T_c T_d V_g_2 V_B_2 V_C_2) (tptp.c_Fun_Obij__betw T_b T_d (tptp.hAPP (tptp.c_Fun_Ocomp T_c T_d T_b V_g_2) V_f_2) V_A_2 V_C_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Ocurry T_b T_c tptp.tc_HOL_Obool V_f_2) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Ocurry T_b T_c tptp.tc_HOL_Obool V_f_2) V_a_2) V_b_2)) (tptp.hBOOL (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_c) V_a_2) V_b_2))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_Ocurry T_c T_d T_b V_f_2) V_a_2) V_b_2) (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_d) V_a_2) V_b_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_3) (tptp.c_Orderings_Otop__class_Otop _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_F_2 $$unsorted) (V_h_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_h_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_c tptp.tc_HOL_Obool))) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_h_2) V_F_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_F_2))))) (forall ((V_f_2 $$unsorted) (V_B_H_2 $$unsorted) (V_A_H_2 $$unsorted) (T_c $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_2) V_A_2) V_B_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_A_H_2) V_B_H_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_A_H_2) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_B_2 V_B_H_2) (tptp.c_Fun_Obij__betw T_b T_c V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2) V_A_2) V_B_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_A_H_2) V_B_H_2))))))))) (forall ((V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_C_2 V_D_2) (=> (= (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__inf__class_Oinf _let_1) V_B_2) V_D_2) (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Fun_Obij__betw T_b T_c 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_C_2) (tptp.hAPP (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1) V_B_2) V_D_2))))))) (forall ((T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (tptp.c_Fun_Obij__betw T_b T_b (tptp.c_Fun_Oid T_b) _let_1 _let_1))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (= (tptp.c_Fun_Obij__betw T_b T_b (tptp.c_Fun_Oid T_b) V_A_2 V_B_2) (= V_A_2 V_B_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2))))) (forall ((V_B_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_B_2)) (= (exists ((B_f $$unsorted)) (tptp.c_Fun_Obij__betw T_b T_c B_f V_A_2 V_B_2)) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2)))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_c tptp.tc_HOL_Obool))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (= (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_c) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (tptp.c_Fun_Obij__betw T_c T_b (tptp.c_Fun_Othe__inv__into T_b T_c V_A_2 V_f_2) V_B_2 V_A_2))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_1)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_B_2) (and (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$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)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_2 _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) _let_2) _let_1))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 _let_1) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool))) _let_1)))) (forall ((V_B_H_2 $$unsorted) (V_B_2 $$unsorted) (V_A_H_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_A_2 V_A_H_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) V_A_2)) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_B_2) V_B_H_2) (tptp.c_Fun_Obij__betw T_b T_c V_f_2 V_B_2 V_B_H_2))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$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)))) (= (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_2 _let_1) (and (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_2) (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) _let_2) _let_1)))))) (forall ((V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 _let_2) (=> (= (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) _let_2) _let_1) (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_2 _let_1)))))) (forall ((V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Ocurry T_b T_c T_d (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) V_f_2)) V_f_2)) (forall ((V_f_2 $$unsorted) (T_d $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.hAPP (tptp.c_Product__Type_Oprod_Oprod__case T_b T_c T_d) (tptp.c_Product__Type_Ocurry T_b T_c T_d V_f_2)) V_f_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_d $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Set_Oimage T_b T_c V_f_2))) (let ((_let_2 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_3) (tptp.c_Orderings_Otop__class_Otop _let_2)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_3) V_A_2) V_B_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_d _let_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_3 _let_2 T_d) _let_1) V_B_2)))))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 (tptp.c_Orderings_Otop__class_Otop _let_2) (tptp.c_Orderings_Otop__class_Otop _let_1)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_2) (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) 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.