%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SCT105+1 : TPTP v8.1.2. Released v5.2.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n004.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Thu Aug 31 14:12:03 EDT 2023

% Result   : Theorem 0.54s 0.86s
% Output   : Proof 0.67s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.17  % Problem    : SCT105+1 : TPTP v8.1.2. Released v5.2.0.
% 0.11/0.19  % Command    : do_cvc5 %s %d
% 0.19/0.40  % Computer : n004.cluster.edu
% 0.19/0.40  % Model    : x86_64 x86_64
% 0.19/0.40  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.40  % Memory   : 8042.1875MB
% 0.19/0.40  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.19/0.40  % CPULimit   : 300
% 0.19/0.40  % WCLimit    : 300
% 0.19/0.40  % DateTime   : Thu Aug 24 15:39:08 EDT 2023
% 0.19/0.40  % CPUTime    : 
% 0.25/0.58  %----Proving TF0_NAR, FOF, or CNF
% 0.54/0.86  ------- convert to smt2 : /export/starexec/sandbox2/tmp/tmp.HXVJ0NfoxL/cvc5---1.0.5_13775.p...
% 0.54/0.86  ------- get file name : TPTP file name is SCT105+1
% 0.54/0.86  ------- cvc5-fof : /export/starexec/sandbox2/solver/bin/cvc5---1.0.5_13775.smt2...
% 0.54/0.86  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 0.54/0.86  % SZS status Theorem for SCT105+1
% 0.54/0.86  % SZS output start Proof for SCT105+1
% 0.54/0.86  (
% 0.54/0.86  (let ((_let_1 (forall ((B_x $$unsorted)) (not (tptp.c_member (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt) (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_x B_x) tptp.v_L))))) (let ((_let_2 (tptp.c_Relation_Otrans tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.v_L))) (let ((_let_3 (forall ((B_x $$unsorted)) (not (tptp.c_member (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt) (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_x B_x) tptp.v_L))))) (let ((_let_4 (not (= (and _let_2 _let_3 (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt))) (=> (not (= B_x B_y)) (or (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_y B_x) tptp.v_L) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_x B_y) tptp.v_L)))))) (and _let_2 _let_1 (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt))) (=> (not (= B_x B_y)) (or (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_x B_y) tptp.v_L) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_y B_x) tptp.v_L)))))))))) (let ((_let_5 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_6 (tptp.tc_fun _let_5 tptp.tc_HOL_Obool))) (let ((_let_7 (tptp.tc_prod _let_6 _let_6))) (let ((_let_8 (tptp.tc_fun _let_7 tptp.tc_HOL_Obool))) (let ((_let_9 (tptp.c_Orderings_Obot__class_Obot _let_6))) (let ((_let_10 (tptp.c_Set_Oinsert _let_7 (tptp.c_Product__Type_OPair _let_6 _let_6 _let_9 _let_9) (tptp.c_Orderings_Obot__class_Obot _let_8)))) (let ((_let_11 (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt))) (or (= B_x B_y) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_y B_x) tptp.v_L) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_x B_y) tptp.v_L)))))) (let ((_let_12 (and _let_2 _let_3 _let_11))) (let ((_let_13 (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt))) (or (= B_x B_y) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_x B_y) tptp.v_L) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_y B_x) tptp.v_L)))))) (let ((_let_14 (and _let_2 _let_1 _let_13))) (let ((_let_15 (EQ_RESOLVE (ASSUME :args (_let_4)) (MACRO_SR_EQ_INTRO :args (_let_4 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_16 (NOT_EQUIV_ELIM1 _let_15))) (let ((_let_17 (not _let_12))) (let ((_let_18 (ALPHA_EQUIV :args (_let_3 (= B_x B_x))))) (let ((_let_19 (TRANS (ALPHA_EQUIV :args (_let_11 (= B_x B_x) (= B_y B_y))) (MACRO_SR_PRED_INTRO :args ((= (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt))) (or (= B_x B_y) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_y B_x) tptp.v_L) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt B_x B_y) tptp.v_L)))) _let_13) SB_DEFAULT SBA_SEQUENTIAL RW_EXT_REWRITE))))) (let ((_let_20 (not _let_14))) (let ((_let_21 (MACRO_RESOLUTION_TRUST _let_16 (REORDERING (CNF_AND_POS :args (_let_14 0)) :args ((or _let_2 _let_20))) (REORDERING (CNF_AND_POS :args (_let_12 0)) :args ((or _let_2 _let_17))) :args (_let_2 true _let_14 true _let_12)))) (let ((_let_22 (MACRO_RESOLUTION_TRUST (CNF_AND_NEG :args (_let_14)) _let_21 (REORDERING (EQUIV_ELIM1 _let_19) :args ((or _let_13 (not _let_11)))) (REORDERING (EQUIV_ELIM1 _let_18) :args ((or _let_1 (not _let_3)))) (REORDERING (CNF_AND_POS :args (_let_12 2)) :args ((or _let_11 _let_17))) (REORDERING (CNF_AND_POS :args (_let_12 1)) :args ((or _let_3 _let_17))) _let_16 :args (_let_14 false _let_2 false _let_13 false _let_1 false _let_11 false _let_3 false _let_12)))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (CNF_AND_NEG :args (_let_12)) (MACRO_RESOLUTION_TRUST (EQUIV_ELIM2 _let_19) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_14 2)) :args ((or _let_13 _let_20))) _let_22 :args (_let_13 false _let_14)) :args (_let_11 false _let_13)) (MACRO_RESOLUTION_TRUST (NOT_EQUIV_ELIM2 _let_15) _let_22 :args (_let_17 false _let_14)) _let_21 (MACRO_RESOLUTION_TRUST (EQUIV_ELIM2 _let_18) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_AND_POS :args (_let_14 1)) :args ((or _let_1 _let_20))) _let_22 :args (_let_1 false _let_14)) :args (_let_3 false _let_1)) :args (false false _let_11 true _let_12 false _let_2 false _let_3)) :args ((forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (= (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x))) (= V_f_2 V_g_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Otrans T_a V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_y B_z) V_r_2) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a 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_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_Relation_Otrans T_a V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_r_2) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_c_2) V_r_2)))))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.c_member T_a V_c_2 V_B_2))) (let ((_let_2 (tptp.c_member T_a V_c_2 V_A_2))) (=> (= V_A_2 V_B_2) (=> (=> _let_2 (not _let_1)) (not (=> (not _let_2) _let_1))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.hAPP (tptp.c_FunDef_Oin__rel T_a T_b V_R_2) V_x_2) V_y_2)) (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_x_2 V_y_2) V_R_2))) (forall ((T_a $$unsorted) (V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_y_H_2 $$unsorted) (T_b $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (=> (tptp.c_member (tptp.tc_prod T_b T_b) (tptp.c_Product__Type_OPair T_b T_b V_y_H_2 V_y_2) (tptp.hAPP V_R_2 V_x_2)) (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 (tptp.c_Product__Type_OPair T_a T_b V_x_2 V_y_H_2) (tptp.c_Product__Type_OPair T_a T_b V_x_2 V_y_2)) (tptp.c_Recdef_Osame__fst T_a T_b V_P_2 V_R_2)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Nitpick_Orefl_H T_a V_r_2) (forall ((B_x $$unsorted)) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_x) V_r_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_La_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt))) (=> (tptp.c_member (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) V_La_2 tptp.c_Arrow__Order__Mirabelle_OLin) (=> (not (= V_x_2 V_y_2)) (= (not (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt V_x_2 V_y_2) V_La_2)) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair tptp.tc_Arrow__Order__Mirabelle_Oalt tptp.tc_Arrow__Order__Mirabelle_Oalt V_y_2 V_x_2) V_La_2)))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (V_P_2 $$unsorted)) (= (forall ((B_x1 $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x1))) (forall ((B_a $$unsorted) (B_b $$unsorted)) (tptp.hBOOL (tptp.hAPP V_P_2 (tptp.c_Product__Type_OPair T_a T_b B_a B_b)))))) (forall ((V_b_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2) (tptp.c_Product__Type_OPair T_a T_b V_a_H_2 V_b_H_2)) (and (= V_a_2 V_a_H_2) (= V_b_2 V_b_H_2)))) (forall ((V_b_H $$unsorted) (V_a_H $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (= (tptp.c_Product__Type_OPair T_a T_b V_a V_b) (tptp.c_Product__Type_OPair T_a T_b V_a_H V_b_H)) (not (=> (= V_a V_a_H) (not (= V_b V_b_H)))))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_x_2 V_A_2) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (=> (= V_a_2 V_b_2) (=> (tptp.c_member T_a V_b_2 V_A_2) (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted)) (=> (= V_x_2 V_y_2) (= (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_y_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (= (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f1_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Oprod_Oprod__rec T_b T_c T_a V_f1_2 (tptp.c_Product__Type_OPair T_b T_c V_a_2 V_b_2)) (tptp.hAPP (tptp.hAPP V_f1_2 V_a_2) V_b_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Oirrefl T_a V_r_2) (forall ((B_x $$unsorted)) (not (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_x) V_r_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (V_b_H_2 $$unsorted) (V_a_H_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (= (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2) (tptp.c_Product__Type_OPair T_a T_b V_a_H_2 V_b_H_2)) (tptp.c_Wellfounded_Olex__prod T_a T_b V_r_2 V_s_2)) (or (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_a_H_2) V_r_2) (and (= V_a_2 V_a_H_2) (tptp.c_member (tptp.tc_prod T_b T_b) (tptp.c_Product__Type_OPair T_b T_b V_b_2 V_b_H_2) V_s_2)))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Ointernal__split T_b T_c T_a V_c_2 (tptp.c_Product__Type_OPair T_b T_c V_a_2 V_b_2)) (tptp.hAPP (tptp.hAPP V_c_2 V_a_2) V_b_2))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2) (tptp.c_member T_b V_b_2 (tptp.c_Relation_ORange T_a T_b V_r_2)))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Relation_OId__on T_a V_A_2)) (and (= V_x_2 V_y_2) (tptp.c_member T_a V_x_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (=> (= V_a_2 V_b_2) (=> (tptp.c_member T_a V_a_2 V_A_2) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (tptp.c_Relation_OId__on T_a V_A_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2) (tptp.c_member T_a V_a_2 (tptp.c_Relation_ODomain T_a T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_y B_z) V_r_2) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_x B_z) V_r_2)))))) (tptp.c_Relation_Otrans T_a V_r_2))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Ototal__on T_a V_A_2 V_r_2) (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_A_2) (forall ((B_xa $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member T_a B_xa V_A_2) (=> (not (= B_x B_xa)) (or (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_x B_xa) V_r_2) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_xa B_x) V_r_2)))))))))) (forall ((V_f_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.hAPP (tptp.c_Relation_Oinv__image T_b T_a V_r_2) V_f_2)) (tptp.c_member (tptp.tc_prod T_b T_b) (tptp.c_Product__Type_OPair T_b T_b (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) V_r_2))) (forall ((V_R2_2 $$unsorted) (T_b $$unsorted) (V_R1_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Otrans T_a V_R1_2) (=> (tptp.c_Relation_Otrans T_b V_R2_2) (tptp.c_Relation_Otrans (tptp.tc_prod T_a T_b) (tptp.c_Wellfounded_Olex__prod T_a T_b V_R1_2 V_R2_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_ORange T_a T_a (tptp.c_Relation_OId__on T_a V_A_2)) V_A_2)) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_ODomain T_a T_a (tptp.c_Relation_OId__on T_a V_A_2)) V_A_2)) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Otrans T_a (tptp.c_Relation_OId__on T_a V_A_2))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Otrans T_a V_r_2) (tptp.c_Relation_Otrans T_b (tptp.hAPP (tptp.c_Relation_Oinv__image T_a T_b V_r_2) V_f_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_a_2 (tptp.c_Relation_ODomain T_a T_b V_r_2)) (exists ((B_y $$unsorted)) (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_a_2 B_y) V_r_2)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_a_2 (tptp.c_Relation_ORange T_b T_a V_r_2)) (exists ((B_y $$unsorted)) (tptp.c_member (tptp.tc_prod T_b T_a) (tptp.c_Product__Type_OPair T_b T_a B_y V_a_2) V_r_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Order__Relation_Ostrict__linear__order__on T_a V_A_2 V_r_2) (and (tptp.c_Relation_Otrans T_a V_r_2) (tptp.c_Relation_Oirrefl T_a V_r_2) (tptp.c_Relation_Ototal__on T_a V_A_2 V_r_2)))) (forall ((V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) V_c_2 (tptp.c_Relation_OId__on T_a V_A_2)) (not (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_A_2) (not (= V_c_2 (tptp.c_Product__Type_OPair T_a T_a B_x B_x)))))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_b_2 (tptp.c_Relation_ORange T_b T_a V_r_2)) (not (forall ((B_x $$unsorted)) (not (tptp.c_member (tptp.tc_prod T_b T_a) (tptp.c_Product__Type_OPair T_b T_a B_x V_b_2) V_r_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 (tptp.c_Relation_ODomain T_a T_b V_r_2)) (not (forall ((B_y $$unsorted)) (not (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_a_2 B_y) V_r_2)))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_a_2 (tptp.c_Wellfounded_Oacc T_a V_r_2)) (forall ((B_x $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x V_a_2) V_r_2) (tptp.c_member T_a B_x (tptp.c_Wellfounded_Oacc T_a V_r_2)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_a V_r_2))) (=> (tptp.c_member T_a V_b_2 _let_1) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (tptp.c_member T_a V_a_2 _let_1))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.c_member T_a V_a_2 V_A_2) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_a_2) V_r_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (tptp.c_member T_a V_x_2 V_A_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (tptp.c_member T_a V_y_2 V_A_2)))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_F_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Recdef_Owfrec__rel T_a T_b V_R_2 V_F_2 V_a1_2 (tptp.hAPP (tptp.hAPP V_F_2 V_a2_2) V_a1_2)) (exists ((B_g $$unsorted)) (and (= (tptp.hAPP (tptp.hAPP V_F_2 V_a2_2) V_a1_2) (tptp.hAPP (tptp.hAPP V_F_2 B_g) V_a1_2)) (forall ((B_z $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_z V_a1_2) V_R_2) (tptp.c_Recdef_Owfrec__rel T_a T_b V_R_2 V_F_2 B_z (tptp.hAPP B_g B_z)))))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Orefl__on T_a V_A_2 (tptp.c_Relation_OId__on T_a V_A_2))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Order__Relation_Opreorder__on T_a V_A_2 V_r_2) (and (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (tptp.c_Relation_Otrans T_a V_r_2)))) (forall ((V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.c_member T_a V_x_2 (tptp.c_Wellfounded_Oacc T_a V_R_2))) (not (forall ((B_z $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_z V_x_2) V_R_2) (tptp.c_member T_a B_z (tptp.c_Wellfounded_Oacc T_a V_R_2))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_y V_x_2) V_r_2) (tptp.c_member T_a B_y (tptp.c_Wellfounded_Oacc T_a V_r_2)))) (tptp.c_member T_a V_x_2 (tptp.c_Wellfounded_Oacc T_a V_r_2)))) (forall ((V_g_2 $$unsorted) (V_F_2 $$unsorted) (T_b $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_z $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_z V_x_2) V_R_2) (tptp.c_Recdef_Owfrec__rel T_a T_b V_R_2 V_F_2 B_z (tptp.hAPP V_g_2 B_z)))) (tptp.c_Recdef_Owfrec__rel T_a T_b V_R_2 V_F_2 V_x_2 (tptp.hAPP (tptp.hAPP V_F_2 V_g_2) V_x_2)))) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (V_F_2 $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Recdef_Owfrec__rel T_a T_b V_R_2 V_F_2 V_a1_2 V_a2_2) (exists ((B_g $$unsorted)) (and (= V_a2_2 (tptp.hAPP (tptp.hAPP V_F_2 B_g) V_a1_2)) (forall ((B_z $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_z V_a1_2) V_R_2) (tptp.c_Recdef_Owfrec__rel T_a T_b V_R_2 V_F_2 B_z (tptp.hAPP B_g B_z)))))))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_a V_r_2))) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_a_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (tptp.c_member T_a V_a_2 _let_1) (tptp.c_member T_a V_b_2 _let_1))))) (forall ((V_b_2 $$unsorted) (V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Oacc T_a V_r_2))) (=> (tptp.c_member T_a V_a_2 _let_1) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_a_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (tptp.c_member T_a V_b_2 _let_1))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_f_2 (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2))) (tptp.hBOOL (tptp.c_Product__Type_Ocurry T_a T_b tptp.tc_HOL_Obool V_f_2 V_a_2 V_b_2)))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_b))) (=> (tptp.c_Relation_Osingle__valued T_a T_b V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_b V_x_2 V_y_2) V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_b V_x_2 V_z_2) V_r_2) (= V_y_2 V_z_2)))))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Osingle__valued T_a T_b V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b B_x B_z) V_r_2) (= B_y B_z))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.c_Product__Type_Ocurry T_a T_b tptp.tc_HOL_Obool V_f_2 V_a_2 V_b_2)) (tptp.hBOOL (tptp.hAPP V_f_2 (tptp.c_Product__Type_OPair T_a 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_a $$unsorted)) (=> (tptp.hBOOL (tptp.c_Product__Type_Ocurry T_a T_b tptp.tc_HOL_Obool V_f_2 V_a_2 V_b_2)) (tptp.hBOOL (tptp.hAPP V_f_2 (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Osingle__valued T_a T_a (tptp.c_Relation_OId__on T_a V_A_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_c $$unsorted) (T_b $$unsorted)) (= (tptp.c_Product__Type_Ocurry T_b T_c T_a V_f_2 V_a_2 V_b_2) (tptp.hAPP V_f_2 (tptp.c_Product__Type_OPair T_b T_c V_a_2 V_b_2)))) (forall ((V_r_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_a_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (V_p_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 V_p_2 V_r_2) (tptp.c_member _let_1 V_p_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))) (forall ((V_y_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.c_member T_a V_x_2 (tptp.c_Relation_ODomain T_a T_a V_R_2))) (= (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (= V_x_2 V_y_2)))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_Relation_Osingle__valued T_a T_a V_r_2) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) _let_1) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_z_2) _let_1) (or (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_y_2 V_z_2) _let_1) (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_z_2 V_y_2) _let_1)))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_c_2) _let_1) (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_r_2) (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_c_2) _let_1) (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Otrans T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b B_x B_y) V_r_2) (forall ((B_z $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b B_x B_z) V_r_2) (= B_y B_z))))) 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T_a V_a_2 V_c_2) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_R_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (let ((_let_3 (= V_a_2 V_b_2))) (=> (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (or _let_3 (and (not _let_3) (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_R_2))))))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_c_2) V_r_2) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_c_2) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_c_2 $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) _let_1) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_c_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_c_2) _let_1)))))) (forall ((V_z_2 $$unsorted) (V_r_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)) (=> (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_y_2 V_z_2) _let_1) (tptp.c_member _let_2 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_z_2) _let_1)))))) (forall ((T_b $$unsorted) (V_a_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted) (B_f $$unsorted) (B_R $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_a_2) B_R) (= (tptp.hAPP (tptp.c_Recdef_Ocut T_a T_b B_f B_R V_a_2) V_x_2) (tptp.hAPP B_f V_x_2)))) (forall ((V_g_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Recdef_Ocut T_a T_b V_f_2 V_r_2 V_x_2) (tptp.c_Recdef_Ocut T_a T_b V_g_2 V_r_2 V_x_2)) (forall ((B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_y V_x_2) V_r_2) (= (tptp.hAPP V_f_2 B_y) (tptp.hAPP V_g_2 B_y)))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Transitive__Closure_Otrancl T_a V_R_2)) (exists ((B_z $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (and (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 B_z) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_z V_y_2) V_R_2)))))) (forall ((V_R_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Transitive__Closure_Otrancl T_a V_R_2)) (exists ((B_z $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (and (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 B_z) V_R_2) (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_z V_y_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (forall ((B_x $$unsorted)) (not (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_x) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (not (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_x) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) V_r_2) (not (= V_x_2 V_y_2))))) (forall ((V_r_2 $$unsorted) (V_z_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_z_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (=> (not (tptp.c_member _let_2 _let_1 V_r_2)) (not (forall ((B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 B_y) V_r_2) (not (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_y V_z_2) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))))))))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (=> (not (tptp.c_member _let_2 _let_1 V_r_2)) (not (forall ((B_c $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 B_c) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)) (not (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_c V_b_2) V_r_2))))))))))) (forall ((V_x_2 $$unsorted) (V_F_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Nitpick_Owf__wfrec_H T_b T_a V_R_2 V_F_2 V_x_2) (tptp.hAPP (tptp.hAPP V_F_2 (tptp.c_Recdef_Ocut T_b T_a (tptp.c_Nitpick_Owf__wfrec T_b T_a V_R_2 V_F_2) V_R_2 V_x_2)) V_x_2))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_a_2) (tptp.c_Relation_OId T_a))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Otrans T_a (tptp.c_Relation_OId T_a))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Osingle__valued T_a T_a (tptp.c_Relation_OId T_a))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) (tptp.c_Relation_OId T_a)) (= V_a_2 V_b_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (not (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_x) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))) (tptp.c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((V_p_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) V_p_2 (tptp.c_Relation_OId T_a)) (not (forall ((B_x $$unsorted)) (not (= V_p_2 (tptp.c_Product__Type_OPair T_a T_a B_x B_x))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (= (tptp.c_Wellfounded_Oacyclic T_a (tptp.c_Set_Oinsert _let_1 (tptp.c_Product__Type_OPair T_a T_a V_y_2 V_x_2) V_r_2)) (and (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (not (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Oacyclic T_a V_r_2) (tptp.c_Relation_Oantisym T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Otrancl T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a))) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) (tptp.c_Transitive__Closure_Otrancl T_a V_r_2) (tptp.c_Relation_OId T_a)) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_Relation_Oantisym T_a V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_a_2) V_r_2) (= V_a_2 V_b_2)))))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_A_2)) (=> (not (= V_a_2 V_b_2)) (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (=> (not (tptp.c_member T_a V_a_2 V_B_2)) (= V_a_2 V_b_2)) (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_c_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (=> (not (tptp.c_member T_a V_c_2 V_A_2)) (tptp.c_member T_a V_c_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (=> (not (tptp.c_member T_a V_c_2 V_B_2)) (tptp.c_member T_a V_c_2 V_A_2)) (tptp.c_member T_a V_c_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Oantisym T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a))) (tptp.c_Relation_Oantisym T_a V_r_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_A_2) V_A_2)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_A_2)))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2))) (= (tptp.c_Set_Oinsert T_a V_x_2 _let_1) _let_1))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 _let_2) _let_2)))) (forall ((V_A_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Set_Oinsert T_a V_y_2 V_A_2)) (tptp.c_Set_Oinsert T_a V_y_2 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2)) (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2)) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_C_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2) V_C_2) (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_C_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2))))) (forall ((V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (and (tptp.c_member T_a B_x (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (or (exists ((B_x $$unsorted)) (and (tptp.c_member T_a B_x V_A_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (exists ((B_x $$unsorted)) (and (tptp.c_member T_a B_x V_B_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))))) (forall ((V_P_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (and (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_A_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_x)))) (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_B_2) (tptp.hBOOL (tptp.hAPP V_P_2 B_x))))))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (= (tptp.hBOOL (tptp.hAPP (tptp.c_Set_Oinsert T_a V_y_2 V_A_2) V_x_2)) (or (= V_y_2 V_x_2) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))))) (forall ((V_s_2 $$unsorted) (V_B_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Orefl__on T_a V_A_2 V_r_2) (=> (tptp.c_Relation_Orefl__on T_a V_B_2 V_s_2) (tptp.c_Relation_Orefl__on T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 V_s_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_ORange T_b T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool) V_A_2 V_B_2)) (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_ORange T_b T_a V_A_2) (tptp.c_Relation_ORange T_b T_a V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_ODomain T_a T_b (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool) V_A_2 V_B_2)) (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_ODomain T_a T_b V_A_2) (tptp.c_Relation_ODomain T_a T_b V_B_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_ODomain T_a T_b (tptp.c_Set_Oinsert (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2)) (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Relation_ODomain T_a T_b V_r_2)))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_ORange T_b T_a (tptp.c_Set_Oinsert (tptp.tc_prod T_b T_a) (tptp.c_Product__Type_OPair T_b T_a V_a_2 V_b_2) V_r_2)) (tptp.c_Set_Oinsert T_a V_b_2 (tptp.c_Relation_ORange T_b T_a V_r_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_c_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (or (tptp.c_member T_a V_c_2 V_A_2) (tptp.c_member T_a V_c_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_c_2 V_A_2) (tptp.c_member T_a V_c_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_c_2 V_B_2) (tptp.c_member T_a V_c_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2))) (forall ((V_A_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_A_2)) (or (= V_a_2 V_b_2) (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (=> (not (tptp.c_member T_a V_x_2 V_B_2)) (= (= (tptp.c_Set_Oinsert T_a V_x_2 V_A_2) (tptp.c_Set_Oinsert T_a V_x_2 V_B_2)) (= V_A_2 V_B_2))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 V_B_2) (tptp.c_member T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 V_A_2) (= (tptp.c_Set_Oinsert T_a V_a_2 V_A_2) V_A_2))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_S_2))) (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_R_2 V_S_2))))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Oantisym T_a (tptp.c_Relation_OId T_a))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (or (tptp.c_member _let_1 V_x_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2)) (tptp.c_member _let_1 V_x_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_S_2))) (tptp.c_member _let_1 V_x_2 (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_1 tptp.tc_HOL_Obool) V_R_2 V_S_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Oantisym T_a (tptp.c_Relation_OId__on T_a V_A_2))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_R_2 (tptp.c_Relation_OId T_a))) (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) _let_1 (tptp.c_Relation_OId T_a)) _let_1))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Otrans T_a V_r_2) (tptp.c_Relation_Otrans T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Oantisym T_a V_r_2) (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_y B_x) V_r_2) (= B_x B_y))))))) (forall ((T_a $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (=> (not (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2))) (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))) (tptp.hBOOL (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) V_x_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) V_x_2)) (=> (not (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2))) (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b T_a) V_f_2 V_g_2) V_x_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Order__Relation_Opartial__order__on T_a V_A_2 V_r_2) (and (tptp.c_Order__Relation_Opreorder__on T_a V_A_2 V_r_2) (tptp.c_Relation_Oantisym T_a V_r_2)))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) V_P_2 V_Q_2))) (=> (forall ((B_x $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x V_b_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2)) (forall ((B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_y B_x) V_Q_2) (= B_y B_x))))) (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2))))))) (forall ((V_Q_2 $$unsorted) (V_P_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2))) (let ((_let_2 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun _let_2 tptp.tc_HOL_Obool) V_P_2 V_Q_2))) (=> (forall ((B_x $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 B_x) (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2)) (forall ((B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_x B_y) V_Q_2) (= B_x B_y))))) (tptp.c_member _let_2 _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_P_2))))))) (forall ((V_r_2 $$unsorted) (V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (= (tptp.c_Wellfounded_Owf T_a (tptp.c_Set_Oinsert _let_1 (tptp.c_Product__Type_OPair T_a T_a V_y_2 V_x_2) V_r_2)) (and (tptp.c_Wellfounded_Owf T_a V_r_2) (not (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP (tptp.c_Relation_Oinv__image T_a T_b V_r_2) V_f_2)))) (forall ((V_rb_2 $$unsorted) (T_b $$unsorted) (V_ra_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_ra_2) (=> (tptp.c_Wellfounded_Owf T_b V_rb_2) (tptp.c_Wellfounded_Owf (tptp.tc_prod T_a T_b) (tptp.c_Wellfounded_Olex__prod T_a T_b V_ra_2 V_rb_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf T_a (tptp.c_Transitive__Closure_Otrancl T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Oacyclic T_a V_r_2))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (not (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_a_2) V_r_2)))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_x_2) V_r_2) (not (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_a_2) V_r_2)))))) (forall ((V_x_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_x_2) V_r_2) (not (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_a_2) V_r_2)))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (not (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_a_2) V_r_2)))) (forall ((V_x_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_member T_a V_x_2 (tptp.c_Wellfounded_Oacc T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Owf T_a V_r_2) (forall ((B_x $$unsorted)) (tptp.c_member T_a B_x (tptp.c_Wellfounded_Oacc T_a V_r_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_x_2 (tptp.c_fequal V_y_2)) (= V_x_2 V_y_2))) (forall ((V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (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.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_x) V_x))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_b V_a)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_x)))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a _let_1) _let_1)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y))) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x _let_1) _let_1)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y))) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x _let_1) _let_1)))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_b (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_c)) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_b V_c))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z)) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_z))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z)) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_z))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) V_c) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_b V_c))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_z) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_z) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z))))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_A_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) V_x_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_B_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_B_2 V_x_2)) (tptp.hBOOL (tptp.hAPP (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) V_x_2)))) (forall ((V_f_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Wellfounded_Owf T_a (tptp.hAPP (tptp.c_Wellfounded_Omeasure T_a) V_f_2))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Order__Relation_Olinear__order__on T_a V_A_2 V_r_2) (and (tptp.c_Order__Relation_Opartial__order__on T_a V_A_2 V_r_2) (tptp.c_Relation_Ototal__on T_a V_A_2 V_r_2)))) (forall ((V_a_2 $$unsorted) (V_H_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_f_2 $$unsorted)) (=> (= V_f_2 (tptp.c_Recdef_Owfrec T_a T_b V_r_2 V_H_2)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (= (tptp.hAPP V_f_2 V_a_2) (tptp.hAPP (tptp.hAPP V_H_2 (tptp.c_Recdef_Ocut T_a T_b V_f_2 V_r_2 V_a_2)) V_a_2))))) (forall ((V_a_2 $$unsorted) (V_H_2 $$unsorted) (T_b $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Recdef_Owfrec T_a T_b V_r_2 V_H_2))) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (= (tptp.hAPP _let_1 V_a_2) (tptp.hAPP (tptp.hAPP V_H_2 (tptp.c_Recdef_Ocut T_a T_b _let_1 V_r_2 V_a_2)) V_a_2))))) (forall ((T_b $$unsorted) (T_a $$unsorted) (B_M $$unsorted) (B_R $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a B_R) (forall ((B_x $$unsorted)) (let ((_let_1 (tptp.c_Recdef_Owfrec T_a T_b B_R B_M))) (= (tptp.hAPP _let_1 B_x) (tptp.hAPP (tptp.hAPP B_M (tptp.c_Recdef_Ocut T_a T_b _let_1 B_R B_x)) B_x)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Wellfounded_Owf T_a V_r_2) (forall ((B_Q $$unsorted)) (=> (exists ((B_x $$unsorted)) (tptp.c_member T_a B_x B_Q)) (exists ((B_x $$unsorted)) (and (tptp.c_member T_a B_x B_Q) (forall ((B_y $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_y B_x) V_r_2) (not (tptp.c_member T_a B_y B_Q)))))))))) (forall ((T_a $$unsorted) (V_R_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted)) (=> (forall ((B_x $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 B_x)) (tptp.c_Wellfounded_Owf T_b (tptp.hAPP V_R_2 B_x)))) (tptp.c_Wellfounded_Owf (tptp.tc_prod T_a T_b) (tptp.c_Recdef_Osame__fst T_a T_b V_P_2 V_R_2)))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (tptp.c_Wellfounded_OwfP T_a (tptp.c_FunDef_Oin__rel T_a T_a V_R_2)))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Order__Relation_Olinear__order__on T_a V_A_2 V_r_2) (tptp.c_Order__Relation_Ostrict__linear__order__on T_a V_A_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_prod T_a T_a))) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_x B_y) V_r_2) (=> (tptp.c_member _let_1 (tptp.c_Product__Type_OPair T_a T_a B_y B_x) V_r_2) (= B_x B_y))))) (tptp.c_Relation_Oantisym T_a V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (forall ((B_x $$unsorted)) (tptp.c_member T_a B_x (tptp.c_Wellfounded_Oacc T_a V_r_2))) (tptp.c_Wellfounded_Owf T_a V_r_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_c_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (not (=> (tptp.c_member T_a V_c_2 V_A_2) (tptp.c_member T_a V_c_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_c_2 V_A_2) (=> (not (tptp.c_member T_a V_c_2 V_B_2)) (tptp.c_member T_a V_c_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_c_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (and (tptp.c_member T_a V_c_2 V_A_2) (not (tptp.c_member T_a V_c_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_c_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (tptp.c_member T_a V_c_2 V_A_2))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_c_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_c_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2)) (not (tptp.c_member T_a V_c_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_A_2)) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_A_2) V_A_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_A_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_C_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_C_2) (tptp.c_Groups_Ominus__class_Ominus _let_1 V_B_2 V_C_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 _let_2 V_B_2) _let_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Ominus T_a) (= (tptp.hAPP (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun T_b T_a) V_A_2 V_B_2) V_x_2) (tptp.c_Groups_Ominus__class_Ominus T_a (tptp.hAPP V_A_2 V_x_2) (tptp.hAPP V_B_2 V_x_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_member T_a V_x_2 V_B_2) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2) V_B_2) (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2))) (let ((_let_3 (tptp.c_Groups_Ominus__class_Ominus _let_1 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2) V_B_2))) (let ((_let_4 (tptp.c_member T_a V_x_2 V_B_2))) (and (=> _let_4 (= _let_3 _let_2)) (=> (not _let_4) (= _let_3 (tptp.c_Set_Oinsert T_a V_x_2 _let_2))))))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a))) (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_Ototal__on T_a V_A_2 (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a))) (tptp.c_Relation_Ototal__on T_a V_A_2 V_r_2))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Oirrefl T_a (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Relation_Otrans T_a V_r_2) (=> (tptp.c_Relation_Oantisym T_a V_r_2) (tptp.c_Relation_Otrans T_a (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a)))))) (forall ((V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Order__Relation_Owell__order__on T_a V_A_2 V_r_2) (and (tptp.c_Order__Relation_Olinear__order__on T_a V_A_2 V_r_2) (tptp.c_Wellfounded_Owf T_a (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 (tptp.c_Relation_OId T_a)))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_OField T_a V_r_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_ODomain T_a T_a V_r_2) (tptp.c_Relation_ORange T_a T_a V_r_2)))) (forall ((V_f_2 $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.hAPP (tptp.c_Wellfounded_Omeasure T_a) V_f_2)) (tptp.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 ((T_a $$unsorted)) (= (tptp.c_Wellfounded_Omeasure T_a) (tptp.c_Relation_Oinv__image tptp.tc_Nat_Onat T_a tptp.c_Wellfounded_Oless__than))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 (tptp.c_Relation_ORange T_b T_a V_A_2) (tptp.c_Relation_ORange T_b T_a V_B_2)) (tptp.c_Relation_ORange T_b T_a (tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool) V_A_2 V_B_2))))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2) (= V_A_2 V_B_2))))) (tptp.c_Relation_Otrans tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than) (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat tptp.c_Wellfounded_Oless__than) (forall ((V_c_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_member T_a V_c_2 V_A_2) (tptp.c_member T_a V_c_2 V_B_2)))) (forall ((V_D_2 $$unsorted) (V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_D_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_C_2 V_D_2)))))) (forall ((V_B_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_C_2))))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) V_B_2)))) (forall ((V_A_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2)))) (forall ((V_Q_2 $$unsorted) (T_a $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_P_2 V_Q_2) (tptp.hBOOL (tptp.hAPP V_Q_2 V_x_2))))) (forall ((V_x_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_P_2 V_Q_2) (=> (tptp.hBOOL (tptp.hAPP V_P_2 V_x_2)) (tptp.hBOOL (tptp.hAPP V_Q_2 V_x_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool) V_r_2 V_s_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_ODomain T_a T_b V_r_2) (tptp.c_Relation_ODomain T_a T_b V_s_2)))) (forall ((V_R2_2 $$unsorted) (V_R1_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_R1_2 V_R2_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Oacc T_a V_R2_2) (tptp.c_Wellfounded_Oacc T_a V_R1_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_A_2)) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (= V_A_2 V_B_2) (and (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_B_2 V_A_2))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 V_s_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_OField T_a V_r_2) (tptp.c_Relation_OField T_a V_s_2)))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2))))) (forall ((T_a $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (= V_A_2 V_B_2) (not (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (not (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_A_2))))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (= (tptp.c_member (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (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.