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)))))))) (forall ((V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_b tptp.tc_HOL_Obool)))) (=> (tptp.c_Fun_Obij__betw T_b T_c V_f_2 _let_1 (tptp.c_Orderings_Otop__class_Otop (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (= (tptp.hAPP (tptp.c_Set_Ovimage T_b T_c V_f_2) V_A_2) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b (tptp.c_Hilbert__Choice_Oinv__into T_b T_c _let_1 V_f_2)) V_A_2))))) (forall ((V_M_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_M_2)) (exists ((B_h $$unsorted)) (tptp.c_Fun_Obij__betw tptp.tc_Nat_Onat T_b B_h (tptp.c_SetInterval_Oord__class_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_M_2)) V_M_2)))) (forall ((V_g_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Fun_Oinj__on T_b T_c V_f_2 V_A_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_b T_c V_f_2) V_A_2)) V_B_2)) (=> (tptp.c_Fun_Oinj__on T_c T_b V_g_2 V_B_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.c_Set_Oimage T_c T_b V_g_2) V_B_2)) V_A_2)) (exists ((B_h $$unsorted)) (tptp.c_Fun_Obij__betw T_b T_c B_h V_A_2 V_B_2))))))) (forall ((V_M_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_M_2)) (exists ((B_h $$unsorted)) (tptp.c_Fun_Obij__betw tptp.tc_Nat_Onat T_b B_h (tptp.hAPP (tptp.c_SetInterval_Oord__class_OatLeastLessThan tptp.tc_Nat_Onat (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_M_2)) V_M_2)))) (forall ((V_M_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_M_2)) (exists ((B_h $$unsorted)) (tptp.c_Fun_Obij__betw T_b tptp.tc_Nat_Onat B_h V_M_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_Finite__Set_Ocard T_b) V_M_2)))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_c $$unsorted) (V_S_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) V_S_2)) (tptp.c_Relation_Orefl__on T_c (tptp.hAPP V_A_2 B_x) (tptp.hAPP V_r_2 B_x)))) (tptp.c_Relation_Orefl__on T_c (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_S_2) V_A_2) (tptp.hAPP (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OSUPR T_b (tptp.tc_fun (tptp.tc_prod T_c T_c) tptp.tc_HOL_Obool)) V_S_2) V_r_2)))) (forall ((V_Pa_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (= (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.c_Quotient_ORespects T_b V_R_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x1)))))) (forall ((V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Quotient_ORespects T_b V_R_2) V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_x_2) V_x_2)))) (forall ((V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) (tptp.c_Quotient_ORespects T_b V_R_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_R_2 V_x_2) V_x_2)))) (forall ((V_Pa_2 $$unsorted) (V_Q_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_Q_2 B_x)) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))) (=> (exists ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Q_2 B_x1))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) B_x) (tptp.c_Quotient_ORespects T_b V_R_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))))) (forall ((V_Pa_2 $$unsorted) (T_c $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequivp T_b V_R_2) (= (tptp.c_Quotient_OBabs T_b T_c (tptp.c_Quotient_ORespects T_b V_R_2) V_Pa_2) V_Pa_2))) (forall ((V_m_2 $$unsorted) (T_c $$unsorted) (V_p_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_b) V_x_2) V_p_2)) (= (tptp.hAPP (tptp.c_Quotient_OBabs T_b T_c V_p_2 V_m_2) V_x_2) (tptp.hAPP V_m_2 V_x_2)))) (forall ((V_r_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_i $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) (tptp.hAPP V_f_2 (tptp.hAPP tptp.c_Nat_OSuc B_i))) (tptp.hAPP V_f_2 B_i))) (tptp.c_Transitive__Closure_Ortrancl T_b V_r_2)))) (=> (tptp.c_Wellfounded_Owf T_b (tptp.c_Transitive__Closure_Otrancl T_b V_r_2)) (exists ((B_i $$unsorted)) (forall ((B_k $$unsorted)) (= (tptp.hAPP V_f_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) B_i) B_k)) (tptp.hAPP V_f_2 B_i))))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.hAPP (tptp.c_Complete__Lattice_Ocomplete__lattice__class_OINFI T_b _let_1) V_A_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_B_2) V_A_2)) (=> (forall ((B_x $$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 (tptp.tc_fun T_c tptp.tc_HOL_Obool)) (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 _let_1) (tptp.hAPP _let_2 V_f_2)) (tptp.hAPP _let_2 V_g_2)))))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (T_c $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_fun T_c tptp.tc_HOL_Obool))) (=> (forall ((B_x $$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 (tptp.tc_fun T_c tptp.tc_HOL_Obool)) V_C_2) (tptp.hAPP V_B_2 B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq _let_1) V_C_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_Pa_2 $$unsorted) (V_r_2 $$unsorted) (V_by_2 $$unsorted) (V_bx_2 $$unsorted) (V_ay_2 $$unsorted) (V_ax_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_c))) (=> (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 _let_2 V_ax_2) V_ay_2)) (tptp.hAPP (tptp.hAPP _let_2 V_bx_2) V_by_2))) (tptp.c_Transitive__Closure_Ortrancl _let_1 V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_ax_2) V_ay_2)) (=> (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_c))) (=> (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 _let_2 V_ax_2) V_ay_2)) (tptp.hAPP (tptp.hAPP _let_2 B_a) B_b))) (tptp.c_Transitive__Closure_Ortrancl _let_1 V_r_2))) (forall ((B_aa $$unsorted) (B_ba $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_2 (tptp.tc_prod T_b T_c))) (=> (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 B_a) B_b)) (tptp.hAPP (tptp.hAPP _let_1 B_aa) B_ba))) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_a) B_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_aa) B_ba))))))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_bx_2) V_by_2)))))))) (forall ((V_r_2 $$unsorted) (V_zb_2 $$unsorted) (V_za_2 $$unsorted) (V_xb_2 $$unsorted) (V_xa_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_2 (tptp.hAPP (tptp.hAPP _let_1 V_za_2) V_zb_2))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_1 V_xa_2) V_xb_2))) (let ((_let_4 (tptp.tc_prod T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_4 _let_4)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_4 _let_4) _let_3) _let_2)) (tptp.c_Transitive__Closure_Ortrancl _let_4 V_r_2))) (=> (not (= _let_3 _let_2)) (not (forall ((B_a $$unsorted) (B_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 B_a) B_b))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_1 _let_1))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_1 _let_1)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 V_xa_2) V_xb_2)) _let_3)) V_r_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 _let_3) (tptp.hAPP (tptp.hAPP _let_2 V_za_2) V_zb_2))) (tptp.c_Transitive__Closure_Ortrancl _let_1 