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x_2 V_y_2))) (forall ((V_a_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_C_2 V_D_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Set_Oinsert T_a V_a_2 V_C_2) (tptp.c_Set_Oinsert T_a V_a_2 V_D_2))))) (forall ((V_b_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_b_2 V_B_2))))) (forall ((V_a_2 $$unsorted) (V_B_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_B_2 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2))) (forall ((V_x $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) V_x) (not (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_x) (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_x))))))) (forall ((V_d $$unsorted) (V_b $$unsorted) (V_c $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_c) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_d) (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_c V_d)))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_z V_x) (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_y V_z) V_x))))) (forall ((V_b $$unsorted) (V_x $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_x) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_x) (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b) V_x))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_x)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y) V_y)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_b) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x (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)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b))))) (forall ((V_z_2 $$unsorted) (V_y_2 $$unsorted) 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V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Olattice T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x V_y)))) (forall ((V_a $$unsorted) (V_b $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_b) (tptp.c_Orderings_Oord__class_Oless T_a V_x (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)) (=> (tptp.class_Lattices_Osemilattice__sup T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_a) (tptp.c_Orderings_Oord__class_Oless T_a V_x (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_a V_b))))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_x_2 V_A_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((V_x_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_2 V_B_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_member T_a V_x_2 V_B_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) V_A_2))) (forall ((V_B_2 $$unsorted) (V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_C_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_D_2 V_B_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) (tptp.c_Groups_Ominus__class_Ominus _let_1 V_C_2 V_D_2)))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 V_C_2) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 V_B_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_C_2 V_A_2)) V_A_2))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_s_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_2 V_r_2 _let_1) (tptp.c_Orderings_Oord__class_Oless__eq _let_2 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2) _let_1))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_R_2))) (let ((_let_2 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_2 V_R_2 V_S_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_2 V_S_2 _let_1) (= (tptp.c_Transitive__Closure_Ortrancl T_a V_S_2) _let_1)))))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_r_2 V_s_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2) (tptp.c_Transitive__Closure_Ortrancl T_a V_s_2))))) (forall ((V_p_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_p_2 V_r_2) (tptp.c_Wellfounded_Owf T_a V_p_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 V_s_2) (=> (tptp.c_Relation_Oantisym T_a V_s_2) (tptp.c_Relation_Oantisym T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (V_s_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Oacyclic T_a V_s_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) V_r_2 V_s_2) (tptp.c_Wellfounded_Oacyclic T_a V_r_2)))) (forall ((V_s_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool) V_r_2 V_s_2) (=> (tptp.c_Relation_Osingle__valued T_a T_b V_s_2) 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(tptp.c_Groups_Ominus__class_Ominus (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool) V_A_2 V_B_2))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_x))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_f_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_f_2 V_y_2)) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_x_2 V_y_2) (tptp.c_Wellfounded_Omlex__prod T_a V_f_2 V_R_2)))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b) V_f_2 V_g_2) (tptp.c_Orderings_Oord__class_Oless__eq T_b (tptp.hAPP V_f_2 V_x_2) (tptp.hAPP V_g_2 V_x_2))))) (forall ((V_x_2 $$unsorted) (V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b 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V_y_2)) (=> (tptp.c_member _let_2 _let_1 V_R_2) (tptp.c_member _let_2 _let_1 (tptp.c_Wellfounded_Omlex__prod T_a V_f_2 V_R_2))))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (= V_x_2 V_y_2) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (= V_x V_y) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y)))) (forall ((V_x_2 $$unsorted) (V_y_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2) (= (tptp.c_Orderings_Oord__class_Oless__eq T_a 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)) (=> (tptp.class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_c) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c V_b) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c V_a))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (=> (= V_b V_c) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_c))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_a) (=> (= V_b V_c) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c V_a))))) (forall 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(tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (=> (= V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_c))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (= V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_c V_b) (tptp.c_Orderings_Oord__class_Oless T_a V_c V_a))))) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oord T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b) (=> (= V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_c))))) (forall ((V_c $$unsorted) (V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a) (=> (= V_b V_c) (tptp.c_Orderings_Oord__class_Oless T_a V_c V_a))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_z) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_z))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_z V_y) (tptp.c_Orderings_Oord__class_Oless T_a V_z V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (not (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (=> (not (= V_x V_y)) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x))))) (forall ((V_f_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (tptp.c_Wellfounded_Owf T_a (tptp.c_Wellfounded_Omlex__prod T_a V_f_2 V_R_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (= (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2)) (tptp.c_Orderings_Oord__class_Oless T_a V_y_2 V_x_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (or (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (not (= V_x_2 V_y_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (= (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (and (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y_2 V_x_2)))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (= (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (or (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x)))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x)) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (= (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (= V_x_2 V_y_2))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (not (= V_a V_b)) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_b $$unsorted) (V_a $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (not (= V_a V_b)) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_a) (tptp.c_Orderings_Oord__class_Oless T_a V_b V_a))))) (forall ((V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (not (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y)))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Olinorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x_2 V_y_2) (= (not (tptp.c_Orderings_Oord__class_Oless T_a V_x_2 V_y_2)) (= V_x_2 V_y_2))))) (forall ((V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (or (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 $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a V_b) (=> (not (= V_a V_b)) (tptp.c_Orderings_Oord__class_Oless T_a V_a V_b))))) (forall ((V_a $$unsorted) (V_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_b V_a) (=> (not (= V_a V_b)) (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)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_x V_y) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_z) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_z))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_x) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_z V_y) (tptp.c_Orderings_Oord__class_Oless T_a V_z V_x))))) (forall ((V_z $$unsorted) (V_y $$unsorted) (V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Opreorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_x V_y) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_y V_z) (tptp.c_Orderings_Oord__class_Oless T_a V_x V_z))))) (forall ((V_z $$unsorted) (V_x $$unsorted) (V_y $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Oorder T_a) (=> (tptp.c_Orderings_Oord__class_Oless__eq T_a V_y V_x) (=> (tptp.c_Orderings_Oord__class_Oless T_a V_z V_y) (tptp.c_Orderings_Oord__class_Oless T_a V_z V_x))))) (forall ((V_g_2 $$unsorted) (V_f_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (=> (tptp.class_Orderings_Oord T_b) (= (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a T_b) V_f_2 V_g_2) (forall ((B_x $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq T_b (tptp.hAPP V_f_2 B_x) (tptp.hAPP V_g_2 B_x)))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_member (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_m_2 V_n_2) (tptp.c_Transitive__Closure_Otrancl tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_member (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_m_2 V_n_2) (tptp.c_Transitive__Closure_Ortrancl tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (=> (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_b_2) (tptp.c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (tptp.c_Orderings_Oord__class_Oless T_a V_a_2 V_b_2) (tptp.c_Orderings_Oord__class_Oless T_a V_c_2 V_d_2))))) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_Q_2 $$unsorted) (V_P_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_P_2 V_Q_2) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_x_2) V_y_2)) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 V_x_2) V_y_2))))) (forall ((V_Q_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted) (V_y_2 $$unsorted) (V_x_2 $$unsorted) (V_P_2 $$unsorted)) (=> (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_x_2) V_y_2)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a (tptp.tc_fun T_b tptp.tc_HOL_Obool)) V_P_2 V_Q_2) (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_Q_2 V_x_2) V_y_2))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2) (=> (tptp.c_Orderings_Oord__class_Oless _let_1 V_B_2 V_C_2) (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_C_2))))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oab__group__add T_a) (=> (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_b_2) (tptp.c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (= V_a_2 V_b_2) (= V_c_2 V_d_2))))) (= tptp.c_Wellfounded_Oless__than (tptp.c_Transitive__Closure_Otrancl tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat)) (tptp.c_Wellfounded_Owf tptp.tc_Nat_Onat tptp.c_Wellfounded_Opred__nat) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Groups_Oordered__ab__group__add T_a) (=> (= (tptp.c_Groups_Ominus__class_Ominus T_a V_a_2 V_b_2) (tptp.c_Groups_Ominus__class_Ominus T_a V_c_2 V_d_2)) (= (tptp.c_Orderings_Oord__class_Oless__eq T_a V_a_2 V_b_2) (tptp.c_Orderings_Oord__class_Oless__eq T_a V_c_2 V_d_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (and (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (not (= V_m_2 V_n_2))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (= V_m_2 V_n_2)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted)) (= (not (= V_m_2 V_n_2)) (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2)))) (forall ((V_y $$unsorted) (V_x $$unsorted)) (=> (not (= V_x V_y)) (=> (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_x V_y)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_y V_x)))) (forall ((V_n $$unsorted)) (not (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n V_n))) (forall ((V_m $$unsorted) (V_n $$unsorted)) (=> (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.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_s V_t) (not (= V_s V_t)))) (forall ((V_P_2 $$unsorted) (V_n_2 $$unsorted) (V_m_2 $$unsorted)) (let ((_let_1 (tptp.hBOOL (tptp.hAPP (tptp.hAPP V_P_2 V_n_2) V_m_2)))) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2) _let_1) (=> (=> (= V_m_2 V_n_2) _let_1) (=> (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_n_2 V_m_2) _let_1) _let_1))))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_l) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_n) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_m))))) (forall ((V_n $$unsorted) (V_k $$unsorted) (V_j $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_j V_k) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_j V_n) V_k))) (forall ((V_n $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n V_n)) (forall ((V_n $$unsorted) (V_m $$unsorted)) (or (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n V_m))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (= V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_j) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_j V_k) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_k)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_n V_m) (= V_m V_n)))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2 V_m_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2 V_n_2) (= (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m_2 V_k_2) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n_2 V_k_2)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m_2 V_n_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted) (V_k $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k V_m) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k V_n) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_k) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n V_k)) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2 V_m_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2 V_n_2) (= (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m_2 V_k_2) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n_2 V_k_2)) (= V_m_2 V_n_2))))) (forall ((V_n $$unsorted) (V_i $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_i V_n) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n V_i)) V_i))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_l) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n V_l)))) (forall ((V_l $$unsorted) (V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_n) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_l V_m)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m V_n) V_m)) (forall ((V_c $$unsorted) (V_b $$unsorted) (V_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a V_b) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_c V_a) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_a V_c) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_b V_c))))) (forall ((V_n_2 $$unsorted) (V_m_2 $$unsorted) (V_k_2 $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2 V_m_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_k_2 V_n_2) (= (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_m_2 V_k_2) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_n_2 V_k_2)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m_2 V_n_2))))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (or (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (= V_m V_n)) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n) (=> (not (= V_m V_n)) (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n)))) (forall ((V_n $$unsorted) (V_m $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_m V_n) (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_m V_n))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_a_2 V_b_2) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_s_2 V_t_2) (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_a_2 V_s_2) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_b_2 V_t_2)) tptp.c_FunDef_Opair__less))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Relation_OField T_a (tptp.c_Set_Oinsert (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2)) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (tptp.c_Relation_OField T_a V_r_2))))) (forall ((T_a $$unsorted)) (tptp.c_Wellfounded_Owf T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted)) (not (tptp.c_member T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Orderings_Obot__class_Obot _let_1) V_A_2))) (forall ((V_k $$unsorted) (V_j $$unsorted) (V_i $$unsorted)) (= (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i V_j) V_k) (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat (tptp.c_Groups_Ominus__class_Ominus tptp.tc_Nat_Onat V_i V_k) V_j))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Set_Oinsert T_a V_a_2 V_A_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))) (tptp.c_Set_Oinsert T_a V_a_2 V_A_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2)) (tptp.c_Groups_Ominus__class_Ominus _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_B_2)))) (forall ((V_B_2 $$unsorted) (V_a_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_a_2 V_B_2)) (tptp.c_Groups_Ominus__class_Ominus _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_B_2) (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (tptp.c_Wellfounded_Owf _let_5 tptp.c_FunDef_Opair__less) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (forall ((B_x $$unsorted)) (not (tptp.c_member T_a B_x V_A_2))) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (exists ((B_x $$unsorted)) (tptp.c_member T_a B_x V_A_2)) (not (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))) (forall ((V_c_2 $$unsorted) (T_a $$unsorted)) (not (tptp.c_member T_a V_c_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_a_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (=> (= V_A_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.c_member T_a V_a_2 V_A_2)))) (forall ((V_x_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Obot T_a) (= (tptp.hAPP (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b T_a)) V_x_2) (tptp.c_Orderings_Obot__class_Obot T_a)))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Oantisym T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (tptp.c_Relation_Ototal__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_r_2)) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot T_a))) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x_2 V_y_2) _let_1) (and (= V_x_2 _let_1) (= V_y_2 _let_1)))))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a V_x (tptp.c_Orderings_Obot__class_Obot T_a)) V_x))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Lattices_Obounded__lattice__bot T_a) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup T_a (tptp.c_Orderings_Obot__class_Obot T_a) V_x) V_x))) (forall ((T_a $$unsorted)) (tptp.c_Order__Relation_Opartial__order__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((T_a $$unsorted)) (= (tptp.c_Relation_OId__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((T_a $$unsorted)) (tptp.c_Order__Relation_Opreorder__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (= (= (tptp.c_Relation_ODomain T_a T_b V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_r_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))))) (forall ((T_b $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_ODomain T_a T_b (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (= (tptp.c_Relation_ORange T_b T_a V_r_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= V_r_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool))))) (forall ((T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_ORange T_b T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((T_a $$unsorted)) (tptp.c_Relation_Orefl__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((T_a $$unsorted)) (tptp.c_Order__Relation_Olinear__order__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 V_A_2) (tptp.c_Orderings_Obot__class_Obot _let_1)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 _let_2 V_A_2) _let_2)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 V_B_2) _let_2) (and (= V_A_2 _let_2) (= V_B_2 _let_2)))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1)) V_A_2))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 (tptp.c_Orderings_Obot__class_Obot _let_1) V_B_2) V_B_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (=> (= (tptp.c_Set_Oinsert T_a V_a_2 _let_1) (tptp.c_Set_Oinsert T_a V_b_2 _let_1)) (= V_a_2 V_b_2)))) (forall ((V_d_2 $$unsorted) (V_c_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= (= (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Set_Oinsert T_a V_b_2 _let_1)) (tptp.c_Set_Oinsert T_a V_c_2 (tptp.c_Set_Oinsert T_a V_d_2 _let_1))) (or (and (= V_a_2 V_c_2) (= V_b_2 V_d_2)) (and (= V_a_2 V_d_2) (= V_b_2 V_c_2)))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Set_Oinsert T_a V_a_2 V_A_2) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))))) (forall ((V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (not (= (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Set_Oinsert T_a V_a_2 V_A_2)))) (forall ((T_a $$unsorted)) (= (tptp.c_Relation_OField T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((T_a $$unsorted)) (tptp.c_Order__Relation_Owell__order__on T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (forall ((V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Set_Oinsert T_a V_x_2 _let_2))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_3) (or (= V_A_2 _let_2) (= V_A_2 _let_3))))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_b_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= V_b_2 V_a_2))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_b_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (= V_b_2 V_a_2))) (forall ((T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool)))) (= (tptp.c_Transitive__Closure_Otrancl T_a _let_1) _let_1))) (forall ((V_x $$unsorted) (T_a $$unsorted)) (=> (tptp.class_Orderings_Obot T_a) (tptp.c_Orderings_Oord__class_Oless__eq T_a (tptp.c_Orderings_Obot__class_Obot T_a) V_x))) (forall ((V_B_2 $$unsorted) (T_a $$unsorted) (V_A_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1)) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_2) (= V_A_2 _let_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (not (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1))))) (forall ((T_a $$unsorted)) (= (tptp.c_Transitive__Closure_Ortrancl T_a (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (tptp.c_Relation_OId T_a))) (= 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_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_member T_a V_a_2 V_A_2) (= (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))) V_A_2)))) (forall ((V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (not (tptp.c_member T_a V_x_2 V_A_2)) (= (tptp.c_Groups_Ominus__class_Ominus _let_1 (tptp.c_Set_Oinsert T_a V_x_2 V_A_2) (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_A_2)))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_member T_a V_x_2 V_A_2))) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 V_B_2)) (and (=> _let_2 (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_B_2)) (=> (not _let_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2))))))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_B_2) (=> (tptp.c_member T_a V_x_2 V_A_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 V_B_2)))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a_2 V_b_2) (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_a_2 V_s_2) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_b_2 V_t_2)) tptp.c_FunDef_Opair__less)))) (forall ((V_B_2 $$unsorted) (V_x_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_member T_a V_x_2 V_A_2))) (let ((_let_3 (tptp.c_member T_a V_x_2 V_B_2))) (= (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 V_B_2)) (and (=> _let_3 (tptp.c_Orderings_Oord__class_Oless _let_1 V_A_2 V_B_2)) (=> (not _let_3) (and (=> _let_2 (tptp.c_Orderings_Oord__class_Oless _let_1 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_B_2)) (=> (not _let_2) (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 V_B_2)))))))))) (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.tc_prod _let_2 _let_2))) (=> (tptp.c_member _let_1 V_y_2 V_Y_2) (=> (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 V_x_2 V_y_2) tptp.c_FunDef_Opair__less) (=> (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_X_2 V_Y_2) tptp.c_FunDef_Omax__strict) (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 (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.tc_prod _let_2 _let_2))) (=> (tptp.c_member _let_1 V_x_2 V_XS_2) (=> (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 V_x_2 V_y_2) tptp.c_FunDef_Opair__less) (=> (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_XS_2 V_YS_2) tptp.c_FunDef_Omin__strict) (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_XS_2 (tptp.c_Set_Oinsert _let_1 V_y_2 V_YS_2)) tptp.c_FunDef_Omin__strict)))))))) (forall ((V_X_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (not (= V_X_2 _let_2)) (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 V_X_2 _let_2) tptp.c_FunDef_Omin__strict))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_a_2 V_b_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq tptp.tc_Nat_Onat V_s_2 V_t_2) (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_a_2 V_s_2) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_b_2 V_t_2)) tptp.c_FunDef_Opair__leq))))) (forall ((V_t_2 $$unsorted) (V_s_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (=> (tptp.c_Orderings_Oord__class_Oless tptp.tc_Nat_Onat V_a_2 V_b_2) (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_a_2 V_s_2) (tptp.c_Product__Type_OPair tptp.tc_Nat_Onat tptp.tc_Nat_Onat V_b_2 V_t_2)) tptp.c_FunDef_Opair__leq)))) (= tptp.c_FunDef_Opair__leq (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod _let_5 _let_5) tptp.tc_HOL_Obool) tptp.c_FunDef_Opair__less (tptp.c_Relation_OId _let_5))) (forall ((V_XS_2 $$unsorted) (V_x_2 $$unsorted) (V_YS_2 $$unsorted) (V_y_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod _let_2 _let_2))) (=> (tptp.c_member _let_1 V_y_2 V_YS_2) (=> (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 V_x_2 V_y_2) tptp.c_FunDef_Opair__leq) (=> (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_XS_2 V_YS_2) tptp.c_FunDef_Omax__weak) (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 (tptp.c_Set_Oinsert _let_1 V_x_2 V_XS_2) V_YS_2) tptp.c_FunDef_Omax__weak)))))))) (forall ((V_YS_2 $$unsorted) (V_y_2 $$unsorted) (V_XS_2 $$unsorted) (V_x_2 $$unsorted)) (let ((_let_1 (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat))) (let ((_let_2 (tptp.tc_fun _let_1 tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod _let_2 _let_2))) (=> (tptp.c_member _let_1 V_x_2 V_XS_2) (=> (tptp.c_member (tptp.tc_prod _let_1 _let_1) (tptp.c_Product__Type_OPair _let_1 _let_1 V_x_2 V_y_2) tptp.c_FunDef_Opair__leq) (=> (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_XS_2 V_YS_2) tptp.c_FunDef_Omin__weak) (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_XS_2 (tptp.c_Set_Oinsert _let_1 V_y_2 V_YS_2)) tptp.c_FunDef_Omin__weak)))))))) (forall ((V_X_2 $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod tptp.tc_Nat_Onat tptp.tc_Nat_Onat) tptp.tc_HOL_Obool))) (tptp.c_member (tptp.tc_prod _let_1 _let_1) (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_Omin__weak (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_8 (tptp.c_Wellfounded_Omin__ext _let_5 tptp.c_FunDef_Opair__leq) _let_10)) (= tptp.c_FunDef_Omax__weak (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_8 (tptp.c_Wellfounded_Omax__ext _let_5 tptp.c_FunDef_Opair__leq) _let_10)) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Omax__ext T_a V_r_2)))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Wellfounded_Owf T_a V_r_2) (tptp.c_Wellfounded_Owf (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Wellfounded_Omin__ext T_a V_r_2)))) (= tptp.c_FunDef_Omax__strict (tptp.c_Wellfounded_Omax__ext _let_5 tptp.c_FunDef_Opair__less)) (forall ((V_D_2 $$unsorted) (V_C_2 $$unsorted) (V_R_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Wellfounded_Omax__ext T_a V_R_2))) (let ((_let_2 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_3 (tptp.tc_prod _let_2 _let_2))) (=> (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_A_2 V_B_2) _let_1) (=> (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 V_C_2 V_D_2) _let_1) (tptp.c_member _let_3 (tptp.c_Product__Type_OPair _let_2 _let_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2 V_A_2 V_C_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_2 V_B_2 V_D_2)) _let_1))))))) (= tptp.c_FunDef_Omin__strict (tptp.c_Wellfounded_Omin__ext _let_5 tptp.c_FunDef_Opair__less)) (forall ((V_x_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Set_Othe__elem T_a (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) V_x_2)) (forall ((V_y_2 $$unsorted) (V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_x_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_member T_a V_x_2 V_A_2) (=> (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_A_2 (tptp.c_Set_Oinsert T_a V_x_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) V_y_2) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 