V_r_2))))))))))))))))))) (forall ((V_Pa_2 $$unsorted) (V_r_2 $$unsorted) (V_by_2 $$unsorted) (V_bx_2 $$unsorted) (V_ay_2 $$unsorted) (V_ax_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_c))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod _let_1 _let_1)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair _let_1 _let_1) (tptp.hAPP (tptp.hAPP _let_2 V_ax_2) V_ay_2)) (tptp.hAPP (tptp.hAPP _let_2 V_bx_2) V_by_2))) (tptp.c_Transitive__Closure_Ortrancl _let_1 V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_bx_2) V_by_2)) (=> (forall ((B_a $$unsorted) (B_b $$unsorted) (B_aa $$unsorted) (B_ba $$unsorted)) (let ((_let_1 (tptp.tc_prod T_b T_c))) (let ((_let_2 (tptp.c_Product__Type_OPair T_b T_c))) (let ((_let_3 (tptp.hAPP (tptp.hAPP _let_2 B_aa) B_ba))) (let ((_let_4 (tptp.c_Product__Type_OPair _let_1 _let_1))) (let ((_let_5 (tptp.c_member (tptp.tc_prod _let_1 _let_1)))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 (tptp.hAPP (tptp.hAPP _let_2 B_a) B_b)) _let_3)) V_r_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_5 (tptp.hAPP (tptp.hAPP _let_4 _let_3) (tptp.hAPP (tptp.hAPP _let_2 V_bx_2) V_by_2))) (tptp.c_Transitive__Closure_Ortrancl _let_1 V_r_2))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_aa) B_ba)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 B_a) B_b))))))))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Pa_2 V_ax_2) V_ay_2)))))))) (forall ((T_d $$unsorted) (V_f_2 $$unsorted) (V_r2_2 $$unsorted) (T_c $$unsorted) (V_r1_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_y1 $$unsorted) (B_z1 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_y1) B_z1)) V_r1_2)) (forall ((B_y2 $$unsorted) (B_z2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_c T_c)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_c T_c) B_y2) B_z2)) V_r2_2)) (= (tptp.hAPP (tptp.hAPP V_f_2 B_y1) B_y2) (tptp.hAPP (tptp.hAPP V_f_2 B_z1) B_z2)))))) (tptp.c_Equiv__Relations_Ocongruent2 T_b T_c T_d V_r1_2 V_r2_2 V_f_2))) (forall ((T_c $$unsorted) (V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted)) (=> (forall ((B_y $$unsorted) (B_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_prod T_b T_b)) (tptp.hAPP (tptp.hAPP (tptp.c_Product__Type_OPair T_b T_b) B_y) B_z)) V_r_2)) (= (tptp.hAPP V_f_2 B_y) (tptp.hAPP V_f_2 B_z)))) (tptp.c_Equiv__Relations_Ocongruent T_b T_c V_r_2 V_f_2))) (forall ((V_xqq $$unsorted) (V_xq $$unsorted) (T_a $$unsorted)) (= (tptp.c_Lazy__Sequence_Oflat T_a (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert (tptp.tc_Lazy__Sequence_Olazy__sequence T_a) V_xq V_xqq)) (tptp.c_Lazy__Sequence_Oappend T_a V_xq (tptp.c_Lazy__Sequence_Oflat T_a V_xqq)))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.c_List_Olist__all tptp.tc_Nat_Onat V_Pa_2 (tptp.c_List_Oupt V_i_2 V_j_2)) (tptp.c_List_Oall__interval__nat V_Pa_2 V_i_2 V_j_2))) (forall ((T_a $$unsorted)) (= (tptp.c_Lazy__Sequence_Oflat T_a (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty (tptp.tc_Lazy__Sequence_Olazy__sequence T_a))) (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_a))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.c_List_Oall__interval__nat V_Pa_2 V_i_2 V_j_2) (forall ((B_x $$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 V_i_2) V_j_2))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x)))))) (forall ((V_j_2 $$unsorted) (V_i_2 $$unsorted) (V_Pa_2 $$unsorted)) (= (tptp.c_List_Olist__ex tptp.tc_Nat_Onat V_Pa_2 (tptp.c_List_Oupt V_i_2 V_j_2)) (not (tptp.c_List_Oall__interval__nat (tptp.hAPP (tptp.c_Fun_Ocomp tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat tptp.c_fNot) V_Pa_2) V_i_2 V_j_2)))) (forall ((V_f_2 $$unsorted) (V_xq_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.c_Lazy__Sequence_Obind T_c T_b V_xq_2) V_f_2) (tptp.c_Lazy__Sequence_Oflat T_b (tptp.c_Lazy__Sequence_Omap T_c (tptp.tc_Lazy__Sequence_Olazy__sequence T_b) V_f_2 V_xq_2)))) (forall ((V_h_2 $$unsorted) (V_g_2 $$unsorted) (V_A_2 $$unsorted) (V_F_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (=> (tptp.c_Finite__Set_Ofolding__image T_b T_c V_f_2 V_z_2 V_F_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_c) V_A_2)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member T_c) B_x) V_A_2)) (= (tptp.hAPP V_g_2 B_x) (tptp.hAPP V_h_2 B_x)))) (= (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_xq_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_Lazy__Sequence_Omap T_c T_b V_f_2 (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_c V_x_2 V_xq_2)) (tptp.c_Lazy__Sequence_Olazy__sequence_OInsert T_b (tptp.hAPP V_f_2 V_x_2) (tptp.c_Lazy__Sequence_Omap T_c T_b V_f_2 V_xq_2)))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_Lazy__Sequence_Omap T_c T_b V_f_2 (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_c)) (tptp.c_Lazy__Sequence_Olazy__sequence_OEmpty T_b))) (forall ((V_i_2 $$unsorted) (V_xq_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.c_New__DSequence_Opos__map T_c T_b V_f_2 V_xq_2 V_i_2) (tptp.c_Lazy__Sequence_Omap T_c T_b V_f_2 (tptp.hAPP V_xq_2 V_i_2)))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_i $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower tptp.tc_Nat_Onat) V_i))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (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) (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat)) V_i)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)))))) (forall ((V_x $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_x) V_x))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_k $$unsorted)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat))) V_k))) (forall ((V_n $$unsorted) (V_k $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k) 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_k) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_m)) (tptp.hAPP _let_2 V_n))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd 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_na_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))) (forall ((V_f_2 $$unsorted) (V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2))) (=> (and (tptp.class_Groups_Ocomm__monoid__mult T_c) (tptp.class_Rings_Odvd T_c)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_B_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_A_2) V_B_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_c) (tptp.hAPP _let_1 V_A_2)) (tptp.hAPP _let_1 V_B_2)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Int_Onumber__class_Onumber__of T_b))) (let ((_let_2 (tptp.hAPP _let_1 V_x_2))) (let ((_let_3 (tptp.hAPP _let_1 V_y_2))) (=> (and (tptp.class_Divides_Osemiring__div T_b) (tptp.class_Int_Onumber T_b)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) _let_2) _let_3)) (= (tptp.c_Divides_Odiv__class_Omod T_b _let_3 _let_2) (tptp.c_Groups_Ozero__class_Ozero T_b)))))))) (forall ((V_d_2 $$unsorted) (V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_b))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv T_b))) (let ((_let_3 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_4 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (not (= V_a_2 _let_4)) (=> (not (= V_c_2 _let_4)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_a_2) V_b_2)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_c_2) V_d_2)) (= (= (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_a_2) (tptp.hAPP (tptp.hAPP _let_2 V_d_2) V_c_2)) (= (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2) (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_d_2))))))))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (not (= V_a_2 (tptp.c_Groups_Ozero__class_Ozero T_b))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_a_2) V_b_2)) (= (= (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_b) V_b_2) V_a_2) V_c_2) (= V_b_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2) V_a_2))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (=> (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 _let_1 V_m))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_n) V_m)) (tptp.hBOOL (tptp.hAPP _let_1 V_n))))))) (forall ((V_a $$unsorted) (V_n $$unsorted) (V_m $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) (tptp.hAPP _let_1 V_m)) (tptp.hAPP _let_1 V_n))))))) (forall ((V_m $$unsorted) (V_n $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (=> (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_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_m))))))))) (forall ((V_m $$unsorted) (V_b $$unsorted) (V_n $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_a))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_n)) V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) V_m) V_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_m)) V_b)))))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_a_2 V_b_2)) (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_c_2 V_d_2))) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c_2) V_a_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_d_2))))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_i_2) (tptp.c_SetInterval_Oord_OatLeast tptp.tc_Nat_Onat _let_1 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_i_2))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_i_2) (tptp.c_SetInterval_Oord_OatMost tptp.tc_Nat_Onat _let_1 V_k_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_k_2))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_i_2) (tptp.c_SetInterval_Oord_OlessThan tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1))) V_k_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_k_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_i_2)))))))))) (forall ((V_k_2 $$unsorted) (V_i_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member tptp.tc_Nat_Onat) V_i_2) (tptp.c_SetInterval_Oord_OgreaterThan tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1))) V_k_2))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_i_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_i_2) V_k_2)))))))))) (forall ((V_n $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_y) V_x)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_n) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_n)))))))) (forall ((V_k_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_m_2))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_b) V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_k_2)))))) (forall ((V_k_2 $$unsorted) (V_m_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (=> (tptp.class_Rings_Olinordered__idom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Oabs__class_Oabs T_b) V_m_2)) V_k_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_k_2)))))) (forall ((V_k $$unsorted) (V_l $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oabs__class_Oabs T_a))) (=> (tptp.class_Rings_Olinordered__idom T_a) (=> (= (tptp.hAPP _let_1 V_l) (tptp.hAPP _let_1 V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_l) V_k)))))) (forall ((V_m_2 $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_m_2) _let_1)) (= V_m_2 _let_1)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) (tptp.c_Groups_Oone__class_Oone T_a)) V_a)))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Ozero__class_Ozero T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) _let_1) V_a)) (= V_a _let_1))))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_a)))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_a))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (tptp.hBOOL (tptp.hAPP _let_2 V_c)))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (= (= V_x_2 V_y_2) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))) (= _let_2 (or (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))) (= V_x_2 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))) (= (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))) (and _let_2 (not (= V_x_2 V_y_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))) (=> (not (= V_a V_b)) (=> _let_2 (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (= V_x V_y) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_x) V_y)))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (=> _let_2 (or (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (= V_x V_y)))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))) (=> _let_2 (=> (not (= V_a V_b)) (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a))))))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (= 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_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_a))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (= V_b V_c) (tptp.hBOOL (tptp.hAPP _let_1 V_c)))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_n) V_m)) (= V_m V_n))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)) (= V_x V_y))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (=> (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 $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_c))) (=> (= V_a V_b) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_b)))) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_a))))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_z))) (let ((_let_3 (tptp.hAPP _let_1 V_x))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_y)) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_y)))) (and (tptp.hBOOL (tptp.hAPP _let_3 V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_x)))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (and (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)))) (not (= V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (=> (and _let_2 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) _let_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (and (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)))) (not (= V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (and (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)))) (not (= V_y V_x))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_a))) (=> (and (tptp.hBOOL (tptp.hAPP _let_2 V_b)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)))) (=> (= V_b V_c) (and (tptp.hBOOL (tptp.hAPP _let_2 V_c)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_c) V_a))))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_x))) (let ((_let_3 (tptp.hAPP _let_1 V_y))) (=> (and (tptp.hBOOL (tptp.hAPP _let_2 V_y)) (not (tptp.hBOOL (tptp.hAPP _let_3 V_x)))) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_z)) (and (tptp.hBOOL (tptp.hAPP _let_2 V_z)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_z) V_x)))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_b)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_a)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_z))) (let ((_let_3 (tptp.hAPP _let_1 V_x))) (let ((_let_4 (tptp.hAPP _let_1 V_y))) (=> (and (tptp.hBOOL (tptp.hAPP _let_3 V_y)) (not (tptp.hBOOL (tptp.hAPP _let_4 V_x)))) (=> (and (tptp.hBOOL (tptp.hAPP _let_4 V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_y)))) (and (tptp.hBOOL (tptp.hAPP _let_3 V_z)) (not (tptp.hBOOL (tptp.hAPP _let_2 V_x))))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x) V_y)))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y) V_x)))) (=> (and _let_2 (not _let_3)) (not (and _let_3 (not _let_2)))))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv T_b))) (let ((_let_3 (tptp.hAPP _let_1 V_a_2))) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_b_2)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_c_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_a_2)) (tptp.hAPP (tptp.hAPP _let_2 V_c_2) V_a_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_c_2)))))))))) (forall ((V_n $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Power_Opower__class_Opower T_a))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_n)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_n)))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_k))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_m V_n))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 V_m))))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_k))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_a V_m V_n)))))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_c) V_b)) (= (tptp.c_Divides_Odiv__class_Omod T_a (tptp.c_Divides_Odiv__class_Omod T_a V_a V_b) V_c) (tptp.c_Divides_Odiv__class_Omod T_a V_a V_c))))) (forall ((V_m_2 $$unsorted) (V_na_2 $$unsorted) (V_k_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_k_2))) (=> (tptp.class_Divides_Osemiring__div T_b) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_na_2)) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.c_Divides_Odiv__class_Omod T_b V_m_2 V_na_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_m_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_x_2))) (=> (tptp.class_Rings_Ocomm__ring__1 T_b) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_y_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (=> (tptp.class_Rings_Ocomm__ring__1 T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.c_Groups_Ouminus__class_Ouminus T_b) V_x_2)) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k))) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_n)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) V_m) V_n))))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_x))) (=> (tptp.class_Rings_Ocomm__ring__1 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_Groups_Ominus__class_Ominus T_a) V_y) V_z)))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_b)) (= (tptp.c_Divides_Odiv__class_Omod T_a V_b V_a) (tptp.c_Groups_Ozero__class_Ozero T_a))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_a_2) V_b_2)) (= (tptp.c_Divides_Odiv__class_Omod T_b V_b_2 V_a_2) (tptp.c_Groups_Ozero__class_Ozero T_b))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_x))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_y) V_x)) (= (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_y)) (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_y)))))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Ouminus__class_Ouminus T_a))) (=> (tptp.class_Divides_Oring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_y) V_x)) (= (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_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.