V_A_2 (tptp.hAPP (tptp.hAPP V_f_2 V_x_2) V_y_2)))))) (forall ((V_z_2 $$unsorted) (V_f_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_f_2 V_z_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_z_2)) (forall ((V_a2_2 $$unsorted) (V_a1_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_a1_2 V_a2_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)) V_a2_2)) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (let ((_let_3 (tptp.c_Relation_OField T_a V_r_2))) (=> (tptp.c_Order__Relation_Opreorder__on T_a _let_3 V_r_2) (=> (tptp.c_member T_a V_a_2 _let_3) (=> (tptp.c_member T_a V_b_2 _let_3) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_a_2 _let_2)) (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_b_2 _let_2))) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_b_2 V_a_2) V_r_2))))))))) (tptp.c_FunDef_Oreduction__pair _let_6 (tptp.c_Product__Type_OPair _let_8 _let_8 tptp.c_FunDef_Omax__strict tptp.c_FunDef_Omax__weak)) (forall ((V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 (tptp.c_Relation_OImage T_a T_a V_R_2 V_A_2)) (= V_A_2 (tptp.c_Orderings_Obot__class_Obot _let_1)))))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_b_2 (tptp.c_Relation_OImage T_b T_a V_r_2 V_A_2)) (exists ((B_x $$unsorted)) (and (tptp.c_member T_b B_x V_A_2) (tptp.c_member (tptp.tc_prod T_b T_a) (tptp.c_Product__Type_OPair T_b T_a B_x V_b_2) V_r_2))))) (forall ((V_r_2 $$unsorted) (V_b_2 $$unsorted) (T_b $$unsorted) (V_A_2 $$unsorted) (V_a_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_a_2 V_A_2) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2) (tptp.c_member T_b V_b_2 (tptp.c_Relation_OImage T_a T_b V_r_2 V_A_2))))) (forall ((V_A_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_Relation_OImage T_a T_a (tptp.c_Relation_OId T_a) V_A_2) V_A_2)) (forall ((V_A_2 $$unsorted) (V_A_H_2 $$unsorted) (V_r_2 $$unsorted) (V_r_H_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun (tptp.tc_prod T_a T_b) tptp.tc_HOL_Obool) V_r_H_2 V_r_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) V_A_H_2 V_A_2) (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_b tptp.tc_HOL_Obool) (tptp.c_Relation_OImage T_a T_b V_r_H_2 V_A_H_2) (tptp.c_Relation_OImage T_a T_b V_r_2 V_A_2))))) (forall ((V_X_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Orderings_Oord__class_Oless__eq (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_OImage T_a T_a V_r_2 V_X_2) V_X_2) (= (tptp.c_Relation_OImage T_a T_a (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2) V_X_2) V_X_2))) (forall ((V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_OImage T_b T_a V_R_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))) (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_OImage T_b T_a V_R_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_b tptp.tc_HOL_Obool) V_A_2 V_B_2)) (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_OImage T_b T_a V_R_2 V_A_2) (tptp.c_Relation_OImage T_b T_a V_R_2 V_B_2)))) (forall ((V_A_2 $$unsorted) (V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted) (T_b $$unsorted)) (= (tptp.c_Relation_OImage T_b T_a (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_b T_a) tptp.tc_HOL_Obool) V_R_2 V_S_2) V_A_2) (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun T_a tptp.tc_HOL_Obool) (tptp.c_Relation_OImage T_b T_a V_R_2 V_A_2) (tptp.c_Relation_OImage T_b T_a V_S_2 V_A_2)))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (= (tptp.c_member T_a V_b_2 (tptp.c_Relation_OImage T_b T_a V_r_2 (tptp.c_Set_Oinsert T_b V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_b tptp.tc_HOL_Obool))))) (tptp.c_member (tptp.tc_prod T_b T_a) (tptp.c_Product__Type_OPair T_b T_a V_a_2 V_b_2) V_r_2))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Relation_OField T_a V_r_2))) (=> (tptp.c_Order__Relation_Opartial__order__on T_a _let_2 V_r_2) (=> (tptp.c_member T_a V_a_2 _let_2) (=> (tptp.c_member T_a V_b_2 _let_2) (= (= (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_a_2 _let_1)) (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_b_2 _let_1))) (= V_a_2 V_b_2)))))))) (forall ((V_b_2 $$unsorted) (V_a_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))) (let ((_let_2 (tptp.c_Relation_OField T_a V_r_2))) (=> (tptp.c_Relation_Orefl__on T_a _let_2 V_r_2) (=> (tptp.c_Relation_Oantisym T_a V_r_2) (=> (tptp.c_member T_a V_a_2 _let_2) (=> (tptp.c_member T_a V_b_2 _let_2) (= (= (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_a_2 _let_1)) (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_b_2 _let_1))) (= V_a_2 V_b_2))))))))) (tptp.c_FunDef_Oreduction__pair _let_6 (tptp.c_Product__Type_OPair _let_8 _let_8 tptp.c_FunDef_Omin__strict tptp.c_FunDef_Omin__weak)) (forall ((V_B_2 $$unsorted) (V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Relation_OField T_a V_r_2))) (=> (tptp.c_Order__Relation_Opreorder__on T_a _let_2 V_r_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_A_2 _let_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 V_B_2 _let_2) (= (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Relation_OImage T_a T_a V_r_2 V_A_2) (tptp.c_Relation_OImage T_a T_a V_r_2 V_B_2)) (forall ((B_x $$unsorted)) (=> (tptp.c_member T_a B_x V_A_2) (exists ((B_xa $$unsorted)) (and (tptp.c_member T_a B_xa V_B_2) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a B_xa B_x) V_r_2)))))))))))) (forall ((V_a5_2 $$unsorted) (V_a1_2 $$unsorted) (V_a2_2 $$unsorted) (V_a4_2 $$unsorted) (V_a3_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.hAPP (tptp.hAPP V_a3_2 V_a1_2) V_a5_2))) (= (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_a3_2 V_a4_2 V_a2_2 _let_1) (or (and (= V_a2_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (= _let_1 V_a4_2)) (exists ((B_x $$unsorted) (B_y $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (and (= (tptp.hAPP (tptp.hAPP V_a3_2 V_a1_2) V_a5_2) (tptp.hAPP (tptp.hAPP V_a3_2 B_x) B_y)) (tptp.c_member T_a B_x V_a2_2) (tptp.c_Nitpick_Ofold__graph_H T_a T_b V_a3_2 V_a4_2 (tptp.c_Groups_Ominus__class_Ominus _let_1 V_a2_2 (tptp.c_Set_Oinsert T_a B_x (tptp.c_Orderings_Obot__class_Obot _let_1))) B_y)))))))) (forall ((V_f_2 $$unsorted) (T_b $$unsorted) (V_P_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_FunDef_Oreduction__pair T_a V_P_2) (tptp.c_FunDef_Oreduction__pair T_b (tptp.c_FunDef_Orp__inv__image T_a T_b V_P_2 V_f_2)))) (forall ((V_A_2 $$unsorted) (V_r_2 $$unsorted) (T_b $$unsorted) (V_b_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member T_a V_b_2 (tptp.c_Relation_OImage T_b T_a V_r_2 V_A_2)) (not (forall ((B_x $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_b T_a) (tptp.c_Product__Type_OPair T_b T_a B_x V_b_2) V_r_2) (not (tptp.c_member T_b B_x V_A_2))))))) (forall ((V_S_2 $$unsorted) (V_R_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (=> (tptp.c_Wellfounded_Owf T_a V_R_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Relation_Orel__comp T_a T_a T_a V_R_2 V_S_2) V_R_2) (tptp.c_FunDef_Oreduction__pair T_a (tptp.c_Product__Type_OPair _let_1 _let_1 V_R_2 V_S_2)))))) (forall ((V_a_2 $$unsorted) (V_b_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun T_a tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.c_Orderings_Oord__class_Oless__eq _let_1 (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_b_2 _let_2)) (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_a_2 _let_2))) (=> (tptp.c_member T_a V_b_2 V_A_2) (tptp.c_member (tptp.tc_prod T_a T_a) (tptp.c_Product__Type_OPair T_a T_a V_a_2 V_b_2) V_r_2))))))) (forall ((V_s_2 $$unsorted) (V_c_2 $$unsorted) (T_c $$unsorted) (V_r_2 $$unsorted) (V_b_2 $$unsorted) (V_a_2 $$unsorted) (T_b $$unsorted) (T_a $$unsorted)) (=> (tptp.c_member (tptp.tc_prod T_a T_b) (tptp.c_Product__Type_OPair T_a T_b V_a_2 V_b_2) V_r_2) (=> (tptp.c_member (tptp.tc_prod T_b T_c) (tptp.c_Product__Type_OPair T_b T_c V_b_2 V_c_2) V_s_2) (tptp.c_member (tptp.tc_prod T_a T_c) (tptp.c_Product__Type_OPair T_a T_c V_a_2 V_c_2) (tptp.c_Relation_Orel__comp T_a T_b T_c V_r_2 V_s_2))))) (forall ((V_r_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.c_Transitive__Closure_Ortrancl T_a V_r_2))) (= _let_1 (tptp.c_Lattices_Osemilattice__sup__class_Osup (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool) (tptp.c_Relation_OId T_a) (tptp.c_Relation_Orel__comp T_a T_a T_a _let_1 V_r_2))))) (forall ((V_a_2 $$unsorted) (V_r_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (=> (tptp.c_Equiv__Relations_Oequiv T_a V_A_2 V_r_2) (=> (tptp.c_member T_a V_a_2 V_A_2) (tptp.c_member T_a V_a_2 (tptp.c_Relation_OImage T_a T_a V_r_2 (tptp.c_Set_Oinsert T_a V_a_2 (tptp.c_Orderings_Obot__class_Obot (tptp.tc_fun T_a tptp.tc_HOL_Obool)))))))) (forall ((V_C_2 $$unsorted) (V_B_2 $$unsorted) (V_A_2 $$unsorted) (T_a $$unsorted)) (let ((_let_1 (tptp.tc_fun (tptp.tc_prod T_a T_a) tptp.tc_HOL_Obool))) (let ((_let_2 (tptp.c_Orderings_Obot__class_Obot _let_1))) (=> (= (tptp.c_Relation_Orel__comp T_a T_a T_a V_A_2 V_B_2) _let_2) (=> (= (tptp.c_Relation_Orel__comp T_a T_a T_a V_A_2 V_C_2) _let_2) (= (tptp.c_Relation_Orel__comp T_a T_a T_a V_A_2 (tptp.c_Lattices_Osemilattice__sup__class_Osup _let_1 V_B_2 V_C_2)) _let_2)))))) (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_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_Orderings_Opreorder T_1) (tptp.class_Orderings_Opreorder (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_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_Ominus T_1) (tptp.class_Groups_Ominus (tptp.tc_fun T_2 T_1)))) (tptp.class_Lattices_Osemilattice__sup tptp.tc_Nat_Onat) (tptp.class_Orderings_Opreorder tptp.tc_Nat_Onat) (tptp.class_Orderings_Olinorder tptp.tc_Nat_Onat) (tptp.class_Lattices_Olattice tptp.tc_Nat_Onat) (tptp.class_Orderings_Oorder 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_Lattices_Obounded__lattice__bot tptp.tc_HOL_Obool) (tptp.class_Lattices_Osemilattice__sup tptp.tc_HOL_Obool) (tptp.class_Orderings_Opreorder tptp.tc_HOL_Obool) (tptp.class_Lattices_Olattice tptp.tc_HOL_Obool) (tptp.class_Orderings_Oorder tptp.tc_HOL_Obool) (tptp.class_Orderings_Oord tptp.tc_HOL_Obool) (tptp.class_Orderings_Obot tptp.tc_HOL_Obool) (tptp.class_Groups_Ominus tptp.tc_HOL_Obool) (forall ((V_y_2 $$unsorted) (V_x_2 $$unsorted)) (or (not (tptp.hBOOL (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.c_fequal V_x_2) V_y_2)))) _let_4 true)))))))))))))))))))))))))
% 0.67/0.87  )
% 0.67/0.87  % SZS output end Proof for SCT105+1
% 0.67/0.87  % cvc5---1.0.5 exiting
% 0.67/0.87  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------