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m) V_n)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_n) V_m))))))) (forall ((V_m_2 $$unsorted)) (let ((_let_1 (tptp.hAPP tptp.c_Nat_OSuc (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_m_2) _let_1)) (= V_m_2 _let_1)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.hAPP _let_1 V_c_2))) (let ((_let_3 (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_d_2)))) (let ((_let_4 (tptp.hAPP _let_1 V_a_2))) (let ((_let_5 (tptp.hBOOL (tptp.hAPP _let_2 V_a_2)))) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_a_2 V_b_2)) (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_c_2 V_d_2))) (and (or (not (tptp.hBOOL (tptp.hAPP _let_4 V_b_2))) (and _let_5 _let_3 (or (and _let_5 (not (tptp.hBOOL (tptp.hAPP _let_4 V_c_2)))) (and _let_3 (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_d_2) V_b_2))))))) (tptp.hBOOL (tptp.hAPP _let_2 V_d_2)))))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (V_z $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_2 (tptp.c_Groups_Oplus__class_Oplus T_a))) (let ((_let_3 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_z))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_x)) (=> (tptp.hBOOL (tptp.hAPP _let_3 V_y)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) V_z) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_z)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)))))))))) (forall ((V_na_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k_2))) (= (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus tptp.tc_Nat_Onat) V_na_2) V_k_2))) (tptp.hBOOL (tptp.hAPP _let_1 V_na_2))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_c)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_a) V_b) V_c)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (= (tptp.c_Orderings_Obot__class_Obot _let_2) (tptp.c_SetInterval_Oord_OatLeastLessThan tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1))) V_a_2 V_b_2)) (not (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2))))))))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (= (tptp.c_Orderings_Obot__class_Obot _let_2) (tptp.c_SetInterval_Oord_OgreaterThanAtMost tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1))) V_k_2 V_l_2)) (not (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_k_2))))))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (= (tptp.c_SetInterval_Oord_OatLeastLessThan tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1))) V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (not (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2))))))))))) (forall ((V_l_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (= (= (tptp.c_SetInterval_Oord_OgreaterThanAtMost tptp.tc_Nat_Onat _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1))) V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_2)) (not (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_k_2) V_l_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_l_2) V_k_2))))))))))) (forall ((V_k_2 $$unsorted) (V_l_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_l_2) V_k_2)) (= (tptp.c_SetInterval_Oord_OgreaterThanLessThan tptp.tc_Nat_Onat (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_1 _let_1) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_2))) V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_b_2) V_a_2)) (= (tptp.c_SetInterval_Oord_OatLeastLessThan tptp.tc_Nat_Onat _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_1 _let_1) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_2))) V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_k_2 $$unsorted) (V_l_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_2 V_l_2) V_k_2)) (= (tptp.c_SetInterval_Oord_OgreaterThanAtMost tptp.tc_Nat_Onat _let_2 (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_1 _let_1) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_1 _let_1) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_2))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_1 _let_1 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_2))) V_k_2 V_l_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (= (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)) (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_a_2 V_b_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (= (= (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b_2) V_a_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))) (= (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat _let_1 V_a_2 V_b_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))))) (forall ((V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (= (= (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_a_2 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_c_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))) (and (= V_a_2 V_b_2) (= V_b_2 V_c_2)))) (forall ((V_a_2 $$unsorted)) (= (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_a_2 V_a_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted)) (=> (= V_a_2 V_b_2) (= (tptp.c_SetInterval_Oord_OatLeastAtMost tptp.tc_Nat_Onat (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_a_2 V_b_2) (tptp.hAPP (tptp.hAPP (tptp.c_Set_Oinsert tptp.tc_Nat_Onat) V_a_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool)))))) (forall ((V_t_2 $$unsorted) (V_D_2 $$unsorted) (V_d_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Rings_Ocomm__ring T_b) (tptp.class_Rings_Odvd T_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2) V_D_2)) (forall ((B_x $$unsorted) (B_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) B_k) V_D_2))) V_t_2)))))))))) (forall ((V_t_2 $$unsorted) (V_D_2 $$unsorted) (V_d_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Rings_Ocomm__ring T_b) (tptp.class_Rings_Odvd T_b)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2) V_D_2)) (forall ((B_x $$unsorted) (B_k $$unsorted)) (let ((_let_1 (tptp.c_Groups_Oplus__class_Oplus T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_d_2))) (= (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 B_x) V_t_2))) (tptp.hBOOL (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus T_b) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) B_k) V_D_2))) V_t_2)))))))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd 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_na_2))) (or (= V_k_2 (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_m_2) V_na_2))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_b)) (= (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Divides_Odiv__class_Odiv T_a) V_b) V_a)) V_b)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_c) V_b)) (= (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_c)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP _let_1 V_b)) V_c))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd 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_b) V_a)) V_a) 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_Divides_Odiv__class_Odiv T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) V_b)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_b) V_a)) V_c) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)) V_a))))))) (forall ((V_w $$unsorted) (V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Groups_Otimes__class_Otimes T_a))) (let ((_let_2 (tptp.c_Divides_Odiv__class_Odiv T_a))) (let ((_let_3 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Divides_Osemiring__div T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_y) V_x)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_3 V_z) V_w)) (= (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP _let_2 V_x) V_y)) (tptp.hAPP (tptp.hAPP _let_2 V_w) V_z)) (tptp.hAPP (tptp.hAPP _let_2 (tptp.hAPP (tptp.hAPP _let_1 V_x) V_w)) (tptp.hAPP (tptp.hAPP _let_1 V_y) V_z)))))))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) (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 $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_a))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_b)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c))))))) (forall ((V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP _let_1 V_c)) (tptp.hBOOL (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_c))))))) (forall ((V_d $$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_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 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_k $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Odvd T_a) (=> (= V_a (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_b) V_k)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_b) V_a))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a) V_c)))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_a))) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_a) V_a) V_b)) V_c)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_b) V_c)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_2 (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_c_2))) (=> (tptp.class_Rings_Oidom T_b) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 (tptp.hAPP _let_2 V_a_2)) (tptp.hAPP _let_2 V_b_2))) (or (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_b_2 $$unsorted) (V_c_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd T_b))) (let ((_let_2 (tptp.c_Groups_Otimes__class_Otimes T_b))) (=> (tptp.class_Rings_Oidom T_b) (= (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))) (or (= V_c_2 (tptp.c_Groups_Ozero__class_Ozero T_b)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_a_2) V_b_2)))))))) (forall ((V_l_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted)) (=> (and (tptp.class_Rings_Odvd T_b) (tptp.class_Rings_Osemiring__0 T_b)) (= (exists ((B_x $$unsorted)) (tptp.hBOOL (tptp.hAPP V_Pa_2 (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes T_b) V_l_2) B_x)))) (exists ((B_x $$unsorted)) (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_b) V_l_2) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Oplus__class_Oplus T_b) B_x) (tptp.c_Groups_Ozero__class_Ozero T_b)))) (tptp.hBOOL (tptp.hAPP V_Pa_2 B_x))))))) (forall ((V_f_2 $$unsorted) (V_i_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_c) (=> (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_i_2) V_A_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_c) (tptp.hAPP V_f_2 V_i_2)) (tptp.hAPP (tptp.hAPP (tptp.c_Big__Operators_Ocomm__monoid__mult__class_Osetprod T_b T_c) V_f_2) V_A_2))))))) (forall ((V_x $$unsorted) (V_n $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Rings_Ocomm__semiring__1 T_a) (=> (or (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_n)) (= V_x (tptp.c_Groups_Oone__class_Oone T_a))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd T_a) V_x) (tptp.hAPP (tptp.hAPP (tptp.c_Power_Opower__class_Opower T_a) V_x) V_n)))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_m_2) V_na_2)) V_m_2)) (= V_na_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_na_2 $$unsorted) (V_m_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat) (tptp.c_Groups_Ozero__class_Ozero tptp.tc_Nat_Onat)) V_m_2)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Otimes__class_Otimes tptp.tc_Nat_Onat) V_na_2) V_m_2)) V_m_2)) (= V_na_2 (tptp.c_Groups_Oone__class_Oone tptp.tc_Nat_Onat))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun tptp.tc_Nat_Onat tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_fun tptp.tc_HOL_Obool tptp.tc_HOL_Obool))) (let ((_let_4 (tptp.tc_fun tptp.tc_Nat_Onat _let_3))) (=> (tptp.class_Orderings_Oorder T_b) (=> (tptp.c_Orderings_Oorder_Ostrict__mono tptp.tc_Nat_Onat T_b (tptp.hAPP (tptp.hAPP (tptp.c_COMBS tptp.tc_Nat_Onat _let_2 _let_2) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_4 (tptp.tc_fun _let_2 _let_2) tptp.tc_Nat_Onat) (tptp.c_COMBS tptp.tc_Nat_Onat tptp.tc_HOL_Obool tptp.tc_HOL_Obool)) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_4 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool _let_3 tptp.tc_Nat_Onat) tptp.c_fconj)) _let_1))) (tptp.hAPP (tptp.hAPP (tptp.c_COMBB _let_2 _let_2 tptp.tc_Nat_Onat) (tptp.hAPP (tptp.c_COMBB tptp.tc_HOL_Obool tptp.tc_HOL_Obool tptp.tc_Nat_Onat) tptp.c_fNot)) (tptp.hAPP (tptp.c_COMBC tptp.tc_Nat_Onat tptp.tc_Nat_Onat tptp.tc_HOL_Obool) _let_1))) V_f_2) (=> (and (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_x_2) V_y_2)) (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP _let_1 V_y_2) V_x_2)))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless T_b) (tptp.hAPP V_f_2 V_x_2)) (tptp.hAPP V_f_2 V_y_2))))))))))) (forall ((V_k_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite T_b) V_A_2)) (=> (tptp.c_Equiv__Relations_Oequiv T_b V_A_2 V_r_2) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_member (tptp.tc_fun T_b tptp.tc_HOL_Obool)) B_x) (tptp.hAPP (tptp.hAPP (tptp.c_Equiv__Relations_Oquotient T_b) V_A_2) V_r_2))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) B_x))))) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Nat_Onat) V_k_2) (tptp.hAPP (tptp.c_Finite__Set_Ocard T_b) V_A_2))))))) (forall ((V_n $$unsorted) (V_z $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Rings_Odvd__class_Odvd tptp.tc_Int_Oint) V_z) V_n)) (=> (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_n)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Int_Oint) V_z) V_n))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (let ((_let_1 (tptp.c_Rings_Odvd__class_Odvd 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 V_m)) (=> (tptp.hBOOL (tptp.hAPP _let_2 V_n)) (=> (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)))))))) (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_Enum_Oenum T_1)) (tptp.class_Enum_Oenum (tptp.tc_fun T_2 T_1)))) (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_Lazy__Sequence_Osmall__lazy 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_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_Rings_Ocomm__ring 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_Rings_Odvd 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_Rings_Odvd 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_Enum_Oenum tptp.tc_HOL_Obool) (tptp.class_Nat_Osize tptp.tc_HOL_Obool) (forall ((T_1 $$unsorted)) (=> (tptp.class_Lazy__Sequence_Osmall__lazy T_1) (tptp.class_Lazy__Sequence_Osmall__lazy (tptp.tc_List_Olist T_1)))) (forall ((T_1 $$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)) (=> (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_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 $$unsorted)) (tptp.class_Nat_Osize (tptp.tc_Nitpick_Opair__box T_2 T_1))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Lazy__Sequence_Osmall__lazy T_2) (tptp.class_Lazy__Sequence_Osmall__lazy T_1)) (tptp.class_Lazy__Sequence_Osmall__lazy (tptp.tc_prod T_2 T_1)))) (forall ((T_1 $$unsorted) (T_2 $$unsorted)) (=> (and (tptp.class_Finite__Set_Ofinite T_2) (tptp.class_Finite__Set_Ofinite T_1)) (tptp.class_Finite__Set_Ofinite (tptp.tc_prod T_2 T_1)))) (forall ((T_1 $$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_Lazy__Sequence_Osmall__lazy tptp.tc_Product__Type_Ounit) (tptp.class_Finite__Set_Ofinite 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_Nat_Osize tptp.tc_Code__Evaluation_Oterm) (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_Rings_Odvd tptp.tc_Code__Numeral_Ocode__numeral) (tptp.class_Nat_Osize tptp.tc_Code__Numeral_Ocode__numeral) (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_Pa_2 $$unsorted) (T_d $$unsorted) (T_b $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBB T_c T_b T_d) V_Pa_2) V_Q_2) V_R_2) (tptp.hAPP V_Pa_2 (tptp.hAPP V_Q_2 V_R_2)))) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBC T_c T_d T_b) V_Pa_2) V_Q_2) V_R_2) (tptp.hAPP (tptp.hAPP V_Pa_2 V_R_2) V_Q_2))) (forall ((V_R_2 $$unsorted) (V_Q_2 $$unsorted) (V_Pa_2 $$unsorted) (T_b $$unsorted) (T_d $$unsorted) (T_c $$unsorted)) (= (tptp.hAPP (tptp.hAPP (tptp.hAPP (tptp.c_COMBS T_c T_d T_b) V_Pa_2) V_Q_2) V_R_2) (tptp.hAPP (tptp.hAPP V_Pa_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_Pa_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP tptp.c_fNot V_Pa_2))) (not (tptp.hBOOL V_Pa_2)))) (forall ((V_Pa_2 $$unsorted)) (or (not (not (tptp.hBOOL V_Pa_2))) (tptp.hBOOL (tptp.hAPP tptp.c_fNot V_Pa_2)))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted)) (or (not (tptp.hBOOL V_Pa_2)) (not (tptp.hBOOL V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_Pa_2) V_Q_2)))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_Pa_2) V_Q_2))) (tptp.hBOOL V_Pa_2))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fconj V_Pa_2) V_Q_2))) (tptp.hBOOL V_Q_2))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted)) (or (not (tptp.hBOOL V_Pa_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fdisj V_Pa_2) V_Q_2)))) (forall ((V_Pa_2 $$unsorted) (V_Q_2 $$unsorted)) (or (not (tptp.hBOOL V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fdisj V_Pa_2) V_Q_2)))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fdisj V_Pa_2) V_Q_2))) (tptp.hBOOL V_Pa_2) (tptp.hBOOL V_Q_2))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted)) (or (tptp.hBOOL V_Pa_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fimplies V_Pa_2) V_Q_2)))) (forall ((V_Pa_2 $$unsorted) (V_Q_2 $$unsorted)) (or (not (tptp.hBOOL V_Q_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fimplies V_Pa_2) V_Q_2)))) (forall ((V_Q_2 $$unsorted) (V_Pa_2 $$unsorted)) (or (not (tptp.hBOOL (tptp.hAPP (tptp.hAPP tptp.c_fimplies V_Pa_2) V_Q_2))) (not (tptp.hBOOL V_Pa_2)) (tptp.hBOOL V_Q_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Finite__Set_Ofinite tptp.tc_Com_Opname) tptp.v_U)) _let_11 (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat) _let_8) _let_10)) (= (tptp.hAPP _let_9 tptp.v_G) (tptp.hAPP (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat) _let_10) _let_8)) _let_7 (not (tptp.hBOOL (tptp.hAPP _let_6 tptp.v_G))) _let_5 true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 80.20/80.52  )
% 80.20/80.52  % SZS output end Proof for SWW368+1
% 80.20/80.52  % cvc5---1.0.5 exiting
% 80.20/80.53  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------